Th`ese Marouan BOUALI - Sites personnels de TELECOM ParisTech

École Doctorale
d’Informatique,
Télécommunications
et Électronique de Paris
Thèse
présentée pour obtenir le grade de docteur
de l’Ecole Nationale Supérieure des Télécommunications
Spécialité : Signal et Images
Marouan BOUALI
Destriping data from multidetector imaging
spectrometers : a study on the MODIS
instrument
Soutenue le 36 avril 2040 devant le jury composé de
Bidule
Truc Muche
Machin
Chose
Tartampion
Patrice Henry
Nozha Boujemaa
Saïd Ladjal
Président
Rapporteurs
Examinateurs
Directeurs de thèse

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Résumé
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Abstract
In this thesis, we tackle the issue of ...

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Table des matières
1 Introduction 11
2 Remote Sensing with MODIS 15
2.1 The evolution of remote sensing . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Applications of satellite imagery . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Monitoring land changes . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Studying the atmosphere dynamics . . . . . . . . . . . . . . . . . . . 17
2.2.3 Cryosphere and climate change . . . . . . . . . . . . . . . . . . . . . 19
2.2.4 Recent exemples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.4.1 The Eyjafjallajökull . . . . . . . . . . . . . . . . . . . . . . 20
2.2.4.2 Deepwater Horizon oil spill . . . . . . . . . . . . . . . . . . 20
2.3 The importance of oceans in the Earth system . . . . . . . . . . . . . . . . 21
2.3.1 The ocean’s carbon cyle . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.2 Phytoplancton and ocean color . . . . . . . . . . . . . . . . . . . . . 22
2.4 Constraints in remote sensing . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.1 Sensor Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.2 Atmospheric correction . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.3 Sun glint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.4 Cloud coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 The Moderate Resolution Imaging Spectroradiometer (MODIS) . . . . . . . 28
2.5.1 Context and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.2 Technical specifications . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.3 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.4 MODIS products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5.5 Stripe noise on MODIS . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Standard destriping techniques and application to MODIS 39
3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Moment Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Histogram Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Overlapping Field-of-View Method . . . . . . . . . . . . . . . . . . . . . . . 48

8 TABLE DES MATIÈRES
3.5 Frequency filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5.1 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5.2 Band-pass filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.6 Haralick Facet filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.7 Multiresolution approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.7.1 Limitations of fourier transform . . . . . . . . . . . . . . . . . . . . . 57
3.7.2 Multiresolution analysis . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.7.3 Wavelet basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.7.4 Filter banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.7.5 2D wavelet basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.7.6 Destriping with wavelet coefficient thresholding . . . . . . . . . . . . 62
3.8 Assessing destriping quality . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.8.1 Noise Reduction Ratio and Image Distortion . . . . . . . . . . . . . 65
3.8.2 Radiometric Improvement Factors . . . . . . . . . . . . . . . . . . . 66
3.8.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 A Variational approach for the destriping issue 73
4.1 PDEs and variational methods in image processing . . . . . . . . . . . . . . 73
4.2 Rudin, Osher and Fatemi Model . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Striping as a texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.1 Yves Meyer’s model for oscillatory functions . . . . . . . . . . . . . . 83
4.3.2 Vese-Osher’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.3 Osher-Solé-Vese’s Model . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3.4 Other u + v models . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3.5 Experimental results and discussion . . . . . . . . . . . . . . . . . . 88
4.4 Destriping via gradient field integration . . . . . . . . . . . . . . . . . . . . 91
4.5 A unidirectional variational destriping model . . . . . . . . . . . . . . . . . 94
4.6 Optimal regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.6.1 Tadmor-Nezzar-Vese (TNV) hierarchical decomposition . . . . . . . 99
4.6.2 Osher et al. iterative regularization method . . . . . . . . . . . . . . 100
4.6.3 Stopping criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.6.4 Experimental results and discussion . . . . . . . . . . . . . . . . . . 103
5 Application : Restoration of Aqua MODIS Band 6 111
5.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.2 Existing restoration techniques . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2.1 Global interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2.2 Local interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.3 Proposed approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.3.1 Spectral similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.3.1.1 Spectral Correlation Measure . . . . . . . . . . . . . . . . . 115
5.3.1.2 Spectral Angle Measure . . . . . . . . . . . . . . . . . . . . 116

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5.3.1.3 Euclidian Distance Measure . . . . . . . . . . . . . . . . . . 116
5.3.1.4 Spectral Information Divergence Measure . . . . . . . . . . 116
5.3.2 Spectral inpainting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.4.1 Validation of spectral inpainting using NDSI measurements . . . . . 119
5.4.2 Assessing the impact of stripe noise . . . . . . . . . . . . . . . . . . 122
Conclusion 126
Bibliographie 135

10 TABLE DES MATIÈRES

11
Chapitre 1
Introduction
Over the years, climate change and its impact on life have become an alarming question
for science to answer. Historical weather records show evidence of cyclic brutal climate
fluctuations occuring since the formation of Earth and due to volcanic eruptions, meteorite
impacts, solar activity, plate tectonic movements and deviations in the Earth rotational
axis. Today, human industrial activities constitues an extra variable that interfers with
land, ocean and atmosphere natural processes.
To understand, quantifie and model the dynamic interactions between the earth system
components, scientists have to relie on information available over the entire globe with a
satisfactory time frequency. These requirements can only be satisfied by the use of satellite
sensors orbiting the earth at a high altitudes. Multiyear to multidecadal data sets collected
in the ultraviolet, visible, infrared and microwave portions of the energy spectrum, provide
valuable information for researchers to interconnect the Earth’s main geophysical variables.
Nevertheless, the complexity of physical processes involved in global change imposes
data quality standards that satellite instruments fail to meet, despite the continuous technological
innovations in imaging sensors design. Indeed, a given geophysical variable (chlorophyll
concentration for exemple) is determined from raw satellite measurements subsequently
used as input for geolocation, radiometric calibration, atmospheric correction, sun
glint correction, bio-optical models, vicarious calibration, temporal bining...and the list is
certainly not exaustive. This long processing chain makes the estimation of any geophysical
parameter extremely sensitive to initial quantity acquired by satellite instruments.
In simple words, small errors in the instrument response will irreversibly compromise the
accuracy of high level products.
Given the crucial role of oceans in global climate change and the stringent requirements
associated with ocean color remote sensing, space agencies have given high priority
to instrument design and pre-launch/on-orbit calibration procedures. Despite this effort,
many issues related to image quality can still be observed as visual artifacts and require
further processing. These typically include straylight, optical leaks, gaussian noise, speckle
noise (for active sensors), blurring, dead pixels, line dropouts..

12 1. Introduction
A common issue for Earth observing instruments, known as striping effect have persisted
for more then 30 years, i.e, since the launch of Landsat 1. It was since reported on
many imaging spectrometers :
- Landsat Multi Spectral Scanner (MSS) and Thematic Mapper (TM)
- Geostationary Operational Environmental Satellites (GOES)
- Advanced Very High Resolution Radiometer (AVHRR)
- Moderate Resolution Imaging Spectrometer (MODIS)
- Compact High Resolution Imaging Spectrometer (CHRIS)
- HYPERION
- MEdium Resolution Imaging Spectrometer (MERIS)
- Compact Reconnaissance Imaging Spectrometer for Mars (CRISM)
- GLobal Imager (GLI)
- Advanced Land Imager (ALI)
The previous list inludes both whiskbroom and pushbroom sensors which are the two
main techniques used for Earth observation.
Whiskbroom instruments, also known as cross-track scanners acquire a series of lines in
the direction perpendicular to the satellite orbital motion. A scan sweep from one side of
the swath to the other is ensured by a continuously rotating mirror. A limited set of detectors,
sensitive to spectific wavelenghts, then captures the radiation emitted or reflected
by the Earth before its convertion into digital numbers.
Pushbroom sensors (along-track scanners) also exploit the orbital motion of the platform
to generate the second dimension of the acquired signal. They differ from the whiskbroom
design in that the rotating mirror is replaced by a linear array of numerous detectors that
capture simultaneously each pixel of a single scan line.
In both acquisition principles, images are formed by enterlacing or concatanating scan
lines, acquired separately by different detectors. Consequently, the imperfect calibration
of individual detectors induces a sharp pattern across or along the scanning direction that
compromises both visual interpretation and quantitative analysis. More specifically, striping
is disturbingly visible over low-radiance homogeneous regions, which often coincide
with oceanographic areas. In fact, the impact of stripe noise on satellite derived geophysical
variables, including ocean color products, is increasingly attracting the interest of
remote sensing research groups.
Although an extensive litterature has tackled the striping issue, existing techniques are
often not able to satisfie the requirements imposed by several remote sensing applications
namely, complete stripe removal without signal distortion or additional post-processing
artifacts.
The goal of this thesis is to analyse the limitations of standard destriping methods
and to explore the issue of stripe noise removal using variational models. We illustrate the

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results of our work using data from NASA’s MODIS instrument.
Our focus on the MODIS sensor is justified by 1) the complexity and amplitude of
stripe noise visible on its emissive bands and 2) the importance of MODIS data in variety
of remote sensing disciplines
This thesis is organised as follows :
Chapter 1 briefly exposes main aspects and constraints of ocean color remote sensing.
This chapter also describes the Moderate Resolution Imaging Spectrometer (MODIS) and
characterises the striping effect on MODIS products.
Chapter 2 constitutes a non exaustive state-of-the-art of the destriping litterature. Several
approaches are described and illustrated on MODIS data. Experimental results and
comparative study are presented at the end of the chapter. The limitations of existing
techniques are analysed and used as a basis to define the requirements of an optimal
destriping.
Chapter 3 is the main contribution of this work. After an overview of variational methods
for image processing applications, we explore Rudin, Osher and Fatemi total variation
model. Yves Meyers variational models for oscillatory paterns, are used in an attempt to
extract stripe noise as a texture component. Our exploration of the striping issue from a
variational angle eventually comes down to a unidirectional variation based regularizing
model able to efficiently remove all the stripe noise without signal distortion.
Chapter 4 present an application that beneficits from the developed destriping technique.
We propose a new approache to restore Aqua MODIS band 6 based on non-local filters
and spectral similarity.

14 1. Introduction

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Chapitre 2
Remote Sensing with MODIS
2.1 The evolution of remote sensing
Look far and wide... This could well have been one of the underlying rules used by
evolution to produce life as we can see it all over the Earth. From the multiple eyes of
arachnophibia species to the mobile stereoscopic eyes of the chameleon, nature offers a
variety of elegant examples. The very origin of hominid bipedism have been investigated
by many evolutionnary scientists and attributed to a survival need. The ability to stand
on two feet in the hostile environment of prehistory, enabled early Homo-Sapiens to look
beyond savanna’s high grasses and thus avoid any uncomfortable encounters.
Jumping back to comtemporary times, the numerous satellites orbiting the earth can
be also perceived as the result of a long-term survival instinct. Remote sensing aims at
measuring information and determining specific properties of an object or phenomenon
without direct physical contact. Today, this term commonly refers to the set of techniques
used to derive geophysical variables of the Earth (or other planets, stars, galaxies...).
The development of remote sensing is a direct consequence of the technological advances
made in aviation from the begining of twentieth century. The first and second world war
triggered an interest in aviation strong enough to attract heavy investments from most of
the countries involved in war. Planes were not only used as a fast transportation vehicule,
but were also considered a strategic advantage for their ability to monitor simultaneously
wide spatial areas. The invasion of Normandie is a good illustration of how planes were
put to contribution. Aerial photographes were taken and analysed to establish appropriate
landing surfaces and determine the depth and state of coastal waters. More significantly,
infrared filters were already included in the primitive imaging devices to distinguish natural
vegetation from enemy camouflages. The following steps towards modern remote sensing
were achieved as consecutive technological intimidation attempts between USA and USSR
during the cold war, leading to a brutal race to space ; Spoutnik and Explorer, quickly
followed by the creation of NASA, the first men on the moon and the first habited spatial
station (Saliout-1).

16 2. Remote Sensing with MODIS
In 1970, inspired by spatial photographs taken during Mercury and Gemini’s missions,
and motivated by the possibility to monitor agriculture crop yelds, NASA launched
Landsat 1, the first satellite aiming at observing the earth surface and providing data
for scdientists from different disciplines. Following the success of the Landsat series ( 7
sensors since 1972), today’s intense research in remote sensing technologies is motivated
by deeper questions. Climate change and its potential disastrous impact on human life
urges an improved understanding of earth sciences, ranging from carbon cycle modelling
to weather forecasting. Such achievements are stricly limited by the quality of data collected
by satellite /aerial sensors and require a continous increase of spatial, temporal and
spectral resolution.
2.2 Applications of satellite imagery
Satellite and airborne sensors provide a continuous flow of data exploited in fields as
diverse as telecommunications, agriculture, deforestation, geology, hydrology, oceanography,
urban management and fire monitoring to cite a few. For its abilities to capture wide
portions of the globe, satellite derived data offers a high potential for the surveillance of
both local and global changes occuring in the Earth main systems.
2.2.1 Monitoring land changes
Among many human-induced activites, deforestation is highly responsible for both
weather and climate change. Forests play a crucial role in the climate stability as they
absorb and trap large quantities of carbon dioxide (CO2) present in the atmopshere. The
massive destruction of forests for urbanization and agricultural development, inevitably
releases in the atmopshere all the CO2 and other greenhouse gazes stored in trees for
decades. Assessing the amount and rate of carbon emissions from deforestation or other
activities, remotely sensed data can be used to monitor primary variables of the Earth
land surface such as the spectral reflectance, albedo and temperature. In addition, higher
order variables in the form of vegetation indices can provide valuable information on land
change. A well-known variable is the Normalized Difference Vegetation Index (NDVI)
[Rouse et al., 1973], defined as the ratio between the difference and the sum of reflectances
in the visible (red) and near-infrared spectral bands :
NDVI = ρ NIR − ρ Red
ρ NIR + ρ Red
(2.1)
A major disadvantage of he NDVI index is that it tends to saturate over dense vegetation
and is highly dependent on the background soil composition. In the presence of high
biomass, the Enhanced Vegetation Index (EVI) [Huete, 2002] provides an optimized vegetation
signal able to capture high variability. and minimizes the influence of atmospheric

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Figure 2.1 – Deforestation in Brazil, in the region of Mato Grosso, monitored using the
Normalized Difference Vegetation Index (NDVI) (TL) RGB image from Terra MODIS
captured on june 17, 2002 (TR) NDVI index (BL) RGB image from Terra MODIS
captured in the same region on june 28 2006 (BR) NDVI index
effects. The EVI is defined as :
EVI =
2.5ρ NIR − ρ Red
L + ρ NIR + C 1 ρ Red − C 2 ρ Blue
(2.2)
where the coefficient L corrects for canopy background, while C 1 and C 2 reduce the effects
of aerosol. The values selected for the MODIS-EVI product are L = 1, C1 = 6 and
C 2 =7.5.
2.2.2 Studying the atmosphere dynamics
The atmosphere determines the amount of solar radiance actually reaching the Earth
surface and therefore acts as a regulator of local and global climate processes. Depending

18 2. Remote Sensing with MODIS
Figure 2.2 – A dust storm over the Red Sea, monitored using the Normalized Difference
Dust Index (NDDI) (Left) RGB image acquired by Aqua MODIS on May 13, 2005
(Right) NDDI*
on the nature and concentration of its components, the atmosphere can scatter and absorb
a given percentage of the energy coming from the sun. In addition, the solar radiation
reaching the Earth’s surface and reflected back to the atmosphere can be re-scatter or reabsorbed
by the atmposhere. The equilibrium between incoming and outgoing energy is
highly dependent on the atmospheric characteritics in terms of clouds, water vapor and
aerosols content. Clouds for exemple, tend to cool the Earth during the day by reflecting
away incoming solar energy. At night, the presence of clouds blocks the outgoing energy
and consequently warms the Earth. In addition to cloud detection, satellite-based imagery
is vital to measure the type and concentration of aerosols present in the atmosphere. Aerosols
are microscopic particules abundant in desert dust, volcanic ashes, fire smoke, sea
salt and pollution derived gazes. Their impact on the Earth’s radiation budget through albedo
variations results directly from the scattering and absorption of sun light. Indirectly,
aerosols intervene in the formation of clouds as they provide a surface where water vapor
emanating from the Earth surface can condensate and form liquid droplets. Furthermore,
the size of aerosol particules influences greatly the optical properties of clouds. Clouds
containing high concentrations of aerosols emitted from industrial activies, are composed
of smaller liquid water drops than those containing aerosols derived from natural processes.
The resulting high brightness of these clouds increases the portion of solar radiation reflected
back to space and tend to reduce the Earth surface temperature.
The study of the atmosphere through the analysis of multispectral/hyperspectral satellite
imagery is an important step in the understanding of ocean-land-atmosphere physical interactions.
Furthermore, the atmospheric layer between the Earth surface and space acts
as a filter and attenuates the signal measured by satellite intruments. Estimation of the
atmosphere optical characteristics is necessary to compute efficient atmosperic correction
and derive accurately primary variables used in global climate models.

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Figure 2.3 – Monitoring snow cover with the Normalized Difference Snow Index (NDSI)
(Left) RGB image acquired by Terra MODIS on May 5, 2001 in Alaska (Right) NDSI,
notice how the index erroneously indicates snow cover over the ocean
2.2.3 Cryosphere and climate change
The cryosphere designates the Earth system where water can be found in solid form. It
is composed of snow, lake and sea ice, glaciers, ice caps and frozen ground and represents
during the winter seasons, 17 % of the Earth surface. For instance, the Greenland and
Antarctic Ice sheets together are equivalent to 70 m of sea level rise. Given its mass and
latent heat capacity, the cryosphere plays a role in climate change almost as important as
the oceans. First, the high surface reflectivity of snow covered regions have a significant
impact on the Earth albedoand even small variations in snow or ice fractional cover can lead
to short-term climate changes. In addition, the quantity of fresh water stored on cryosphere
can interfer with ocean currents. The variations in sea level, temperature and salinity of
ocean waters due to seasonal snowmelt can modify the interactions between oceans and the
atmosphere. Furthermore, global warming in the polar region induces permafrost melting
which in turn releases large quantities of methane stored in frozen soils. Understanding
the interactions between cryosphere and other parts of the Earth system, requires short
and long-term quantitative measurements provided by satellite instruments.
2.2.4 Recent exemples
The year 2010 was subject to few unfortunate hazards. The political and economical
implications of these events have emphasized the crucial role played by satellite sensors in
the management of such disasters.

20 2. Remote Sensing with MODIS
Figure 2.4 – The ash plume resulting from the Eyjafjallajokull eruption, monitored by
two sensors on April 17, 2010 (Left) Image acquired by NASA’s Aqua MODIS on April 17,
2010 showing a diffuse cloud of volcanic ash and a column plume rising at higher altitudes
(Right) Image from CNES/NASA CALIOP, lidar instrument aboard CALIPSO dedicated
to the study of the atmosphere vertical profile of aerosol.
2.2.4.1 The Eyjafjallajökull
Around 20 Mars 2010, Eyjafjallajökull, a volcano located in the south of Iceland began
its first erupting phase. A more explosive phase started on 14 april 2010. The volcano being
located beneath glacial ice, the melting water flowing back in the volcano vent resulted in
the formation of silica particules and ash, explosively rejected in the atmosphere at heights
reaching 10 km. In addition, the south-easterly path of the Jet Stream, unusually stable
at that specific period, progressively dispersed the ash plume across Europe’s airspace. Although
not specifically quantified by manufacturers, the sensitivity of aircraft engines to
ash particules, forced the Federal Aviation Administration (FAA) to shut down air-traffic
in most european countries. The financial losses caused by airspace closure to the airline
industry were estimated approximatively at 150 ME per day from 15 to 23 April. The
quantity of rejected ash and the trajectory of the resulting plumes were eventually monitored
with numerous satellite-based instruments including CALIPSO, VEGETATION
and MODIS (figure 2.4). As the eruptive phase of the volcano persisted, satellite images
of the ash cloud were used to determine its trajectory and establish low-risk air corridors.
2.2.4.2 Deepwater Horizon oil spill
In April 20, 2010, an offshore drilling platform, Deepwater Horizon, located southeast
of the Louisiana coast exploded and sank in the ocean. The explosion was followed by the
release of 8500 m 3 of crude oil per day and is considered today as the largest oil spill in the
history of petroleum industry. Data collected from several spatial instruments were used
to analyse the extend of the oil spill, and evaluate its spreading evolution with respect
to wind direction. The reflection of sun light off the ocean surface (sun glint) highlights

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Figure 2.5 – RGB images from MODIS showing the evolution of the oil spill after the
Deepwater Horizon platform explosion in 2010 (TL) April 29 (TR) May 4 (BL) May 9
(BR) In 24 May, the oil spill reach the shores of Blind Bay and Redfish Bay at the eastern
edge of the Mississippi River delta. Credit : NASA/MODIS Rapid Response Team
oil contaminated waters, which appear in satellite imagery as bright silvery stains. High
concentrations of oil in the ocean surface tend to reduce the magnitude of waves and the
formation of whitecaps. This change in the ocean surface optical properties can increase
the reflection of light in the sensor viewing direction (see section 2.4.3) specially if the
oil-covered waters are located near the specular reflection.
2.3 The importance of oceans in the Earth system
2.3.1 The ocean’s carbon cyle
Oceans occupie 71% of the earth and constitute a large reservoir of carbon present
in both organic and non-organic form. Their impact on the regulation of our climatic
system flows directly from the fundamental role they play in the global carbon cycle and
justifies the importance given by space agencies to the study of physical and biological
processes along the ocean-atmosphere interface. The carbon cycle designates the process

22 2. Remote Sensing with MODIS
Figure 2.6 – The ocean carbon cycle. Credit : NASA Robert Simmon
of carbon exchange fluxes between the earth’s dynamic systems, namely biosphere, lithosphere,
hydrosphere and atmosphere. Oceans are the earth’s largest active carbon reservoir
and contain approximatively 4.10 19 gC. They hold a double role in the global carbon cycle
refered to as solubility and biological pumps. The exchange of carbon dioxide between
ocean and atmosphere occurs in the air-sea interface and its flux F can be modelled as :
(
)
F = K. PCO Ocean
2
− P Atmosphere
CO 2
(2.3)
where PCO Ocean
2
(respectively P Atmosphere
CO 2
) is the partial pressure of CO 2 in the ocean (resp.
atmosphere) and K a coefficient known as piston velocity. The term PCO Ocean
2
is dependent
on other thermodynamic parameters such as the water temperature and salinity, and thus
dictates the directional exchange of CO 2 while its rate is related to K and to wind driven
turbulence. The depositing of atmospheric CO 2 in the deep layers f the ocean is ensured
by a continuous mechanism known as solubility pump ; Once trapped in surface waters,
the CO 2 is transfered to deeper ocean layers. This process occurs in high latitudes areas,
where overturning circulation is generated by the formation and sinking of dense waters
(figure 2.6).
2.3.2 Phytoplancton and ocean color
The intense biological activity witin the ocean and the resulting photosynthetic production
(also known as primary production) constitutes the fastest exchange of CO 2 between
oceans and atmosphere. In eutophic areas (upper layer of the oceans located between 0-
200m), algal micro-organisms such as phytoplanton, absorb the energy derived from solar

23
radiance and other mineral compounds to convert CO 2 into organic matter in the form of
particulate or dissolved organic carbon. This process is refered to as photosynthesis :
nCO 2 + nH 2 O → (CH 2 O) n + nO 2 (2.4)
Phytoplancton is a major regulator of the glocal carbon cycle. These microscopic algae
are in fact at the base of the marine food chain and ensure the survival of most species
including other micro-organisms such as zooplancton, responsible for the exportation of
CO 2 to the deep ocean. Phytoplancton organisms are composed of pigments with specific
ligh absorption spectra. Chlorophyll-a for example, is an ubiquitous pigment that attributes
the green color to most marine and continental plants. Despite their microscopic
size, the concentration of phytoplancton in the water column determines the ocean color.
From oligotrophic to eutrophic waters, the increase of phytoplancton biomass and its
primary production shifts the color of the ocean from deep blue to green. Consequently,
satellite-derived measurements of the water-leaving radiance can be used to evaluate the
phytoplancton biomass and quantify the ocean carbon fluxes. The spectral analysis of
the radiance emanating from the oceans can be coupled to bio-optical models in order
to retrived the ocean biomass constituents. Chlorophyll-a pigment concentration, a major
geophysical variable in oceanography, can be expressed as :
[Chla] = 10 (c 0+c 1 .ρ+c 2 .ρ 2 +c 3 .ρ 3 ) (2.5)
where c 0 , c 1 , c 2 , c 3 are empirically-derived constants and ρ = ρ 488 /ρ 551 is the ratio of
remote sensing reflectances at wavelenght 488 nm and 551 nm [Aiken et al., 1995].
Another parameter used to study the optical properties of ocean waters is the diffuse
attenuation coefficient K. This coefficient determines the attenuation of light intensity
within the water column. At a wavelenght of 490 nm, K is a direct indicator of water
turbidity and is defined as :
K(490) = 0.016 + 0.156
where L w designates the water leaving radiance.
( )
Lw (490) −1.54
(2.6)
L w (555)
2.4 Constraints in remote sensing
2.4.1 Sensor Calibration
In order to provide reliable information, the response of an instrument needs to be
characteriszed precisely with respect to a lage data set of controlled input signals. This
process known as calibration, occurs during the pre-launch and in-flight stages of the sensor
mission as data quality highly depends on it. The raw counts measured by a sensor
can not be used directly for quantitative studies because they are not associated with

24 2. Remote Sensing with MODIS
any unit. Radiometric calibration converts instrument digital numbers (DN) to physical
radiance values (W.sr −1 .m −2 ). Since CZCS (Coastal Zone Color Scanner), most space
instruments relie on on-board, in-flight internal calibration systems that include a solar
diffuser plate oriented towards the sun. The solar irradiance reaching the diffuser is then
used as a radiance reference to adjust the sensor absolute radiometric calibration. Furthermore,
pre-flight calibrations are not necessarely optimal in the actual space environment
of the sensors and might require further adjustments. On many instruments, the continous
exposure of the diffuser plate to solar radiations and bombardment by space particules,
induces a progressive degradation that needs to be accounted for in the overall system
(see section ). This can be achieved using an additional diffuser to monitor the primary
diffuser degradation (MERIS) or by comparing lunar observations to the light scattered
by the solar diffuser (MODIS and SeaWiFS).
In addition to absolute and relative radiometric calibration, vicarious calibration techniques
are used to minimize errors between satellite-derived data and stable ground targets
with known radiance values. Typical calibration sites include desertic regions, ice sheets,
clouds and ocean targets contaminated with sun glint. Assuming the availability of an
extensive in situ data set and a reliable atmospheric correction, the sensor response can
be adjusted so that the values of a geophysical variable estimated from satellite radiances,
Figure 2.7 – The ocean color from satellite imagery is determined by the organic constituents
of the ocean upper surface. These images acquired by MODIS illustrate phytoplancton
blooms in the South Atlantic Ocean, off of the cost of Argentina. Phytoplancton
blooms offer a wide variety of colors ranging from turquoise blue to dark green. The strong
color variations are due to the pigment composition of each phytoplancton specie and their
depth in the eutophic layer.

25
coincide with those derived from ground measurements.
2.4.2 Atmospheric correction
Prior and after its reflection on the Earth surface, solar radiation is subject to molecular
and aerosol scattering. The cumulation of these processes, combined with the high altitude
of satellites, reduces the magnitude of the radiance reaching the sensor. The received signal,
known as Top-Of-Atmosphere (TOA) can be expressed as a sum of radiance contributions
related to several processes. Atmospheric Correction then estimates the radiative portion
of these effect prior to the generation of surface reflectances. Atmospheric correction is
particularly complex in the case of ocean color remote sensing where the TOA radiance
can be written in a simplified form as :
L T OA = L Rayleigh + L Aerosol + L Rayleigh−Aerosol + TL Glint + tL W ater (2.7)
In the previous equation,
- T and t are the direct and diffuse atmospheric transmittances between the ocean surface
and the instrument. The portion of energy that reaches the sensor is related to the optical
thickness of aerosols contained in the atmopshere
- L Glint is the specular reflection of sun light on the oceanic surface towards the sensor
viewing direction
- L Rayleigh and L Aerosol both translate scattering effects of light and depend on atmospheric
pressure, temperature and polarization.
- L Rayleigh−Aerosol accounts for coupling processes between Rayleigh and Mie scaterring.
- L W ater is the signal to be retrieved after atmopheric correction and corresponds to the
water-leaving radiance
The L W ater term represents a weak portion (less than 10%) of the TOA signal and its
extraction from the combined surface/atmosphere system highly depends on the accuracy
of the atmopheric correction. In fact, a 1% error on the measured L T OA leads to a 10%
error on the estimated L W ater . In addition to scattering effects, many gazes present in the
atmosphere such as ozone, oxygen and water vapor can absord solar radiation in specific
regions of the electromagnetic spectrum.
The effective removal of atmospheric contribution from the TOA signal is a major requirement
for the generation of operational land and ocean colour products.
2.4.3 Sun glint
The greatest obstacle for the generation of ocean color products is the presence of sun
glint. Sun glint is the specular reflection of sunlight off the ocean surface and into the
satellite sensor. This optical phenomena occurs when the ocean surface directs the solar
radiation in the exact viewing direction of the sensor and as such, depends on the sea
surface state, the sun position and the satellite viewing geometry. The high intensity of

26 2. Remote Sensing with MODIS
Figure 2.8 – Sun glint, a major obstacle for the generation of ocean color products appears
on MODIS images as a wide and bright vertical stripe. It results from the reflection of sun
light off the ocean surface.
sun glint radiance L Glint (often close to the sensor saturation) compromises the estimation
of water-leaving radiance and all the derived oceanic geophysical variables. In addition,
atmospheric correction over ocean targets often fails to distinguish sun glint from high
concentrations of white aerosols. Many techniques have been devised to remove the sun
glint contribution from the TOA signal. The most common approach for medium resolution
instruments relie on the well-known Cox and Munk statistical model of sea surface
roughness [Cox and W.Munk, 1954]. For a given sensor viewing geometry, the amount of
sun glint radiance can be expressed as a function of the probability density function of sea
surface slopes which in turn, depend on the wind speed and direction. The reflectance due
to sun glint can be predicted from the sensor viewing geometry and the wind speed with :
ρ Glint (λ, θ s ,θ v ,φ s ,φ v ,W)= P (θ s,θ v ,φ s ,φ v ,W)f(w, λ)
4cos 4 βcosθ v cosθ s
(2.8)
where :
- θ s and φ s are the zenith and azimuth angles of the sun
- θ v and φ v are the zenith and azimuth angles of the sensor
- f(w, λ) is the Fresnel reflectance at the ocean surface for an angle of incidence of w
- W is the wind speed
- P (θ s ,θ v ,φ s ,φ v ,W) is the probability distribution function of Cox/Munk model corresponding
to a 4 th order Gramm-Charlier expansion and often approximated with a Gaussian
distribution.
Although implemented in numerous ocean colour sensors, Cox and Munk based correcting
schemes [Montagner et al., 2003], [Wang and Bailey, 2001], [Fukushima et al., 2007], [Ottaviani
et al., 2008] can only process moderate sun glint. Also, the results are limited by
the accuracy and resolution of available wind data. Motivated by these limitations, Steinmetz,
Deschamps and Ramon, [Steinmetz et al., 2008] recenly introduced a new approach,

27
POLYMER based on neural networks. The TOA reflectance is modelled as :
ρ T OA (λ) =c 0 + c 1 λ −1 + c 2 λ −4 + tρ W ater (λ) (2.9)
where c 0 includes spectrally flat components such as sun glint, c 1 λ −1 accounts for aerosol
effects and c 2 λ −4 represents coupling processes between sun glint and aerosols. The water
reflectance ρ W ater is assumed to be a function of chlorophyll concentration, derived from
bio-optical models. Rather different techniques are employed to process sun glint effects
on high resolution imagery (1-20m)[Hochberg et al., 2003], [Hedley et al., 2005], [Lyzenga
et al., 2006], [Kutser et al., 2009]. They are mostly based on the assumption of neglectable
water-leaving radiance at Near-Infrared (NIR) bands and the linear relationship of reflectances
at visible and NIR bands. An aternative option for sun glint correction is sun glint
avoidance. Several instruments are designed with the ability to tilt the sensors viewing
direction (SeaWiFS) or to include a multi-angle viewing mode (POLDER). The presence
of such mechanisms can reduce considerably the loss of data related to strong sun glint
contamination and increase the spatial coverage of ocean products.
2.4.4 Cloud coverage
Persisting cloud coverage in satelite imagery disturbs the study of many geological and
oceanographic processes because it reduces the availability of exploitable data. More specifically,
it disables the detection of changes occuring at temporal frequencies higher than
the persistence of clouds and, therefore poses a serious challenge for applications related to
disaster management (see section 2.2.4.2). As a reponse to the limited set of measurements
available in high level geophysical variables, many studies have focused on possible techniques
to enhance the daily spatial coverage. A common approach in ocean color remote
sensing is to exploit data measured by many sensors at slightly different times. Under the
assumption of stationnay oceanic processes, measurements acquired within a reasonable
time frame can be merged to fill in the gaps related to clouds. In the context of ocean color
remote sensing, combination of data from multiple instruments was explored for SST in
[Reynolds and Smith, 1994] and have since been deeply investigated by the SIMBIOS project
of NASA [M. E. and Gregg, 2003], [Fargion and McClain], [Kwiatkowska and Fargion,
2002]. Common techniques for the merging of chlorophyll concentration products include :
- Weighted averaging : estimation of missing values is determined from different sensors
using a weighted average where the weights are dependent on the accuracy of observations.
- Statistical objective analysis, also known as optimal interpolation, consists in filling
gaps through spatial and temporal interpolation.
More recently, the Short-term Prediction and Research Transition (SPoRT) program of
NASA has developped a composite product for MODIS SST [Haines et al., 2007] to improve

28 2. Remote Sensing with MODIS
Figure 2.9 – The issue of limited spatial coverage illustrated on Aqua MODIS Level 3
nightime 11 µm Sea Surface Temperature (SST) (Left) Daily product with missing measurements
due to clouds, sun glint and distance between swaths (Right) 8 day composite
product showing a substantial increase in spatial coverage.
regional weather prediction models. The methodology tackles the issue of cloud contamination
using a temporal compositing approach, able to estimate SST missing values while
ensuring global spatial continuity of SST gradients.
2.5 The Moderate Resolution Imaging Spectroradiometer
(MODIS)
2.5.1 Context and objectives
In 1978, the Coastal Zone Color Scanner (CZCS) was launched aboard the spacecraft
Nimbus-7. Designed as a proof-of-concept instrument with a mission duration of one year,
CZCS primary goal was to determine the potential of satellite imagery for the quantification
of ocean chlorophyll concentration and other dissoved and suspended organic
matter. Despite many limitations related to its prelaunch characterisation and on-orbit
calibration, CZCS imposed the basis of modern ocean color satellite sensors. The CZCS
operational period (1978-1986) was followed by the launch of several instruments dedicated
to the study of oceans, MOS, OCTS, POLDER and SeaWiFS, each including specific
technological innovations with increased dynamic range, signal-to-noise ratio and number
of spectral bands. Motivated by the demanding requirements of remote sensing scientists,
the improvement in satellite sensors design reached its climax with the Moderate Resolution
Imaging Spectrometer (MODIS). As a multipurpose mission, MODIS primary goals
are :
- Ensure the continuity of data collection provided by heritage sensors such as AVHRR,
CZCS, SeaWiFS and HIRS
- Deliver products in a variety of disciplines with improved radiometric quality
- Monitor changes that occur at short time-scales combining Terra and Aqua MODIS

29
morning and afternoon observations
- Provide highly consistent time series of observations used to improve our understanding
of climate change at seasonal-to-decadal time scales
The first prototype of MODIS was launched on December 18, 1999 aboard the Terra
EOS-AM-1 platform. The MODIS on the Aqua EOS-PM1 spacecraft was launched on
May 4, 2002.
2.5.2 Technical specifications
MODIS was initially conceptualized as a double instrument MODIS-N (Nadir) and
MODIS-T (Tilt). Aiming improved ocean colour capabilities, MODIS-T was designed with
the ability to tilt away from specular reflection directions and avoid sun glint effects.
Despite its success on SeaWiFS, a tilting mechanism was not retained for the MODIS
project because the combination of data collected from Terra and Aqua MODIS was proven
to provide almost similar spatial coverage [Gregg, 1992], [Gregg and Woodward, 2007],
with the additional advantage of both morning and afternoon observations. MODIS is
composed of 36 spectral bands ranging from the visible (0.4µm) to the far infrared (14µm)
and centered at wavelenghts dedicated to three major applications. Bands 1-7 are devoted
to the study of land remote sensing, cloud detection and aerosol estimation. These bands
are centered at wavelenghts similar to Landsat TM and measure data at spatial resolutions
of 250 m for bands 1-2 and 500 m for bands 3-7. Stringent requirements associated with
ocean color monitoring and studies conducted on CZCS and SeaWiFS instruments lead to
nine spectral bands on MODIS (8-16). Compared to SeaWiFS, MODIS ocean color bands
are narrower (average of 10 nm width compared to 20 nm on SeaWiFS), and, therefore
allow more reliable atmospheric correction with higher signal-to-noise ratio values. Most
of the remaining bands 17-26 were spectrally positioned with respect to HIRS, AVHRR
and ATRS. To provide accurate Sea Surface Temperatures (SST), two split-windows at
mid-wave infrared (MWIR) (bands 23-24) and long-wave infrared (LWIR) (bands 31 and
32) were included. The split-window composed of bands 31-32 enables the derivation of
day time SST measurements, because channels 23 and 24 are contaminated with sun glint
effects, still persisting in the MWIR portion of the electromagnetic spectrum. MODIS
was designed as a whiskbroom sensor ; it uses the obital motion of the satellite to acquire
successive lines using a scanning mirror that rotates at ± 55˚. MODIS swath reaches 2330
km and allows a global coverage of the entire earth every one to two days.
2.5.3 Components
Compared to its predecessors, MODIS includes many components (figure 2.10), each
playing a specific role in the acquisition process. As we shall see, emphasis was given to the
calibration of the instrument. We present here a brief description of the main subsytems.

30 2. Remote Sensing with MODIS
Main application Band Spectral Center Spectral Width
Land/Cloud/Aerosols Boundaries
1 645 nm 25 nm
2 856 nm 15 nm
3 469 nm 20 nm
4 555 nm 20 nm
Land/Cloud/Aerosols Properties 5 1240 nm 20 nm
6 1640 nm 24 nm
7 2130 nm 50 nm
8 412 nm 15 nm
9 443 nm 10 nm
10 493 nm 10 nm
11 531 nm 10 nm
Ocean Color/Phytoplankton 12 551 nm 10 nm
13 667 nm 10 nm
14 678 nm 10 nm
15 748 nm 10 nm
16 869 nm 15 nm
17 905 nm 30 nm
Atmospheric Water Vapor 18 936 nm 10 nm
19 940 nm 50 nm
20 3.750 µm 0.180 µm
Surface/Cloud Temperature
21 3.959 µm 0.030 µm
22 3.959 µm 0.030 µm
23 4.050 µm 0.060 µm
Atmospheric Temperature
24 4.465 µm 0.065 µm
25 4.515 µm 0.067 µm
26 1.375 µm 0.030 µm
Cirrus Clouds Water Vapor 27 6.715 µm 0.360 µm
Cloud Properties 29 8.550 µm 0.300 µm
Ozone 30 9.730 µm 0.300 µm
Surface/Cloud Temperature
31 11.030 µm 0.500 µm
32 12.020 µm 0.500 µm
33 13.335 µm 0.300 µm
Cloud Top Altitude
34 13.635 µm 0.300 µm
35 13.935 µm 0.300 µm
36 14.235 µm 0.300 µm
Table 2.1 – MODIS spetral bands
More details are provided in (Barnes et al., 1998).

31
Figure 2.10 – Main Components of the MODIS instrument
Scan mirror assembly The energy from Earth radiation is reflected into the focal
plane assembly via a double sidded scan mirror, composed of nickel-plated Beryllium. The
rotation of the mirror at 20.3 rpm is ensured by a 2-phase brushless DC motor and the
stability of its speed is monitored by a 14-bit optical encoder with an accuracy of 11-
microradian . The system mirror-motor-encoder has been limited to a weight of 4.3 kg
and only requires a power of 2.9 W.
Focal plane assembly Dichroic beamsplitters are used to separate the scene radiation
into 4 Focal Plane Assemblies (FPA) associated with each spectral region (VIS, NIR,
SWIR-MWIR, LWIR). Each FPA contains detector arrays of pixels which size ranges
from 135 to 540 µm. Bands with spatial resolution of 1 km have 10 detectors array, while
bands at 250m and 500m, have 20 and 40 detectors. This is illustrated in figure 2.11.
The FPAs are highly responsible for the presence of stripe noise in the images as striping
originates partly from the miscalibration and non-linear responses of the pixels contained
in the detector arrays.
The onboard calibration system includes four complementary components :
Solar Diffuser The solar diffuser (SD) is a pressed plate located in the forward part
of the instrument and used for the calibration of the reflective bands. Periodically (once
per orbit, at the north and south pole to avoid loss of data), the SD provides diffuse
solar illumination to the scan mirror. Assuming a good characterization of the SD surface
reflectance and precise estimation of the sun position, the solar diffuser radiance provides
a stable source for absolute radiometric calibration.

32 2. Remote Sensing with MODIS
Figure 2.11 – Four Focal Plane Assemblies composed of detector arrays
Solar Diffuser Stability Monitor Due to extended exposure to sun radiation during
the mission, the solar diffuser can be subject to a progressive deterioration that translates
as slight variations in its bidirectional reflectance distribution function (BRDF). Such
degradation can impact the radiometric calibration of the sensor and is monitored independently
with the solar diffuser stability monitor (SDSM). Deviations of the SD response
can be deduced by comparing measurements from its solar-illumated surface with measurements
obtained directly from the sun. This is done with a spherical integrating source
(SIS) composed of nine filtered detectors that succcessively view the SD, a dark scene
(space view) and the sun.
Spectral Radiometric Calibration Assembly The Spectral Radiometric Calibration
Assembly (SRCA) provides on orbit radiometric, spectral and spatial calibration without
interfering with the sensor current acquisition. An integrating sphere with four lamps
directs light on a toroidal relay mirror that reflects it towards a Czerny-Turner monochromator.
A grating/mirror assembly then points the monocromatic light into the VIS, NIR
and SWIR bands to evaluate spectral response deviations. Radiometric response is evaluated
when the entrance and exit aperture of the SRCA are open and a mirror replaces
the grating. The SRCA integrating sphere provides six-level radiometric sources used for
the radiometric calibration of the reflective bands. Band-to-Band spatial registration is
achieved with reticule patterns placed at the exit of the monochromator and projected
into MODIS optical system.

33
Blackbody : The calibration of MODIS emissive bands is achieved with the Blackbody,
a component with zero reflectivity and an effective emissivity above 0.992. The temperature
of the Blackbody is determined with a precision of ±0.1 K and is measured with twelve
thermistors. For every scan line (1.47s), a two-point calibration of the thermal bands is
achieved with the scan mirror successively viewing space and the blackbody.
2.5.4 MODIS products
The generation of MODIS products from instrument raw data is done at NASA’s
DAAC (Distributed Active Archive Center). The distribution of the products is then
ensured by NASA’s Goddard Space Flight Center, the United States Geological Survey’s
(USGS) EROS Data Center and the NOOA’s National Snow and Ice Data Center
(NSIDC). MODIS products are organized in three levels.
Level 1A products are composed of digital counts and data related to spacecraft and
instrument telemetry. L1A data is processed at the EOS Data and Operations Systems
(EDOS), then stored in 5-min granules containing all necessary information for further
geolocation and calibration.
Level 1B products are deduced from L1A and correspond to Top of the Atmosphere
(TOA) calibrated radiances.
Level 2 products result from the application of atmospheric correction and bio-optical
algorithms to L1B data. L2 data represents geolocated geophysical variables.
Level 3 products correspond to spatially binned L2 data accumulated during a time
period of one day, 3 days, 8 days, a mounth or an entire year. L3 data are mapped in an
Equidistant Cylindrical Projection grid of the earth and available at resolutions of 4 km
and 9 km.
MODIS geophysical products belong to four major disciplines, Ocean, Land, Atmosphere
and Cryosphere (Table 2.2).
2.5.5 Stripe noise on MODIS
The analysis of MODIS level 1B data derived from reflective and emmissive bands reveals
the presence of striping with slightly different aspects. [Gumley, 2002] was the first to
point out that three types of stripes can actually be seen on MODIS (figure 2.12). Mirror
side stripes (mirror banding) affects most reflective bands and some emissive bands located
below the MWIR range. However, they are visible only over bright targets. Mirror banding
appears to be the result of a quasi constant offset between forward and backward scans
(we recall that MODIS scanning mirror is double sided) and is systematically located in
regions displaying radiance values high enough to bring the sensor close to its saturation

34 2. Remote Sensing with MODIS
Figure 2.12 – Three types of stripe noise on Terra MODIS Level 1B geolocated calibrated
radiances (Left) Detector-to-detector stripes (Band 27) (Center) Mirror side stripes
(Band 9) (Right) Random stripes (Band 33). All the images are 200×200 pixels in size
with a resolution of 1km
mode. Typical cases of mirror banding can be seen in homogeneous areas contaminated
with sun glint or high concentrations of aerosol. The analysis of oceanographic data indicates
that mirror banding reaches a maximal amplitude in the specular direction, (highest
level of sun glint) and tends to fade away as the sun glint intensity decreases. This shows
that mirror side stripes are dependent on the signal level and can be correlated with the
scan angle.
Detector-to-detector stripes take the form of a periodic pattern of stripes and unlike mirror
banding they cover entire MODIS swaths. Studies related to other sensors Horn and
Woodham [1979], attribute the presence of these periodic stripes to a poor gain/offset
calibration of the indivual detectors composing the sensor. Furthermore, detectors responses
display strong non linear effects in the low radiance range (figure 3.4).
The third type of stripes are random and appear clearly on Terra MODIS band 33 as
black and white stripes with a limited lenght over a given scan line. Noisy stripes are
presumably due to errors in the internal system and random noise.
The processing of level 1B data to level 2 geophysical products does not include a correcting
algorithm for striping. The most recent version of MODIS data (collection 5),
includes a destriping procedure [] limited only to land surface reflectances. As atmospheric
correction and bio-optical algorithms combine the information from multiple spectral
bands to generate level 2 data, striping tends to be emphazised in the derived geophysical
products. Mirror banding clearly affects ocean colour products such as normalized
water leaving radiance and chlorophyll concentration (figure 2.13). Level 2 daytime Sea
Surface Temperature (SST) is computed from bands 31 and 32, and shows evidence of
detector-to-detector stripes. Going further in MODIS processing chain, striping appears
to persist even in level 3 products, altough this effect migh also be originating from the
poor resolution of level 3 data (4km). It is clear, that stripe noise on level 1B data impacts
the quality of higher level products and needs to be corrected efficiently before the gene-

35
ration of level 2 and level 3 products. Because the stripe noise does not affect similarly
all spectral bands, the derivation of geophysical variables often results in a stripe noise
amplification. For instance, striping can hardly be seen on bands 31 and 32. Yet, SWIR
and LWIR SST products display stripe noise. MODIS SST is obtained with an NLSST
(Non linear SST) algorithm that can be expressed as :
SST = a 0 + a 1 .BT 11 + a 2 .(BT 11 − BT 12 )+a 3 .(BT 11 − BT 12 ).( 1 − 1) (2.10)
µ
where a 0 , a 1 , a 2 , a 3 are time dependent coefficients determined by the Rosentiel School of
Marine and Atmospheric Science from in-situ measurements. BT 11 and BT 12 are brightness
temperatures of bands 31 and 32, and µ is the cosine of the sensor zenith angle. The
difference between bands 31 and 32 is used to correct atmospheric effects due to waper
vapor absorption and is responsible for the introduction of striping in the SST. The amplification
of striping can be stronger on products generated from algorithms that include
multiplicative operations between spectral bands, which is the case of many bio-optical
models used to estimate oceanographic variables.

36 2. Remote Sensing with MODIS
Figure 2.13 – Stripe noise on Aqua MODIS Level 2 ocean products (TL) Remote sensing
reflectance at 412 nm (TR) Chlorophyll concentration (BL) Diffuse attenuation coefficient
at 490 nm (BR) Sea surface temperature at 11 µm

37
Discipline Product ID Product name
MOD01 Level-1A Radiance Counts
– MOD02 Level-1B Calibrated Geolocation Radiances
MOD03 Geolocation Data Set
MOD04 Aerosol Product
MOD05 Total Precipitable Water
Atmosphere
MOD06 Cloud Product
MOD07 Atmospheric Profiles
MOD08 Gridded Atmospheric Product
MOD035 Cloud Mask
Cryosphere
MOD010 Snow Cover
MOD029 Sea Ice Cover
MOD09 Surface Reflectance (Atmospheric Correction)
MOD011 Land Surface Temperature and Emissivity
MOD012 Land Cover/Land Cover Change
MOD013 Gridded Vegetation Indices (NDVI & EVI)
MOD014 Thermal Anomalies - Fires and Biomass Burning
Land MOD015 Leaf Area Index (LAI) and FPAR
MOD016 Evapotranspiration
MOD017 Vegetation Production, Net Primary Productivity (NPP)
MOD040 Gridded Thermal Anomalies
MOD043 Surface Reflectance BRDF/Albedo Parameter
MOD044 Vegetation Cover Conversion
MOD018 Normalized Water-leaving Radiance
MOD019 Pigment Concentration
MOD020 Chlorophyll Fluorescence
MOD021 Chlorophylla Pigment Concentration
MOD022 Photosynthetically Available Radiation (PAR)
MOD023 Suspended-Solids Concentration
MOD024 Organic Matter Concentration
MOD025 Coccolith Concentration
Ocean MOD026 Ocean Water Attenuation Coefficient
MOD027 Ocean Primary Productivity
MOD028 Sea Surface Temperature
MOD029 Sea Ice Cover
MOD031 Phycoerythrin Concentration
MOD032 Processing Framework and Match-up Database
MOD036 Total Absorption Coefficient
MOD037 Ocean Aerosol Properties
MOD039 Clean Water Epsilon
Table 2.2 – MODIS products

38 2. Remote Sensing with MODIS

39
Chapitre 3
Standard destriping techniques
and application to MODIS
3.1 Data
The characteristics of stripe noise on the MODIS instrument are more complex than
those observed on other imaging spectrometers. We shall see that early destriping techniques
developped to improve the quality of Landsat MSS/TM images, provide unsatisfactory
results for MODIS data. As illustrated in figure 2.12, images extracted from MODIS
are contaminated with three different types of stripes, ubiquitous on few spectral bands.
Although this increases the difficulties associated with destriping, it constitutes - in addition
to the importance of MODIS in earth science research - a complementary motivation
to explore new techniques.
Despite a growing number of destriping methodologies, very few take into account particular
cases of stripe noise. In fact, a major concern on MODIS is the presence of random
stripes, so far discussed only in [Rakwatin et al., 2007]. In this chapter, we explore several
techniques and we analyse the results obtained on images extracted from specific emissive
bands of MODIS.
The algorithms described in each section are applied on both Terra and Aqua level 1B
TOA calibrated radiances. Given the considerable size of MODIS swaths (2330 km cross
track ), we illustrate most of the results on 512 × 512 sub-images to allow accurate visual
examination of small scale features. We specifically selected bands 27, 30 and 33 for Terra
MODIS and bands 27, 30 and 36 for Aqua MODIS as they are representative of an extreme
case due to severe striping with strong non-linear effects. All these bands display periodic
stripes (mainly detector-to-detector), except band 33 of Terra MODIS which contains
mostly random stripes. The data set used for this study is composed of three images for
Terra MODIS and three images for Aqua MODIS, each extracted from a different spectral
band (see figure 3.1). The scene captured by Terra MODIS was acquired on July 1 st , 2009
in the Mediterannean Sea. The images from Aqua MODIS were acquired on November

40 3. Standard destriping techniques and application to MODIS
Figure 3.1 – MODIS data set used for the analysis of destriping algorithms and composed
of level 1B TOA calibrated radiances (TL) Terra band 27 (TC) Terra band 30 (TR) Terra
band 33 (BL) Aqua band 27 (BC) Aqua band 30 (BR) Aqua band 36
10, 2009, in the Pacific Ocean on the west coast of USA.
3.2 Moment Matching
The need for post-processing algorithms and the development of destriping methods
quickly followed the distribution of data collected by Landsat MSS. The MSS incorporates
a rotating scan mirror that redirects the scene energy to six separate detectors.
After completion of an entire scan sweep, the orbital motion of the satellite ensures the
acquisition of six additional lines and so forth. The six individual detectors of MSS are
not perfectly identical and the corresponding transfer functions are subject to slight deviations
from each other. During the acquisition process, the radiometric drift between
detectors produces an undesirable striping effect in the image(*add*). The removal of
this artifact requires an accurate characterization of the detectors response in term of
deviations from nominal operational mode. Assuming that input/output functions of the
detectors established during the pre-launch calibration stage remain the same during onorbit
operations, a simple inversion of these functions could provide a reliable correction

41
for striping. The drift observed on MSS detectors response was attributed in [Horn and
Woodham, 1979] to the degradation of photomultipliers with respect to exposure time.
Despite the calibration system on-board LANDSAT, MSS images still contain stripes and
additional radiometric correction were developped by [Strome and Vishnubhatla, 1973]
and [Sloan and Orth, 1977]. The resulting method is referred to as moment matching and
is based on the assumption of both linear and time invariant behavior of the detectors.
Stripe noise is assumed to be the result of relative uncorrected gain and offset coefficients.
Hereafter, we denote d k the signal acquired by detector number k. In the case of MODIS,
k =1..10 (for FPAs with 1 km resolution), images are formed by interlacing signals from
the 10 detectors. Under the assumption of linear responses, the signal from detector k can
be written as :
d k = G k .d c k + O k (3.1)
where G k and O k are gain and offset parameters, d k is the noisy signal and d c k
is the true
signal for detector k. Once G k and O k are determined, values of detector k are corrected
as :
d c k = d k − O k
(3.2)
G k
Unless calibration targets with known radiance values are used, absolute values for G k and
O k remain undetermined. Nevertheless, destriping only requires relative gain/offset coefficients.
These can be estimated statistically from the entire image. In fact, the acquisition
process of whiskbroom systems makes it reasonnable to assume that signals acquired by
each detector share similar statistical characteristics. The validity of this hypothesis holds
for images with high dimensions. Relative gain/offsets can then be determined from the
mean value and standard deviation of a signal d ref acquired by a predetermined reference
detector. Let us denote µ k and σ k , - respectively µ ref et σ ref - the mean value and standard
deviation of d k , - resp. d ref -. Moment matching adjusts the response of a noisy detector
by forcing its mean value and standard deviation to coincide with those derived from the
reference detector. Using equation (3.1), this translates to :
and gain and offset are obtained as :
µ k = G k .µ ref + O k
σ k = G k .σ ref
(3.3)
G k =
σ k
σ ref
O k = µ k − µ ref . σ k
σ ref
(3.4)
Replacing equations (3.4) in (3.2), the signal from detector k is corrected with :
ˆd c k = σ ref .(d k − µ k )
σ k
− µ ref (3.5)

42 3. Standard destriping techniques and application to MODIS
Detector Terra Band 27 Terra Band 30 Terra Band 33
– Mean value Std deviation Mean value Std deviation Mean value Std deviation
d 1 12982 1043.6 16933 2441.9 23072 1660.9
d 2 12982 1043.6 16933 2441.4 23072 1660.9
d 3 12982 1043.7 16934 2442.5 23072 1661.1
d 4 12981 1043.7 16934 2442.1 23072 1661.3
d 5 12982 1043.9 16934 2442.1 23072 1661.6
d 6 12982 1043.9 16934 2442.1 23072 1661.0
d 7 12984 1044.1 16934 2442.0 23072 1661.0
d 8 12981 1044.0 16934 2442.0 23072 1660.9
d 9 12980 1044.0 16934 2442.0 23072 1660.9
d 10 12980 1043.9 16934 2442.0 23072 1660.6
Table 3.1 – Mean/standard deviation values (DN) of 10 detectors for Terra
MODIS bands 27, 30 and 33. The small statistical deviation between detectors
discredits the application of moment matching on MODIS data
The moment matching technique only rectifies first/second order statistics and as such
requires a careful selection of the reference signal. If a detector with strong statistical
deviations from others (a detector responsible for random stripes for exemple) is used
as a reference, moment matching can result in very poor destriping. Many strategies based
on the analysis of inter-detector response statistics can be used (see section 3.4). A
common option is to rely on mean/standard deviation values computed over the entire
scene. Altough the simplicity of its implementation makes it a popular technique, moment
matching suffers from many limitations described in [Horn and Woodham, 1979]. A major
issue pointed out on MSS is the impact of non-linearities in photomultipliers responses,
presumably abscent in more sophisticated imaging devices. Mean/standard deviation values
reported in tables 3.1, 3.2 and experiments conducted on both Terra and Aqua MODIS
data, illustrate how moment matching only results in partial destriping (figure 3.3). Indeed,
the gain/offset model only modifies the affine response of the detectors and fails to
take into account non-linear effects, highly responsible for residual stripes. Local analysis
of MODIS swaths reveals a dependency of detectors linear response to the signal intensity.
This can be verified experimentally by estimating gains/offsets from swaths covering
different levels of radiances (oceans and clouds). Furthermore, first order statistics make
the method very sensitive to the geophysical content of the images. In many cases, small
clouds or other highly reflective targets, are only visible by a limited set of detectors and
the resulting statistical bias is not accounted for in the moment matching procedure.
3.3 Histogram Matching
The hypothesis of linear and stationnary response exploited by the moment matching
method is too strong to provide reliable results on MODIS data. Investigation of
tables 3.1, 3.2 shows that on Terra/Aqua bands severely contaminated with stripes, detectors
have very similar mean/standard deviation values. The visual examination of su-

Figure 3.2 – Sub-images acquired by each of the 10 detectors of Terra MODIS in band
27. From Top to bottom and left to right, signals from detectors d 1 to d 10 . Visual analysis
indicates that stripes are not only due to differences between mean values and variances
of adjacent detectors. Images derived from some individual detectors also display strong
striping. This is clearly visible for detectors 1 and 6
43

44 3. Standard destriping techniques and application to MODIS
Figure 3.3 – (Left) Noisy image from Terra MODIS band 27 (Right) Image destriped
with moment matching
Detector Aqua Band 27 Aqua Band 30 Aqua Band 36
– Mean value Std deviation Mean value Std deviation Mean value Std deviation
d 1 14026 3508.2 14684 2872.0 22497 1420.3
d 2 14026 3508.1 14685 2871.7 22497 1420.4
d 3 14027 3507.8 14685 2871.2 22498 1420.3
d 4 14027 3508.3 14685 2871.0 22498 1420.5
d 5 14027 3508.4 14685 2870.9 22497 1420.7
d 6 14028 3508.3 14685 2870.9 22498 1420.8
d 7 14028 3508.2 14686 2870.6 22498 1420.8
d 8 14027 3507.2 14685 2870.6 22498 1420.6
d 9 14029 3508.3 14686 2870.3 22498 1420.6
d 10 14030 3508.5 14687 2870.2 22499 1420.6
Table 3.2 – Mean/standard deviation values of 10 detectors for Aqua MO-
DIS bands 27, 30 and 36
bimages extracted from individual detectors (figure 3.2) shows evidence of striping. This
means that suddle deviations occur within a single detector between two consecutive scan
sweeps. Given the short time period between two scans, 1.477 seconds, these intra-detector
deviations cannot be related to a potential physical degradation but are very likely to be
induced by the on-board calibration procedure itself ; For instance, the blackbody (section
) provides a two-point calibration for all emmissive bands every 1.477s.
The inefficiencies of moment matching due to non linear effects on MSS, gave rise to an
improved method. To overcome the limitations of first order statistics, [Horn and Woodham,
1979] introduced a refined approach where the probability distribution function of
scene radiances captured by each detector is assumed to be identical. In fact, radiations
emmitted or reflected by the earth reach all the detectors with approximatively the same

45
magnitude. The derived technique, histogram matching, then consists in adjusting the empirical
cumulative distribution function (ECDF) of each detector to an ECDF selected as
reference. Let us consider the signal measured by detector k as a discrete random variable
X k , with a probability distribution p k (x). The ECDF of X k is defined as :
and computed as :
P k (x) =P (X k ≤ x) (3.6)
P k (x) =
x∑
p k (i) (3.7)
We consider P ref to be the ECDF of a reference detector, x is a value measured by detector
k and x ′ its corresponding corrected value. Forcing detector k to have the same ECDF as
the reference detector, the following relation holds :
i=0
P ref (x ′ )=P k (x) (3.8)
The ECDF P ref is a non increasing function and can be inversed to determine an estimate
value of x ′ as :
x ′ = P −1
ref (P k(x)) (3.9)
When applied to the set of acquired values, equation (3.9) provides a normalization lookup
table that associates to every signal value x, its corrected value x ′ (figures 3.4 and 3.5).
Analogously to moment matching, the reference detector can be determined experimentally
from the analysis of each detector or by selecting the ECDF of the entire swath.
Following the results obtained on Landsat MSS, the histogram matching method was
later used in 1985 by [Poros and Peterson, 1985] for the destriping of Landsat TM. The limitations
of histogram matching were first discussed in [Wegener, 1990] and a modification
of the original approach was introduced to reinforce the assumption of similar detectors
ECDFs. The considerable size of captured swaths systematically garantees the acquisition
of geophysical data diverse enough in terms of radiances to contradict the hypothesis of
statistically identical ECDFs. Wegener suggested to constrain the estimation of detectors
statistics to the only homogeneous regions of the swath. In his approach, a noisy image
is decomposed into subimages which size is a multiple of the number of detectors in the
instrument. In the case of Landsat MSS, images are fragmented into blocs of 12 × 12
pixels (MSS has 6 detectors), and used for the computation of ECDFs if they satisfy a
homogeneous criteria established by Bienaymé-Tchebychev inequality :
P (|x − µ| > kσ) ≤ k −2 (3.10)
where µ and σ are mean/standard deviation values of a subimage, and k ≤ 1 a real number
that determines the percentage of rejected sub-images.
Another interesting implementation of histogram matching was introduced by [Weinreb
et al., 1989], for the destriping of meteorological data acquired by GOES (Geostationary

46 3. Standard destriping techniques and application to MODIS
ECDF
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
d1
d7
d8
d9
d10
9000 10500 12000 13500 15000 16500
Radiance values
Corrected Radiances
16000
14000
12000
10000
8000
d1
d7
d8
d9
d10
6000
6000 8000 10000 12000 14000 16000
Measured Radiances
ECDF
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
d1
d3
d5
d6
d8
12000 14000 16000 18000 20000 22000
Radiance values
Corrected Radiances
16000
14000
12000
10000
8000
d1
d3
d5
d6
d8
6000
6000 8000 10000 12000 14000 16000
Measured Radiances
ECDF
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
d1
d2
d5
d7
d9
20000 21500 23000 24500 26000
Radiance values
Corrected Radiances
25000
22000
19000
16000
13000
d1
d2
d5
d7
d9
10000
10000 13000 16000 19000 22000 25000
Measured Radiances
Figure 3.4 – (Left) From top to bottom, Empirical Cumulative Distribution Functions
for noisy detectors of Terra MODIS bands 27, 30 and 33. (Right) From top to bottom,
Normalization Look-up table for the same detectors obtained with the histogram matching
technique (IMAPP) and used to correct striping. Nonlinear effects are highly present in
the low-radiance range
Operational Environmental Satellites). The methodology was dedicated to the pre-launch
calibration of the 8 detectors composing GOES I-M (launched by NOAA in 1990). While
the original method of [Horn and Woodham, 1979] derives and applies a normalization

47
ECDF
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
d3
d6
d8
d9
d10
6000 9000 12000 15000 18000 21000
Radiance values
Corrected Radiances
18000
15000
12000
9000
6000
d3
d6
d8
d9
d10
3000
3000 6000 9000 12000 15000 18000
Measured Radiances
ECDF
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
d3
d4
d8
d9
d10
7000 10000 13000 16000 19000 22000
Radiance values
Corrected Radiances
17000
14000
11000
8000
d3
d4
d8
d9
d10
5000
5000 8000 11000 14000 17000
Measured Radiances
ECDF
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
d1
d2
d3
d4
d8
18000 19500 21000 22500 24000 25500
Radiance values
Corrected Radiances
24000
22000
20000
18000
d1
d2
d3
d4
d8
16000
16000 18000 20000 22000 24000
Measured Radiances
Figure 3.5 – (Left) From top to bottom, Empirical Cumulative Distribution Functions
for noisy detectors of Aqua MODIS bands 27, 30 and 36. (Right) From top to bottom,
Normalization Look-up table for the same detectors obatined with the histogram matching
technique (IMAPP). Comparison with Terra normalization look-up table indicates
a improved pre-launch calibration
look-up table to a single image, Weinreb suggests using the same table to process independent
acquisitions. The normalization table is generated from a specific sample covering
a wide dynamic range of radiances. An image acquired by GOES-7 on May 18, 1988 was

48 3. Standard destriping techniques and application to MODIS
Figure 3.6 – (Left) Noisy image from Terra MODIS band 27 (Right) Image destriped
with histogram matching (IMAPP)
used to establish the 8 detectors normalization curves. The look-up table was then successfully
applied to destripe an independent image collected two weeks later on June 1,
1988.
More recently, this approach was adopted by [Xiaoxiang et al., 2007] for the destriping of
CMODIS (SZ-3 Chinese MODIS) data.
It is currently implemented in the NASA IMAPP (International MODIS/AIRS Processing
Package) software and distributed by the university of Madison-Wisconsin for MODIS
users (http ://cimss.ssec.wisc.edu/imapp/). Additional features specific to MODIS data
have been included in the software. Two sets of look-up tables were developed for both
Terra and Aqua MODIS, and their application depends on the acquisition date of images to
be detriped. The comparative study of section 3.8 uses the histogram matching technique
implemented in the NASA IMAPP software.
3.4 Overlapping Field-of-View Method
The Overlapping Field-of-View method (OFOV) was specifically designed for MODIS.
Its basic principle was introduced by [Antonelli et al., 2004] and recently examined by
[di Bisceglie et al., 2009]. This technique relies on another artifact of MODIS, the bowtie
effect. When the scan angle of the mirror increases, the actual resolution of pixels also
increases due to the earth curvature. The nominal resolution of a pixel is 1 × 1 km at
Nadir and reaches 4.8 × 2km at the begining and ending of every scan, which correspond
to scan angles of ±55˚. As a result of pixel resolution growth, a swath will cover an
extended area of 20km along track at scan extremums, causing overlaps with previous
and next swaths (figure 3.7a). Areas affected with the bowtie effect will display identical

49
Figure 3.7 – (Left) Bowtie effect due to the growth of pixel size in the cross track
direction of a swath (Right) Image from Terra MODIS band 17, extracted from an area
with a scan angle raging from 47˚ to 55˚. Bowtie effect appears as overlapping fields of
view.
geometrical features in successive scans. This is clearly visible along land/ocean transitions
(figure 3.7b). The OFOV method then exploits the information redundancy to equalize
the detectors responses. In the bowtie region (areas corresponding to scan angles higher
than ±25˚), some of MODIS detectors are viewing exactly the same scene and, therefore
thus deviations in the detectors reponses are only due to stripe noise. The OFOV algorithm
is divided in two steps. Following the classification of the detectors as in-family and outof
family detectors, a reference detector is selected and equalization functions are derived
using only bow-tie measurements. The equalization curves are then applied to the detectors
over the entire swath. To assess the accuracy of relative inter-detectors calibration, a Bowtie
Based Detector Distance (BTBDD) between detectors i and j is defined as :
d i,j =
∑ Ni,j
n=1 |I i(n) − I j (n)| .W (n)
∑ Ni,j
n=1 W (n) (3.11)
where N i,j is the number of OFOVs of detectors i and j, with an overlapping percentage
above η, a value fixed to 65%. I i (n) (respectively I j (n)) is the radiance measured by
detector i (respectively j) on the n th OFOV and W (n) is the overlapping percentage. In
addition to the BTBDD distances, the similarity between detectors ECDFs is computed
using the Kolmogorov-Smirnov distance :
D i,j = 1
N obs
N obs
∑
|P i (r(n)) − P j (r(n))| (3.12)
n

50 3. Standard destriping techniques and application to MODIS
Table 3.3 – Mean Kolmokorov-Smirnov detectors distance for bands 27, 30
and 33 for Terra MODIS and 27, 30 and 36 for Aqua MODIS
– Terra Aqua
Detector Band 27 Band 30 Band 33 Band 27 Band 30 Band 36
d 1 0.3544 0.1108 0.0413 0.0265 0.0114 0.0189
d 2 0.1637 0.0253 0.0150 0.0253 0.0113 0.0142
d 3 0.1624 0.0261 0.0149 0.0338 0.0120 0.0192
d 4 0.1604 0.0253 0.0140 0.0246 0.0130 0.0139
d 5 0.1562 0.0312 0.0181 0.0245 0.0111 0.0114
d 6 0.1579 0.0273 0.0130 0.0278 0.0111 0.0106
d 7 0.1973 0.0259 0.0154 0.0255 0.0106 0.0118
d 8 0.2188 0.0429 0.0142 0.0958 0.0125 0.0135
d 9 0.3195 0.0260 0.0218 0.0353 0.0134 0.0134
d 1 0 0.2587 0.0248 0.0152 0.0514 0.0379 0.0140
where P i and P j are the ECDFs of detectors i and j, and r(n) is the set of radiances
observed by both detectors. The Kolmogorov-Smirnov distance measures the distributional
distance for every couple of detectors and can be averaged for a single detector i ≠ j as :
D i =
1
card(M i ) − 1
∑
D i,j
(3.13)
where M i is the set of detectors. [Antonelli et al., 2004] and [di Bisceglie et al., 2009],
specify that the classification of detectors and the selection of a reference one is determined
from both BTBDD and Kolmogorov-Smirnov distances. However, it is not mentioned how
this is done. As an alternative, we will rely only on the Kolmogorov-Smirnov distances ;
the weakest value D M i
i
determines the reference detector to be used for the radiometric
equalization. Let us denote by B r (n) and B i (n) the set of radiances measured in the
bowtie region by a reference detector r and a noisy detector i. The equalization function
is obtained with a polynomial regression between B r (n) and B i (n) :
j∈M i
j≠i
B r (n) =p 0 + p 1 .B i (n)+p 2 .B 2 i (n)+... + p N .B N i (n) (3.14)
where p 0 , p 1 ...p N are the coefficients of the best fitting polynome. The radiometric equalisation
of the signal I i (n) measured by the detector i over the entire swath is then corrected
as :
Î i (n) =p 0 + p 1 .I i (n)+p 2 .I 2 i (n)+... + p N .I N i (n) (3.15)
All the other detectors are corrected with the same procedure. Destriping results obtained
with the OFOV technique are illustrated in figure 3.7. When the equalization is based on
polynomial functions of order 1, the OFOV method might be comparable to the moment
matching technique because only the affine response of the detectors is modified. However,
while moment matching is based on statistical assumptions satisfied only over homogeneous
areas, the OFOV method relies entirely on the bowtie effect to equalize the detectors

51
Figure 3.8 – (Left) Noisy image from Terra MODIS band 27 (Right) Image destriped
with the IFOV method
response. To this extent, the OFOV method can be viewed as a radiometric calibration
procedure. It is worth mentionning that only the range of radiances contained in the bowtie
region is corrected and additional processing is required to cover the entire dynamic
range of the acquired signal. In our implementation, the equalization functions were derived
from measurements with scan angles higher than ±25˚. Linear and quadratic fitting
computed on the selected images showed little differences and only first order polynomes
were used. Radiance values not contained in the bowtie region were not equalized.
3.5 Frequency filtering
Frequency filtering methods has been widely used for the elimination of stripes on
satellite images. [Srinivasan et al., 1988] proposed a new approach for Landsat TM and
MSS based on a spectral analysis of the stripe noise. Despite extensive research in this
direction [Crippen, 1989], [Simpson et al., 1995], [Simpson et al., 1998], [Chen et al., 2003]
and good visual results on GOES and Landsat sensors, frequency filtering can only be
viewed as a cosmetic improvement. The smoothed results obtained with low-pass filtering
can nonetheless be used as a reference to compare the performances of other detriping
techniques. In this subsection, we briefly explore the application of frequency filtering on
MODIS data.
3.5.1 Spectral Analysis
In most cases, striping can be considered as a periodic noise. The application of a lowpass
filter (an averaging filter in the spatial domain for exemple) reduces the visual impact

52 3. Standard destriping techniques and application to MODIS
Figure 3.9 – (Left) Noisy image from Terra MODIS band 27 (Right) Module of its
fourier transform, showing lobes in the vertical axis of the fourier domain
of stripes but inevitably removes fine structures of the signal and, therefore compromisies
further quantitative analysis. To minimize the loss of high-frequency information, the
filtering procedure shall take into account the periodic nature of stripes prior to the design
of a band-pass filter. Stripe-related frequencies are determined from the spectrum of the
noisy image I s , which we consider as a two dimensional vector of size M × N. Its discrete
fourier transform is given by :
I F s (u, v) =
M∑
N∑
i=1 k=1
( ( jui
I s (i, k)exp −
2πM + jvk ))
2πN
(3.16)
where j is the imaginary unity (j = √ −1). Periodic stripes on images translate in the
fourier spectrum as a signature taking the form of multiple lobes, concentrated along the
vertical axis (figure 3.9b). An accurate estimation of periodic stripe frequencies requires a
distinction from the frequencies related to the true signal structures. For instance, the frequency
power spectrum estimated on the striped signal values, barely reveals the presence
of periodic noise (figure 3.10a). The spectral components of stripes can be highlighted
using the averaging method of
3.5.2 Band-pass filtering
The spectral analysis of MODIS data described above, identifies the frequencies to be
reduced. [Simpson et al., 1995] proposed several possible implementations of Finite Impulsional
Response (FIR) filters. The approach we retain here is based on a two-dimensional
filter in fourier space composed of wells centered at stripe frequencies. The band-pass filter

53
12
11
10
10
Power spectrum
8
6
4
2
0
Power spectrum
9
8
7
6
5
4
−2
0 0.1 0.2 0.3 0.4 0.5
Normalized frequency
3
0 0.1 0.2 0.3 0.4 0.5
Normalized frequency
Figure 3.10 – (Left) Power spectrum computed directly on the noisy image, frequncies
of striping are not visible (Right) Power spectrum computed as an average of the columns
periodograms ; striping frequencies appear as distinct peaks
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
u
u
v
v
Figure 3.11 – Frequency response of the filter H 1 . As the value of σ increases, the wells
(represented here as lobes for visual clarity) overlap. When frequencies located near the
center of fourier domain are reached by the lobes, the band-pass filter H 1 becomes a high
pass filter
takes the form :
H 1 (u, v) =1− ∑
(u s,v s)
exp
(− (u − u s) 2 +(v − v s ) 2 )
σ 2
(3.17)
where u s and v s are the center coordinates of the wells in the fourier domain and σ
controls the sharpness of the wells. The design of the FIR filter H 1 in the fourier 2D
domain, implicitly takes into account the unidirectional spatial property of stripes noise
by placing the wells along the vertical axis, centered at the same coordinates as the stripe
lobes observed in figure 3.8. With increasing values of σ, the wells start to overlap with
each other and eventually affect low-frequencies (figure 3.11b). For high values of σ, the
filter H 1 shifts from a band-pass filter to a high-pass filter that removes all the low-pass

54 3. Standard destriping techniques and application to MODIS
Figure 3.12 – (Left) Original image from Terra MODIS band 27 (Right) Image destriped
with the filter H 1
components. This results in high-pass filtered images containing mostly stripe noise. In
addition, the mean value of filtered images differs from the original signal. To overcome
this issue, we consider the following low-pass filter :
H B (u, v) =exp
(− u2 + v 2 )
σ 2 B
(3.18)
where σ B is small enough so that the filter H B mainly retrieves the mean value of the
noisy image. The destriped image is then obtained with the filter :
H 2 (u, v) =H 1 (u, v)+H B (u, v) (3.19)
An alternative implementation of the FIR filter can be done in the spatial domain by
convolving each column with a unidimensional spatial FIR filter kernel derived from the
Parks-McClellan (PM) algorithm [Parks and Mcclellan, 1972]. Destriping in the spatial
domain overcomes many limitations compared to the 2D fourier filtering. It removes the
constraints related to images dimensions being a power of two and slightly reduces horizontal
edge effects.
Another FIR filtering approach was presented in [Srinivasan et al., 1988] for the destriping
of Landsat. The fourier response of the proposed filter is composed of concentric
rings centered at stripe frequencies. This method however, smoothes data equally in both
directions and introduces blurring and ringing artefacts along the discontinuities of the
images.

55
Image distortion ID
1
0.98
0.96
0.94
0.92
0.9
0.88
0.86
0.84
0
0 10 20 30 40 50
σ
ID
NR
2000
1600
1200
800
400
Noise reduction NR
Image distortion ID
1
0.98
0.96
0.94
0.92
0.9
0.88
0.86
0.84
ID
NR
30
25
20
15
10
0
0 10 20 30 40 50
σ
5
Noise reduction NR
Figure 3.13 – ID and NR indexes as a function of σ in the filter H 2 for images of (Left)
Terra band 27 and (Right) Terra band 30. The value of σ B is fixed to 5. Images from band
27 are representative of smooth atmospheric effects and therefore, small scale structures
in the images are mostly due to striping. In (Left) for σ > 21, the filter H 2 behaves
as a high-pass filter that retains only high-frequency components, coincident with stripe
noise, hence the decrease in NR. This effect is not visible on images from band 30 where
high-frequencies are composed of land-ocean-clouds discontinuities.
3.6 Haralick Facet filtering
Destriping techniques described this far are only dedicated to only periodic stripes
and fail to process random stripes. Application on images extracted from Terra MODIS
band 33 do not display any improvement (and therefore are not illustrated). The study
presented in [Rakwatin et al., 2007] was the first to tackle the issue of random stripes on
MODIS. Rakwatin proposed an hybrid approach that combines histogram matching for
the removal of detector-to-detector stripes and mirror banding, with an iterated weighted
least squares facet filtering for random stripes. The facet model introduced in [Haralick and
Watson, 1981] decomposes a given image into connected facets. A given pixel is contained
in K 2 different blocks each composed of K × K pixels.
For a pixel i, we denote W i,k a resolution cellcomposed of k = {1, .., K × K} neighbooring
pixels. A given pixel is contained in several resolution cells as they overlap with each other.
Let us denote by J ik (r, c) the signal value at row r and column c in the resolution cell
W ik . Haralick’s sloped facet model assumes that J ik (r, c) can be expressed as :
J ik (r, c) =α ik r + β ik c + γ ik + n ik (r, c) (3.20)
where α ik , β ik and γ ik represent the slope plan coefficients of the facet model in the
cell W ik and n ik is the noise. Denoting (−L, −L) (resp. (L, L)) the relative row-column
coordinates of a cell upper-left corner (resp. lower-right corner), the values of a resolution
cell slope plane coefficients can be estimated by minimizing the following quadratic energy

56 3. Standard destriping techniques and application to MODIS
functional :
E(α ik ,β ik ,γ ik )=
L∑
L∑
r=−L c=−L
(α ik r + β ik c + γ ik − J ik (r, c)) 2 (3.21)
Least square estimates of the facet model coefficients are obtained by considering the
partial derivatives of the functional E with respect to α ik , β ik and γ ik and lead to the
following :
α ik =
β ik =
γ ik =
3
L(L + 1)(2L + 1) 2
3
L(L + 1)(2L + 1) 2
1 ∑
L
(2L + 1) 2
L∑
r=−L c=−L
L ∑
r=−L
L ∑
c=−L
r
c
L∑
c=−L
L∑
r=−L
J ik (r, c)
J ik (r, c)
J ik (r, c)
(3.22)
Let us consider the particular case where L = 1, which corresponds to blocks composed
of 3 × 3 pixels. Using the least square estimates of α ik , β ik and γ ik from equation (3.22)
with L = 1 , Ĵ ik (r, c) can be written as :
Ĵ ik (r, c) = 1 18 [(−3r − 3c + 2)J ik(−1, −1) + (−3r + 2)J ik (−1, 0) + (−3r +3c + 2)J ik (−1, 1)
+(−3c + 2)J ik (0, −1) + 2J ik (0, 0) + (3c + 2)J ik (0, 1)
+(3r − 3c + 2)J ik (1, −1) + (3r + 2)J ik (1, 0) + (3r +3c + 2)J ik (1, 1)]
(3.23)
Each 3 × 3 pixels block is represented by a set of 9 coefficients and is associated with an
estimate error defined as :
L
ɛ 2 ik (r, c) =
∑
L∑
r=−L c=−L
) 2 (Ĵik (r, c) − J ik (r, c)
(3.24)
In the original approach introduced in [], the value of a pixel is determined using the slope
coefficients of the block which induces minimal error ɛ ik . Another alternative proposed in
Li and Tam [2000] is to estimate the value at pixel i with an iterative procedure :
Ĵ (n+1)
ik
=
L∑
L∑
r=−L c=−L
w ik (r, c)Ĵ (n)
ik
(3.25)
The weighting coefficients w ik depend on the estimate errors obtained for each bloack and
are computed as :
(
)
L∑ L∑
w ik (r, c) =1/ ɛ ik (r, c) ɛ −1
ik (r, c) (3.26)
r=−L c=−L

57
Figure 3.14 – (Left) Image from Terra MODIS band 33 affected mostly with random
stripes (Right) Destriped result obtained after histogram matching and Haralick’s sloped
facet model filtering.
The application of Haralick’s model is not devoted to the correction of periodic stripes and
Rakwatin et al. suggest using the iterative facet filtering procedure only on specific pixels.
Detector-to-detector stripes are initially corrected via the histogram matching technique.
Lines acquired by noisy detectors are then visually detected and processed with the sloped
facet model. In pratice, we selected a facet window of size 3 × 3 pixels. As the number of
iterations of the weighted facet filtering procedure increases (3.25), the visual impact of
random stripes decreases. For the image from Terra MODIS band 33, 10 iterations were
required to achieve a cosmetic improvement (see figure 3.14).
3.7 Multiresolution approach
3.7.1 Limitations of fourier transform
The Fourier transform is a remarquable tool in signal processing. Shifting to the frequency
domain offers the possibility to extract information that would otherwise be unperceptible
in the spatial or temporal domain. Nevertheless, the fourier transform has a
major drawback. The shift in fourier domain is inevitably followed by a loss of temporal/spatial
information ; The frequency of a given event can only be known at the expense
of it occuring times. A compromise can be achieved with the short-term fourier transform
(STFT). It consists in limiting the computation of fourier transform to local portions of
the signal, using a fixed size sliding analysing window. The Heiseinberg principle then
highlights the limitations of the STFT ; For small sized windows, a good temporal localisation
is achieved with approximative frequency localisation. For increasing windows size,

58 3. Standard destriping techniques and application to MODIS
one eventually converges to the original fourier transform where temporal information is
completely lost.
3.7.2 Multiresolution analysis
Multiresolution analysis, and more precisely wavelet transform constitutes a powerful
alternative to fourier transform, exploited in numerous signal processing applications.
Multiresolution analysis allows the decomposition of a function f ∈ L 2 (R 2 ), as a sum of
approximations associated with different resolution levels. In [Mallat, 2000], Mallat defines
an multiresolution approximation as a set of closed vector sub-spaces {V j } j∈Z that verify
several properties among which :
∀j ∈ Z, ⊂ V j+1 ⊂ V j ⊂ L 2 (R) (3.27)
∀j ∈ Z,f(t) ∈ V j ⇔ f( t 2 ) ∈ V j+1 (3.28)
The embedding property (3.27) of vector spaces {Vj} ensures that the approximation of
the function f at resolution 2 j is obtained from an approximation at a higher resolution
2 j+1 . The property (3.28) garantees that the projection of f on the space V j contains
twice as much details as the projection on the space V j+1 . Denoting φ, a function ∈ L 2 (R)
which translations {φ(t − k)} k∈Z form an orthonormal basis of the space V 0 , it is shown
that the family of functions {φ j,k } k∈Z obtained as dilatations and translations of φ is an
orthonormal basis of V j where :
φ j,k = 1 √
2 j φ ( t
2 j − k )
(3.29)
The function φ, known as scale function or father wavelet, is dilated and translated to
form an orthonormal basis of the space V j , where the orthorgonal projection of a function
f defines its approximation at the resolution 2 −j .
3.7.3 Wavelet basis
Going from a resolution 2 j to a lower resolution 2 j+1 , leads to a loss of information.
Details visible in the approximation of resolution 2 j are lost at the resolution 2 j+1 . The
embedding property of multiresolution approximations and the inclusion of vector space
V j in V j−1 can be used to define the space of details W j as the orthogonal complementary
of V j in V j−1 :
V j−1 = V j ⊕ W j (3.30)
Similarly to the vector space of approximations V 0 , it is shown that the space of details W 0
is generated from an orthonormal basis composed of translated version {ψ(t − k)} k∈Z of a

59
function ψ, the mother wavelet. The family of functions {ψ j,k } k∈Z obtained as dilatations
and translations of ψ is an orthonormal basis of W j and is expressed as :
ψ j,k = √ 1 ( ) t
ψ
2 j 2 j − k (3.31)
The wavelet transform of a function f then provides the approximation coefficients denoted
a j [k] and the details coefficients d j [k]. These are obtained by projecting f on the vector
spaces V j and W j :
a j [k] =< f, φ j,k > (3.32)
d j [k] =< f, ψ j,k > (3.33)
where the symbol < ., . > denotes the scalar product in L 2 (R).
3.7.4 Filter banks
The property (3.27) implies that the vector space V j is included in V j−1 . Then, any
function in V j−1 can be written as a linear combination of a function in V j . If we consider
a given sequence h[k], the function √ 1
2
φ ( t
2)
can be expressed with respect to the family
of functions {φ(t − k)} k∈Z as :
(
1 t
∞∑
√ φ = h[k]φ(t − k) (3.34)
2 2)
where :
k=−∞
h[k] =< √ 1 φ( t ),φ(t − k) > (3.35)
2 2
W j being also a sub-space of V j−1 , the function √ 1
2
ψ ( t
2)
can be expressed with respect to
the family of functions {φ(t − k)} k∈Z and a different sequence g[k] :
where :
1
√
2
ψ
( t
=
2)
∞∑
k=−∞
g[k]φ(t − k) (3.36)
g[k] =< √ 1 ψ( t ),φ(t − k) > (3.37)
2 2
Equations (3.34) and (3.36) are known as the two scale equations and are used to expresss
the inter-scale relationship between the scaling function and the mother wavelet
with respect to their translations and the coefficients of filters h[k] and g[k]. Through the
combination of equations (3.32), (3.33), (3.34) and (3.36) it is shown that wavelet decomposition
and reconstruction are computed as a series of discrete convolutions with filters h
and g. In the decomposition stage, approximations and details coefficients are given with :
∞∑
a j+1 [k] = h[n − 2k]a j [n] =a j ⋆ h[2k] (3.38)
n=−∞

60 3. Standard destriping techniques and application to MODIS
d j+1 [k] =
∞∑
n=−∞
g[n − 2k]d j [n] =d j ⋆ g[2k] (3.39)
where h[k] =h[−k] and ⋆ is the convolution symbol. The approximation coefficients a j+1
result from the convolution of approximations a j with the low-pass filter h, followed by
a sub-sampling of factor 2. The sub-sampling operation consists in preserving only one
coefficient out of two. Details coefficients d j+1 are deduced from the convolution of d j
with the high-pass filter g. The reconstruction of the signal is obtained with the inverse
wavelet transform :
∞∑
∞∑
a j [k] = h[k − 2n]a j+1 [n]+ g[k − 2n]d j+1 [n]
(3.40)
n=−∞
=ă j+1 ⋆h + ˘d j+1 ⋆g
n=−∞
The successive approximations a j are obtained with a convolution of signals ă j+1 and ˘d j+1
with the filters h and g. ă results from an up-sampling of factor 2 of a, where zeros are
inserted between succesive samples as :
ă[2n] =a[n]
ă[2n + 1] = 0
(3.41)
The decomposition and illustration in filter banks is illustrated in figure . Filters ¯h, ḡ,
h and g are quadrature mirror filters due to the orthogonality relation between h and g :
3.7.5 2D wavelet basis
g[k] = (−1) 1−n h[1 − k] (3.42)
The wavelet decomposition of a bidimensional signal f ∈ L 2 (R 2 ) can be seperately
computed along its dimensions, using separable wavelet basis generated from the tensor
product of unidimensional orthogonal wavelet basis. In the 2D case, the wavelet decomposition
is computed first on the lines and then on the columns (or inversely). The sequence
{Vj 2}
j∈Z defined as Vj 2 = V j ⊗ V j is a separable multiresolution approximation of L 2 (R 2 ).
Similarly to the unidimensional case, the space of details Wj
2 is defined as the orthogonal
complementary of Vj 2 in Vj−1 2 : Vj−1 2 = Vj 2 ⊕ Wj 2 (3.43)
An orthonormal wavelet basis of L 2 (R 2 ) can be constructed from the scaling functions
φ and the mother wavelet ψ. To this prupose, let us define ∀(x, y) ∈ R 2 , the following
wavelets :
ψ 1 (x, y) =φ(x)φ(y)
ψ 2 (x, y) =ψ(x)φ(y)
ψ 3 (x, y) =ψ(x)ψ(y)
(3.44)

61
Figure 3.15 – (Left) Noisy image from Terra MODIS band 30 (Right) Wavelet decomposition
at two resolution levels showing that (1) stripe noise is isolated in the only
horizontal details d 1 j (2) Wavelet coefficients associated with stripes have a magnitude of
the same order than those related to the image edges
We then consider the dilated and translated versions of these wavelets defined for k =1, 2, 3
as :
ψj,m,l k = 1 ( x − 2 j m
2 j ψk 2 j , y − )
2j l
2 j (3.45)
The wavelet family {ψj,m,l 1 ,ψ2 j,m,l ,ψ3 j,m,l } (m,l)∈Z 2 is an orthonormal basis of the details
vector space Wj 2 and the family {ψ1 j,m,l ,ψ2 j,m,l ,ψ3 j,m,l } (j,m,l)∈Z3 is an orthonormal basis of
L 2 (R 2 ).
The separability of bidimensional wavelet basis is an interesting property for the multiresolution
analysis of images. In fact, the wavelet family {ψj,m,l 1 ,ψ2 j,m,l ,ψ3 j,m,l } (m,l)∈Z 2 allows
the extraction of details in the horizontal, vertical and diagonal directions. This particular
feature is indeed the main motivation behind the use of wavelet analysis for the striping issue.
Wavelet decomposition and reconstruction of a function f ∈ L 2 (R 2 ) is computed with
separable bidimensional convolutions. Using the conjugate mirror filters h and g of the
mother wavelet ψ, wavelet coefficients at level 2 j+1 are obtained from the approximation
at level 2 j with :
a j+1 [m, l] =a j ⋆ ¯h[m]¯h[l]
d 1 j+1[m, l] =a j ⋆ ¯h[m]ḡ[l]
d 2 j+1[m, l] =a j ⋆ ḡ[m]¯h[l]
d 3 j+1[m, l] =a j ⋆ ḡ[m]ḡ[l]
(3.46)
The bidimensional wavelet decomposition of an image contaminated with stripe noise is
illustrated in figure 3.15.

62 3. Standard destriping techniques and application to MODIS
3.7.6 Destriping with wavelet coefficient thresholding
As exposed in the previous section, the discrete dyadic wavelet transfom offers a sparse
representation of a signal, well suited for many image processing applications. In the case
of denoising, a major limitation not mentionned this far is the translation invariance condition,
not satisfied by the discrete dyadic wavelet transfom. This drawback does not affect
the wavelet decomposition in itself, however, if the wavelet coefficients are modified, the
wavelet reconstruction introduces artifacts along the image discontinuities. These artifacts,
visible as pseudo-Gibbs oscillations come from the decimation of wavelet coefficients between
successive decomposition levels. To avoid these effects, we will rely on the stationary
wavelet transform [Nason and Silverman, 1995]. The multiscale decomposition of an image
provides wavelet coefficients with a magnitude dependent on the local regularity of the
signal in a given resolution. Assuming a moderate amount of noise in the observed signal,
strong wavelet coefficients will be distributed along the edges of an image while weak coefficients
will be located in homogenenous areas. The noise can then be reduced from the
image by considering only specific wavelet coefficients in the reconstruction process. The
seminal work of [Donoho and Johnstone, 1994] describes two denoising techniques based
on the thresholding of wavelet coefficients. A first intuitive approach consists in setting
to zero all wavelet coefficients with an amplitude lower than a fixed threshold λ. This
operation is known as hard thresholding and is represented by a function S hard that takes
the following form :
{ 0 if d
Sλ hard (d k k
j )=
j ≤ λ
d k j if d k j >λ (3.47)
where {d k j } k=1,2,3 are detail coefficients in the horizontal (k = 1), diagonal (k = 2) and
vertical (k = 3) directions, obtained at the resolution 2 j .
An alternative approach in the selection of wavelet coefficients is based on soft thresholding
or wavelet shrinkage ; Coefficients that exceed a threshold are attenuated by the value of
the threshold. The resulting thresholding function S soft
λ
is given by :
{
S soft
0 if d
λ
(d k k
j )=
j ≤ λ
d k j − sign(dk j )λ if dk j >λ (3.48)
The quality of the reconstructed signal highly depends on the choice of the threshold which
can be the same for all resolutions or vary from one resolution to the other. A common
methodology relies on the universal thresholding of [Donoho and Johnstone, 1994] where
the value of the threshold is fixed as :
λ =ˆσ √ 2 log(M × N) (3.49)
where M × N is the number of pixels in the image and ˆσ is an estimation of the noise
variance, determined from the detail coefficients of the highest resolution with :
λ
ˆσ = median{dk 0 } k=1,2,3
0.6745
(3.50)

63
The highest resolution is used for the estimation of σ because its wavelet coefficients are
mostly related to the noisy component of the signal.
Going back to the striping issue, let us underscore an important point. Most denoising
technique presented in the litterature and based on wavelet coefficients thresholding are
devotd to the elimination of isotropic noise (gaussian, poissonian, speckle noise). Although
the basic principle of wavelet thresholding remains valid for our study, specific aspects such
as the choice of the threshold have to be revised and adapted for the case of stripe noise.
Destriping via wavelet thresholding have already been used on MODIS in [Yang et al.,
2003]. However, the proposed methodology is based on a soft thresholding applied to detail
coefficients of all directions. A refinement of this approach was introduced in [Torres and
Infante, 2001] for the destriping of Landsat MSS images. This technique, which we apply
here to MODIS data, exploits the unidirectional signature of striping and its impact on the
multiresolution decomposition. As illustrated in figure 3.15, the presence of stripe noise
affects only the horizontal component of the image. It is then reasonnable to restrain the
manipulation of wavelet coefficients to the horizontal details d 1 j . Figure 3.15 also indicates
that unlike white gaussian noise, striping translates as wavelet coefficients with very high
intensity. The amplitude of wavelet coefficients associated with the image edges is actually
dominated by stripe noise and application of classic thresholding strategies cannot be
considered in our case. The strategy developed in [Torres and Infante, 2001] consists in
eliminating all horizontal wavelet coefficients of the m highest resolution levels, m being
determined heuristically. This procedure is equivalent to a hard thresholding where all
horizontal coefficients are set to zero. Its thresholding function is :
{ 0 if k = 1 and 1 ≤ j ≤ m
S λ (d k j )=
otherwise
d k j
(3.51)
The destriping quality depends on the parameter m. When m is small, only small scale
details of the striping effect are removed. If m takes high values, the thresholding function
3.51 removes the horizontal low-frequency component of the image and introduces strong
blurring. The visual analysis of successive approximations shows that the impact of striping
tends to diminish through lower resolutions. The hard thresholding should then be limited
to resolutions where the stripe noise signature is still persistent (see figure 3.17). Wavelet
destriping through hard thresholding can provide good visual results depending on the
manipulation of horizontal coefficients. However it inevitably eliminates details related to
the image sharp structures.
3.8 Assessing destriping quality
In the previous sections, several approaches have been described and applied to MODIS
data severely affected with stripe noise. Visual examination of destriped results indicates
that equalization methods, either based on statistical or radiometric considerations, fail
to completely remove the noise in that a considerable amount of residual stripes are still

64 3. Standard destriping techniques and application to MODIS
Figure 3.16 – (Left) Noisy image from Terra MODIS band 30 (Right) Destriped result
setting to zero vertical details d 1 1 , d1 2 and d1 3 (m=3) prior to wavelet reconstruction
Image distortion ID
0.93
0.925
0.92
0.915
0.91
0.905
0.9
0.895
0.89
0.885
0.88
ID
NR
7000
6000
5000
4000
3000
2000
1 2 3 4 5 6 7 8 0 1000
m (db2)
Noise reduction NR
Image distortion ID
0.935
0.93
0.925
0.92
0.915
0.91
0.905
ID
NR
7000
6000
5000
4000
3000
2000
0.9
1000
0.895
1 2 3 4 5 6 7 8 0
m (db4)
Noise reduction NR
Figure 3.17 – Image distortion (ID)* and Noise reduction (NR)* as a function of the
thresholding parameter m for Terra MODIS band 27. The selected wavelets are (Left)
db2 and (Right) db4. The breaking points visible in the NR curves at m = 5 for the
wavelet db 2 and m = 3 for db 4 , translate the reduction of striping from higher to lower
resolution scales. Depending on the number of vanishing moments of the selected wavelet,
the striping will be concentrated in the wavelet coefficients of higher resolutions and its
impact on vertical wavelet coefficients eventually fades below at given scale, hence the
constant NR in lower resolutions.
visible. This is due to non-linear effects and random stripes, mostly present on Terra MO-
DIS. Despite uncomplete destriping, equalization techniques are often prefered for their
ability to preserve the signal radiometry.
On the other hand, filtering methods (frequency filtering, facet filtering and wavelet thre-

65
sholding) can be tuned to regulate the amount of distortion introduced in the restored
image. A rigourous comparison of these methods requires additional criteria other than
visual interpretation. In this section, we recall the definition of several qualitative indexes
used to evaluate the destriping quality.
3.8.1 Noise Reduction Ratio and Image Distortion
The spectral analysis conducted in section 3.5.1, takes advantage of the unidirectional
aspect of striping to highlight its spectral signature. Periodic stripes related to detectorto-detector
and mirror banding introduces peaks, clearly visible in the ensemble averaged
power spectrum down the columns. A reliable destriping should reduce the stripe peaks
without interfering with the global shape of the original signal’s power spectrum. Let us
denote P 0 -resp. P 1 - the power spectrum obtained as an average of the periodograms of
the noisy image -resp. the destriped image- columns. We define :
N 0 =
∑
P 0 (f)
f∈BW N
N 1 =
∑
P 1 (f)
f∈BW N
(3.52)
where BW N is the noisy part of the frequency spectrum (f ∈ BW N corresponds to f ∈
{0.1, 0.2, 0.3, 0.4, 0.5} in the case of detector-to-detector stripes). N 0 and N 1 contain the
spectral components of stripes in both the noisy image and the restored one. The Noise
Reduction ratio (NR) is then defined as :
NR = N 0
N 1
(3.53)
The NR index measures the attenuation of the frequency peaks in the power spectrum and
hence can only be used for images affected with periodic stripes. Furthermore, evaluation
of a destriping techniques with the NR index has to be imperatively completed with a
distortion index that quantifies the blur introduced in the destriped result. Denoting :
S 0 =
∑
P 0 (f)
f∈BW S
S 1 =
∑
P 1 (f)
f∈BW S
(3.54)
where BW S is the noise-free portion of the spectrum, the image distortion index used in
[Pan and Chang, 1992] is defined as :
ID = N 0
N 1
(3.55)

66 3. Standard destriping techniques and application to MODIS
The previous definition contrains the computation of the ID index to the only portion of
the frequency spectrum not affected with noise and, therefore tends to over estimate its
value. In addition, the sums in the terms N 0 and N 1 are dominated by the amplitude of
low-frequency power spectrum, which are approximatively the same for both noisy and
destriped images. This results in ID values very close to unity even in the presence of blur.
An alternative option is to focus on the periodograms of lines. Denoting Q 0 -resp. Q 1 -,
the ensemble averaged power spectrum down the lines of the noisy -resp. destriped imagewe
define :
S 0 =
∑
Q 0 (f)
S 1 =
f∈BW
∑
f∈BW
Q 1 (f)
(3.56)
where BW represents the entire frequency spectrum (BW = BW N + BW S ). The Image
Distortion (ID) index is then defined as :
ID =1−
1
card(BW)
∑
f∈BW
3.8.2 Radiometric Improvement Factors
|Q 0 (f) − Q 1 (f)|
Q 0 (f)
(3.57)
Results achieved with standard destriping techniques display considerable residual
stripes despite the reduction of the spectral component related to periodic noise. In fact,
both indexes NR and ID are defined in the fourier domain and cannot measure the spatial
regularity of the restored image. In the spatial domain, periodic and random stripes
translate as strong fluctuations of the image values along the vertical axis. Let us denote
m Is [j] and mÎ[j] the unidimensional signals associated with the mean values of lines j in
the noisy image I s and the destriped image Î. The destriping quality in the spatial domain
can then be determined by comparing the values of m Is [j] and mÎ[j]. To this purpose, let
us consider a reference signal I ref with a smooth cross-track profile. I ref is obtained from
the noisy image I s with a low-pass filter given by :
H(u, v) =exp
(− u2 + v 2 )
σ 2 (3.58)
Since the filter H is only used to provide a reference regularized cross-track profile, it can
also be replaced by a simple averaging filter in the spatial domain. Denoting :
d s [j] =m Is [j] − m Iref [j]
d e [j] =mÎ[j] − m Iref [j]
the first radiometric improvement factor is defined as :
(∑ )
j
IF 1 = 10log d2 s[j]
10 ∑
j d2 e[j]
(3.59)
(3.60)

67
A secondary index, independent of I ref , considers the radiometric errors :
∆ s [j] =m Is [j] − m Isj−1[j]
∆ e [j] =mÎ[j] − mÎ[j − 1]
The second order radiometric improvement factor is then given by :
(∑ )
j
IF 2 = 10log ∆2 s[j]
10 ∑
j ∆2 e[j]
(3.61)
(3.62)
3.8.3 Conclusion
Destriping techniques presented in this chapter can be classified in two major categories.
The first class is composed of statistical or radiometric equalization techniques such
as moment matching, histogram matching and the OFOV method. All these approaches
rely on the acquisition principle of pushbroom imaging systems to equalize the response of
individual detectors. Their application to MODIS data illustrates a significant amount of
residual stripes. Figure 3.20 shows that even after destriping, rapid fluctuations can still
be seen in the cross-track profiles of the denoised data. The analysis of column ensemble
averaged power spectrums (figure ) indicates that the amplitude of peaks related to periodic
stripes is reduced. To improve visual analysis of noise reduction, power spectrums
are plotted with a logarithmic scale, as a function of normalized frequency.
Equalization techniques do not account for non linear effects which are predominant in
MODIS data. Residual stripes in the images affect its power spectrum in two ways. Strong
non linear effects result in poor detector-to-detecotr stripes reduction and appear as persisting
peaks with a lower magnitude located at the same frequencies of 0.1, 0.2, 0.3, 0.4 and
0.5 pixels per cycles. Random stripes translate as strong variations of the power spectrum
in the high frequency range. Despite residual stripes, equalization techniques maintain the
image distortion index close to 1.
Alternative approaches to equalization are based on filtering techniques. The periodicity
of stripes can be exploited by a band-pass filter that cuts-off stripe related frequencies.
This can be achieved by placing narrow wells in the fourier response of the filter, centered
on the coordinates associated with periodic stripes. As seen in figure e, periodic peaks
are properly removed from the power spectrums. However, the overall shape of the column
power spectrum is strongly distorted. This is even more visible on the line ensemble
averaged power spectrum, which should be identical for both noisy and destriped images.
Wavelet analysis also offers an interesting perspective on the striping issue. The multdirectional
representation provided by wavelet transform is well suited to process striping
since only horizontal wavelet coefficients are contaminated with stripes. Setting to zero the
wavelet coefficients of the m highest resolution scales prior to wavelet reconstruction can
reduce the visual impact of stripe noise. The choice of the parameter m is fixed according
to the NR and ID indexes values and highly depends on the number of vanishing moments

68 3. Standard destriping techniques and application to MODIS
of the selected wavelet. If the analysing wavelet has enough vanishing moments, regular
areas will have small wavelet coefficients and the stripe noise will be isolated in the highest
resolution scales (figure 3.17b). On the contrary, if the number of vanishing moments is
weak, stripe noise will also affect wavelet coefficients of lower resolutions (figure 3.17a)
and therefore, wavelet destriping provides images with a weak ID index.
The facet filtering model proposed in [Rakwatin et al., 2007] is used over the histogram
matching techniques to process random stripes. Due to its hybrid aspect, it is not compared
to other techniques.
Filtering techniques provide results that are visually better than equalization methods because
the removal of stripes can be tuned with σ for band-pass frequency filtering or m for
wavelet thresholding. However, blurring and ringing artifacts introduced in the corrected
signal discards any further quantitative analysis based on radiometric values.
The limitations of standard techniques exposed in this chapter serve as a basis for
the development of a robust destriping technique. An optimal destriping algorithm should
satisfy the following requirements :
- Complete removal of stripe noise, whether periodic or random
- Minimization of the distortion introduced in the restored image
From the definition of the NR and ID indexes, an optimal destriping algorithm increases
NR while leaving ID close to 1. To achieve such results, we explore in the next chapter
the striping issue from a variational perspective.

Figure 3.18 – Destriped results on Terra MODIS band 30 (TL) Original image (TC)
Moment matching (CL) Histogram matching (CR) IFOV method (BL) Frequency filtering
(BR) Wavelet thresholding. Reflectances over oceanic regions are highlighted to
emphasize the presence of residual stripes.
69

70 3. Standard destriping techniques and application to MODIS
13500
13500
Mean value
13000
12500
Mean value
13000
12500
12000
0 100 200 300 400 500
Line Number
12000
0 100 200 300 400 500
Line Number
13500
13500
Mean value
13000
12500
Mean value
13000
12500
12000
0 100 200 300 400 500
Line Number
12000
0 100 200 300 400 500
Line Number
13500
13500
Mean value
13000
12500
Mean value
13000
12500
12000
0 100 200 300 400 500
Line Number
12000
0 100 200 300 400 500
Line Number
Figure 3.19 – Cross track profiles computed for (TL) Original image (TR) Moment
matching (CL) Histogram matching (CR) IFOV method (BL) Frequency filtering (BL)
Wavelet thresholding.

71
10
10
9
9
Power spectrum
8
7
6
5
4
Power spectrum
8
7
6
5
4
3
3
2
0 0.1 0.2 0.3 0.4 0.5
Normalized frequency
2
0 0.1 0.2 0.3 0.4 0.5
Normalized frequency
10
10
9
9
Power spectrum
8
7
6
5
4
Power spectrum
8
7
6
5
4
3
3
2
0 0.1 0.2 0.3 0.4 0.5
Normalized frequency
2
0 0.1 0.2 0.3 0.4 0.5
Normalized frequency
10
10
9
9
Power spectrum
8
7
6
5
4
Power spectrum
8
7
6
5
4
3
3
2
0 0.1 0.2 0.3 0.4 0.5
Normalized frequency
2
0 0.1 0.2 0.3 0.4 0.5
Normalized frequency
Figure 3.20 – Ensemble averaged power spectrums downn the columns for (TL) Original
image (TR) Moment matching (CL) Histogram matching (CR) IFOV method (BL)
Frequency filtering (BR) Wavelet thresholding.

Table 3.4 – Noise Reduction (NR), Image distortion (ID), radiometric Improvement Factors IF 1 and
IF 2 for the Terra MODIS band 27, 30 and 33
– Terra band 27 Terra band 30 Terra band 33
index NR ID IF 1 IF 2 NR ID IF 1 IF 2 NR ID IF 1 IF 2
moment matching 45.8895 0.6305 7.3085 11.5732 9.4280 0.9963 6.2656 6.6818 – 0.9995 0.1637 0.1724
histogram (IMAPP) 101.0629 0.9459 8.5463 20.1003 15.3316 0.8816 5.7455 12.8977 – 0.9773 4.8320 11.1566
IFOV 100.5751 0.9692 5.8563 13.7338 5.4195 0.8991 4.6190 5.7684 – 0.9388 -0.1046 0.5466
Frequency filtering 1784 0.9868 9.8866 21.5316 25.5615 0.9884 7.3092 14.8566 – – – –
Wavelet thresholding 5749 0.9326 9.8649 38.3728 48.9917 0.9050 6.2421 18.6632 – 0.8892 4.7349 17.0134
Table 3.5 – Noise Reduction (NR), Image distortion (ID), radiometric Improvement Factors IF 1 and
IF 2 for the Aqua MODIS band 27, 30 and 36
– Aqua band 27 Aqua band 30 Aqua band 36
index NR ID IF 1 IF 2 NR ID IF 1 IF 2 NR ID IF 1 IF 2
moment matching 12.643 0.984 10.4393 12.0686 4.2363 0.9991 6.4180 12.6573 1.90 0.9997 3.1844 2.3252
histogram (IMAPP) 33.8517 0.9828 11.5966 18.1798 4.4847 0.9839 5.5991 12.6135 1.9593 0.9957 3.2537 2.4242
IFOV 1 12.1310 0.9890 9.3335 11.9212 1.8756 0.9646 -3.6762 8.9016 1.1396 0.8627 -1.8263 2.3097
frequency filtering 510.5828 0.9854 10.3039 24.8529 7.8478 0.9874 6.35 17.8977 3.2048 0.9868 0.7912 10.6393
wavelet thresholding 1131 0.9280 9.6601 25.4008 13.4574 0.9288 6.1015 17.9192 5.2528 0.8918 -0.1918 11.3379
72 3. Standard destriping techniques and application to MODIS

73
Chapitre 4
A Variational approach for the
destriping issue
4.1 PDEs and variational methods in image processing
Many disciplines from physical sciences such as thermodynamics and fluids mechanics
have inspired the application of Partial Differential Equations (PDEs) in the field
of image processing. Early denoising methods are mostly based on smoothing operations,
either computed directly in the spatial domain via a convolution with a filter, or in the
frequency domain. Assuming the noise to be contained in the high frequencies of the observed
signal, a common restoration technique consists in convolving the noisy image with
a linear operator. Denoting u 0 the noisy signal defined in a bounded domain Ω of R 2 , an
estimate of the true image u is obtained by considering the scale-space generated by u 0
as :
∫
u(x, y, t) = G(x − ξ, y − η, t)u 0 (ξ, η)dxdy, (x, y) ∈ Ω (4.1)
Ω
Typically, the operator G is a bidimensional gaussian kernel :
G(x, y, t) = 1 ( −(x 2
4πt exp + y 2 )
)
4t
(4.2)
where the variance σ 2 =2t of the gaussian operator controls the degree of smoothing in
the restored image u. The work of [Koenderink, 1984] reformulates the convolution with
a gaussian kernel as a diffusion process where the value of a pixel can be expressed with
respect to a neighboorhood which size is determined by the variance σ 2 of the operator
G. The noise free image u can be seen as the solution of a simple parabolic PDE :
∂u
∂t (x, y, t) u
=∂2 ∂ 2 x (x, y, u
t)+∂2 ∂ 2 (x, y, t)
y
u(x, y, 0) = u 0 (x, y)
(4.3)

74 4. A Variational approach for the destriping issue
The previous PDE, also known as the heat equation, translates an isotropic diffusion process
where all directions are smoothed identically. The isotropic property of the heat
equation is a major drawback for denoising applications. In fact, noise in homogeneous
regions is effectively removed, however in heterogeneous areas, sharp discontinuities related
to edges are processed similarly to noise. The corresponding high gradient magnitude
is therefore reduced which results in a significant loss of contrast in the restored image.
This limitation was first explored by [Perona and Malik, 1990] by means of anisotropic
diffusion model, where the intensity of the diffusion process is dependent on the local
gradient value. The degree of smoothing is inversely proportional to the gradient value
so that homogeneous areas are strongly diffused while edge-discontinuities are preserved.
The anisotropic diffusion PDE proposed by Perona and Malik can be formulated as :
∂u
∂t (x, y, t) =div (φ(|∇u(x, y, t)|)∇u(x, y, t)) (4.4)
where div is the divergence operator, ∇ designates the spatial gradient operator and ψ is
a decreasing function. The decreasing property of φ ensures that the smoothing process is
weaker in regions with strong gradient values, and stronger in flat areas. Common choices
for φ include exponentially decreasing functions as :
φ(|∇u|) =exp(− |∇u|2
k 2 ) (4.5)
If φ is a constant function, the PDE equation (4.4) of Perona and Malik is reduced to :
∂u
∂t (x, y, t) =div (∇u(x, y, t)) (4.6)
which is another formulation of the isotropic heat equation. The anisotropic diffusion (4.4)
also faces limitations. The presence of noise introduces strong oscillations which can be
considered as edges and be subsenquently preserved. To circumvent this issue, alternative
techniques were proposed independently in [Catté et al., 1992] and [Nitzberg and Shiota,
1992]. The gradient of the image is replaced by a smoothed version and the PDE (4.4)
becomes :
∂u
∂t (x, y, t) =div (φ(|∇(G σ ∗ u(x, y, t))|)∇u(x, y, t)) (4.7)
where G σ is gaussian operator with variance σ. Further research in the field of anisotropic
diffusion introduced in [Alvarez et al., 1992] resulted in a non linear PDE :
∂u
∂t (x, y, t) =g (|∇(G σ ∗ u(x, y, t))|) |∇u(x, y, t)| div
( ∇u(x, y, t)
|∇u(x, y, t)|
)
(4.8)
where g is a decreasing function that tends to 0 when ∇u tends to infinity. The( second )
derivative of u in the direction orthogonal to the gradent ∇u is the term |∇u|div ∇u
|∇u|
,
and coincides with the image level lines. The intensity of the diffusion process is regulated

75
Figure 4.1 – (Left) Noisy image from Terra MODIS band 30 (Center) Application of the
heat equation (ID=0.41) (Right) Application of the Perona-Malik algorithm (ID=0.84).
by the term g (|∇G σ ∗ u|). In homogeneous areas, strong values of g (|∇G σ ∗ u|) induce an
anisotropic diffusion in the direction orthogonal to the gradient. Along the edges of the
images (strong gradient values), the weak weighting due to the descreasing property of g
disables the diffusion process.
In [Nordstrom, 1990], the original PDE of Perona and Malik is formulated in a way that
forces the estimated solution to be close to the observed image u 0 . The resulting PDE
takes the following form :
∂u
∂t (x, y, t) − div(φ(|G σ ∗∇u(x, y, t)|)∇u(x, y, t)) = β (u(x, y, t) − u 0 (x, y)) (4.9)
where the coefficient β regulates the fidelity of the estimated solution to the orignal noisy
sinal u 0 .
In the unifying framework proposed in [Deriche and Faugeras, 1995], the authors show
that image restoration based on the resolution of PDEs can be reformulated as the minimization
of energy functionals. Such formulation is adequate to many ill-posed inverse
problems encountered in computer vision applications such as denoising, deconvolution or
inpainting. According to Hadamard’s definition, a mathematical problem can be considered
as ill-posed if at least one of the following conditions is not satisfied :
- A solution exists
- If a solution exists, it is unique
- The solution is stable ; A small perturbation in the observed data induces a small perturbation
in the estimated solution.
Ill-posed inverse problems are systematically regularized to impose well-posedness and
are tackled under the assumption of a specific image formation model. Let us denote by
K a linear transformation that accounts for the sensor multiple degradations (diffraction,
defocalisation, mouvement blur). The observed image f resulting from the acquisition

76 4. A Variational approach for the destriping issue
process is given by :
f = Ku + n (4.10)
where n is generally assumed to be a zero mean gaussian noise. Equation (4.10) is refered to
as the direct problem and the inverse problem then consists in determining the real image
u from the noisy observation f. This can not be achieved through the inversion of the
operator K, since existence and stability of the inverse of K are not garanteed. Instead,
a stable approximation of K −1 can be considered by introducing a priori regularizing
information to enlarge the space of admissible solutions. The estimation of the true image
u from (4.10) reduces to an optimization problem based on the minimization of an energy
or cost function as described in (Deriche and Faugeras, 1995) :
E(u) =E 1 (u)+λE 2 (u) (4.11)
The term E 1 (u) translates the fidelity of the solution to the observed image while E 2 (u)
is a regularizing term that smoothes the estimated solution and is often related to the
gradient of u. The parameter λ balances the tradeoff between a good fit to the observed
image and a regularized solution. Typically,
E 1 (u) =‖f − Ku‖ 2
∫
E 2 (u) = φ(|∇u|)
Ω
(4.12)
In the particular case (4.12), an estimate of the true image û is obtained as the solution
to the following optimization problem :
∫
û = argmin u ‖u − f‖ 2 + λ φ(|∇u|)dΩ (4.13)
The selection of φ(|∇u|) =|∇u| 2 corresponds to the well-known Tikhonov regularization.
4.2 Rudin, Osher and Fatemi Model
Among many variational based image restoration procedures, the total variation regularization
introduced in the seminal work of [Rudin et al., 1992] provides efficient results
for image denoising because it is well suited for piecewise constant signals and therefore
allows the retrieval of sharp discontinuities. Total variation regularization was first proposed
in the context of image denoising, and have been successively applied to a variety
of fields including image recovery [Blomgren et al., 1997], image interpolation [Guichard
and Malgouyres, 1998], image decomposition (see section 4.3), scale estimation [Luo et al.,
2007], image inpainting [Chan et al., 2005] and super resolution [D.Babacan et al., 2008].
Let us recall the image formation model where we assume the linear operator K to be
unity :
f = u + n (4.14)
Ω

77
The goal is to find the true image u from the noisy observation f assuming n to be white
gaussian noise with a known variance σ 2 . This ill-posed problem can be regularized by
assuming that the true image u belongs to the space of functions of bounded variations,
denoted BV and defined as :
BV (Ω) =
{
u ∈ L 1 (Ω)|sup ϕ∈C 1 c ,|ϕ|≤1
(∫
) }
u divϕ < ∞
(4.15)
The total variation of a signal u ∈ BV (Ω) is equivalent to the L 1 norm of its gradient
norm and is given by :
{∫
}
TV(u) =sup u divϕ, ϕ ∈Cc 1 , |ϕ| ≤ 1
(4.16)
Hereafter, we consider u to be in BV (Ω) ∩C 1 , and the total variation simplifies into :
∫
TV(u) = |∇u|dΩ (4.17)
To retrieve u from f, Rudin, Osher and Fatemi propose to solve a constrained optimisation
problem :
minimize
TV(u)
subject to ∫ Ω f = ∫ Ω u, and ‖u − f‖2 = σ 2 (4.18)
In the previous formulation the equality constraint is not convex and can be replaced by
an inequality constraint as :
Ω
minimize
TV(u)
subject to ∫ Ω f = ∫ Ω u, and ‖u − f‖2 ≤ σ 2 (4.19)
Chambolle and Lions have shown in [Chambolle and Lions, 1997] that if ‖f− ∫ Ω fdΩ‖2 >σ 2
then problems (4.18) and (4.19) are equivalent. (4.19) can then be reformulated as the
minimization of :
E(u) =λ‖u − f‖ 2 + TV(u) (4.20)
where the lagrangian multiplier λ> 0 regulates the compromise between the fidelity term
and the regularizing term. For a given value of λ, Karush-Kuhn-Tucker conditions ensure
the equivalence between (4.19) and (4.20). The energy functional (4.20) will be refered to
hereafter as the ROF model. The minimization of ROF model is obtained via its Euler-
Lagrange equation :
⎛
⎛
−2λ(u − f)+ ∂
∂x
⎜
⎝
√ ( ∂u
∂x
∂u
∂x
) 2
+
(
∂u
∂y
⎞
⎟
) 2 ⎠ + ∂ ⎜
∂y ⎝
√ ( ∂u
∂x
∂u
∂x
) 2
+
(
∂u
∂y
⎞
⎟
) 2 ⎠ =0 (4.21)

78 4. A Variational approach for the destriping issue
Figure 4.2 – (Left) Noisy image from Terra MODIS band 30 (Right) Denoising with
TV regularization (ID=0.51)
which can take the simple form :
−2λ(u − f) + div
( ) ∇u
= 0 (4.22)
|∇u|
From a computational point of view, the non-differentiability of the term |∇u| for ∇u =0
is problematic. In [Acar and Vogel, 1994], the authors suggest a smoothed version of the
energy functional where the total variation norm is relaxed with a small positive parameter
ɛ. The original ROF energy functional (4.20) becomes :
∫
√
E ɛ (u) =λ‖u − f‖ 2 + |∇u| 2 + ɛdΩ (4.23)
which leads to the following Euler-Lagrange Equation :
(
)
∇u
−2λ(u − f) + div √ = 0 (4.24)
|∇u| 2 + ɛ 2
From the following inequality :
∫
∫
∀u ∈ L 1 (Ω), |∇u|dΩ ≤
Ω
Ω
Ω
√
∫
|∇u| 2 + ɛdΩ ≤
Ω
|∇u|dΩ+ √ ɛ|Ω| (4.25)
it follows that :
√
∫
lim |∇u|
ɛ→0
∫Ω
2 + ɛdΩ = |∇u|dΩ (4.26)
Ω
When the value of ɛ tends to zero, the solution of the optimisation problem (4.20) converges
to that of (4.25).

79
Fast and exact minimization algorithms for TV-based energy functionals have been subject
to intense research, and many methods can be used to solve ROF model. In their original
work, Rudin, Osher and Fatemi relied on a fixed step gradient descent. Quasi-Newton
schemes have been investigated in [Chambolle and Lions, 1997], [Dobson and Vogel, 1997],
and [Nikolova and Chan, 2007]. The dual formulation of TV minimization was explored
in [Cha] and later lead to Antonin Chambolle’s projection algorithm [Chambolle, 2004].
The approach developed by Chambolle was the first to provide a solution to the exact
TV minimization problem (4.20) instead of a smoothed version. More recently, techniques
based on graph cuts have proved successfull in [Boykov et al., 2001] and [Dar]. In this
section, we propose a fixed point technique based on a Gauss-Seidel iterative method. The
image domain Ω is discretized into a grid where (x i = ih) and (y i = jh), h being the cell
size. Let us denote D + , D − and D 0 the forward, backward and central finite differences.
In the x-direction for exemple, we can write :
(D ±x u) i,j = ± u i±1,j − u i,j
h
(D 0x u) i,j = u i+1,j − u i−1,j
2h
(4.27)
The divergence operator is implemented using the non-negative discretization which offers
more stability than a discretization based solely on central differences. The discret form
of Euler-Lagrange equation (4.24) is :
u i,j = f i,j + 1
2λ D −x
+ 1
2λ D −y
(
(
)
D +x u i,j
√
(D+x u i,j ) 2 +(D 0y u i,j ) 2 + ɛ 2
)
D +y u i,j
√
(D0x u i,j ) 2 +(D +y u i,j ) 2 + ɛ 2
(
)
= f i,j + 1
u i+1,j − u i,j
2λh 2 √
(D+x u i,j ) 2 +(D 0y u i,j ) 2 + ɛ − u i,j − u i−1,j
√ 2 (D−x u i,j ) 2 +(D 0y u i−1,j ) 2 + ɛ 2
(
)
+ 1
u i,j+1 − u i,j
2λh 2 √
(D0x u i,j ) 2 +(D +y u i,j ) 2 + ɛ − u i,j − u i,j−1
√ 2 (D0x u i,j−1 ) 2 +(D −y u i,j ) 2 + ɛ 2
(4.28)
The linearization of the previous equation leads to :
⎛
⎞
u n+1
i,j
= f i,j + 1
u
⎝
n i+1,j − un+1 i,j
u n+1
i,j
− u n i−1,j
2λh 2 − √
⎠
√(D +x u n i,j )2 +(D 0y u n i,j )2 + ɛ 2 (D −x u n i,j )2 +(D 0y u n i−1,j )2 + ɛ 2
⎛
⎞
+ 1
u
⎝
n i,j+1 − un+1 i,j
u n+1
i,j
− u n i,j−1
2λh 2 − √
⎠
√(D 0x u n i,j )2 +(D +y u n i,j )2 + ɛ 2 (D 0x u n i,j−1 )2 +(D −y u n i,j )2 + ɛ 2
(4.29)

80 4. A Variational approach for the destriping issue
To simplifie notations, we introduce :
1
C 1 = √
(D +x u n i,j )2 +(D 0y u n i,j )2 + ɛ 2
1
C 2 = √
(D −x u n i,j )2 +(D 0y u n i−1,j )2 + ɛ 2
(4.30)
1
C 3 = √
(D 0x u n i,j )2 +(D +y u n i,j )2 + ɛ 2
1
C 4 = √
(D 0x u n i,j−1 )2 +(D −y u n i,j )2 + ɛ 2
The solution of (4.24) is obtained with the following iterative scheme :
u n+1
i,j
= 2λh2 f i,j + C 1 u n i+1,j + C 2u i−1,j + C 3 u n i,j + C 4u n i,j−1
2λh 2 + C 1 + C 2 + C 3 + C 4
(4.31)
To satisfy Neumann boundary condition ∂u
∂n = 0, u i,j is extended by reflection outside the
domain Ω.
Since it’s introduction in 1992, total variation regularization has grown very popular in the
field of image processing. Nevertheless, for denoising purposes, its application is limited
to the removal of isotropic noises such as gaussian or speckle noise [Sheng et al., 2005].
Given the unidirectionality of stripe noise, tackling the striping issue with TV regularization
might not be appropriate. The wavelet analysis conducted in chapter 2 underscored
another geometrical feature of striping ; on most of MODIS emisssive bands, the amplityde
of stripe noise is of the same order as the edges of the image. Discontinuities due to
striping are perceived as image sharp structures and are therefore preserved by the TV
model. Distinction between stripe noise and image edges can not be achieved directly with
the TV model and regardless the choice of the lagrange multiplier λ, reduction of striping
effect is inevitably followed by a loss of contrast (figure 4.2).
Nonetheless, an alternative use of ROF model is conceivable. Recently, a variational approach
was proposed in the context of image destriping and inpainting. It is based on a
maximum a posteriori (MAP) algorithm applied to a modified image formation model
where the noisy observation f is given by :
f = Au + B + n (4.32)
where Au is a point to point multiplication. The degradation process is assumed to be
linear and includes the parameters A and B associated with gain and offset values for
every pixel. A and B are matrices of the same size as the image. n is assumed to be zero
mean gaussian noise. The true image u is deduced from a MAP estimate :
û = argmax
u
p(u|f) (4.33)

81
Using Bayes’rule the previous equations becomes :
û = argmax
u
p(f|u)p(u)
p(f)
(4.34)
The a posteriori term p(u|f) being independent on p(f), problem (4.33) reduces to :
û = argmax
u
Application of the logarithm function on (4.35) then gives :
p(f|u)p(u) (4.35)
û = argmax
u
{log (p(f|u)) + log (p(u))} (4.36)
Under the assumption of white additive gaussian noise, the conditional density of f given
u is given by :
p(f|u) = 1 (
Z exp − 1 )
2 ‖K− 1 2 (f − Au − B)‖
2
(4.37)
where Z is a normalizing constant and K is a diagonal matrix containing the variance
values of the noise n. The prior density probability function p(u) is dependent on the prior
contraint imposed on the image. The Markov prior for example is given by :
⎛
⎞
p(u) = 1 Z exp ⎝− 1 ∑ ∑
ρ(d c (u i,j )) ⎠ (4.38)
2λ
where C is the set of image cliques and d c (u i,j ) is a spatial measure of u at pixel (i, j),
expressed as a function of first or second order differences. In [Shen et al., 2008], the
authors selected an edge-preserving prior based on the Huber potential function defined
as :
{
x
ρ(x) =
2 if |x| ≤ µ
2µ|x|− µ 2 (4.39)
if |x| >µ
The maximization of the posterior probablility distribution (4.36) is equivalent to the
minimization of the following energy :
i,j
c∈C
E(u) =λ‖K − 1 2 (f − Au − B)‖ 2 + ∑ i,j
∑
ρ(d c (u i,j ))
c∈C
(4.40)
The work of [Shen et al., 2008] can be directly transposed to the TV framework. Since
the Huber-Markov model is used mainly for its edge-preserving ability, the prior density
probability function p(u) can also relie on the TV norm as :
p(u) = 1 exp (−λT V (u)) (4.41)
Z

82 4. A Variational approach for the destriping issue
Figure 4.3 – (Left) Image from Terra MODIS band 30 denoised with TV regularization
(ID=0.5750) (Right) Denoised with the variational model () (ID=0.7768)
If we discard the presence of gaussian noise n or assume it is implicitly accounted for in
the gain and offset parameters A and B, the covariance matrix K reduces to the identity
matrix and can be discarded. Then, a destriping approach similar to [Shen et al., 2008]
can be achieved by minimizing :
E(u) =λ‖(f − Au − B)‖ 2 + TV(u) (4.42)
Destriping via the minimization of (4.42) relies on the observational model (4.32)) where
A and B are assumed to be known. This is however not the case in pratice and a preprocessing
stage is required to estimate the values of A and B. In their work, [Shen et al.,
2008] used the moment matching technique to evaluate the values of A and B from the
image mean value and variance. The limitations of such approach are directly attached to
its hybrid aspect. In fact, the TV or Huber-Markov regularization acts as a post-processing
stage, smoothing residual stripes that moment matching or any other technique based on
linear adjustment (OFOV method for exemple) fails to remove. Nonetheless, the destriped
results are much less distorted than those obtained with a direct application of the ROF
model (figure 4.3).
4.3 Striping as a texture
The main characteristic of stripping effect lies on it’s unidirectional aspect. To this
extent, stripe noise can be considered as a structured texture with sharp fluctuations
along a single axis of the image. This is particularly the case on MODIS spectral bands
where striping is periodic. Texture discriminating variational models inspired from Yves

83
Meyer’s work can then be used in hope of isolating the stripe noise from other structures
present in the true scene.
4.3.1 Yves Meyer’s model for oscillatory functions
The total variation regularization used in the previous section as a denoising technique
can be seen from a different perspective as an image decomposition model, where
the observed image f is approximated by a sketchy version u that lies in the BV space
and the component v = f − u contains small sace details such as noise and/or texture.
Many variational methods have the explicit goal of extracting an image u composed of
homogeneous areas separated by sharp discontinuities but do not retain the component
v as it is considered to be noise. This can be problematic for images containing texture
because noise and texture are both oscillatory functions, processed equally with the TV
regularization. From an image decomposition perspective, ROF model can be formulated
as the following minimization :
inf
(u,v)∈BV (Ω)×L 2 (Ω)/u+v=f
(
)
TV(u)+λ‖v‖ 2 L 2 (Ω)
(4.43)
In his investigation of the standard ROF model [Meyer, 2002], Yves Meyer pointed out
that small values of λ can remove fine details related to texture. To overcome this issue,
he proposes a different decomposition, where the classical L 2 norm associated with the
residual v = f − u is replaced by a weaker-norm, more sensitive to oscillatory functions.
Yves Meyer suggests finding a component v in the space G defined as the Banach space
composed of all distributions v which can be written as :
v(x, y) =∂ x g 1 (x, y)+∂ y g 2 (x, y) (4.44)
where g 1 and g 2 both belong to the space L ∞ (R 2 ). The space G is endowed with the norm
‖v‖ G defined as the lower bound of all L ∞ norms of the functions |⃗g| where ⃗g =(g 1 ,g 2 )
and |⃗g(x, y)| = √ g 1 (x, y) 2 + g 2 (x, y) 2 . Additionnaly to G, which can be viewed as the
dual space of BV , Meyer introduces the spaces E and F also suited to model texture.
The space E (dual of Ḃ 1,1
1 ) is defined similarly to G except that g 1,g 2 are in the space
of bounded mean oscillations functions denoted by BMO(R 2 ). For the space F (dual of
H 1 ), g 1 ,g 2 belong to Besov space B∞
−1,∞ (R 2 ). When the texture component v is assumed
to lie in the space G, Yves Meyer proposes the following decomposition model :
(
TV(u)+λ‖v‖G(Ω ))
2 (4.45)
inf
(u,v)∈BV (Ω 2 )×G(Ω 2 )/u+v=f
The ‖.‖ G -norm can efficiently capture oscillating patterns because it is weaker than the
‖.‖ 2 -norm (L 2 (Ω) ⊂ G(Ω) ). However, due to its mathematical form, the Euler-Lagrange
equation of (4.45) cannot be expressed explicitly and several u + v decomposition models
have been later introduced as an approximation to Yves Meyer model.

84 4. A Variational approach for the destriping issue
4.3.2 Vese-Osher’s Model
Vese and Osher were the first to overcome the difficulty of Meyer’s functional minimization.
They proposed a pratical resolution in [Vese and Osher, 2002] where they
approximate the L ∞ norm of |⃗g| with :
√
∥ ‖|⃗g|‖ L ∞ =
∥ g1 2 + ∥∥∥ g2 2∥ = lim
√g 2
L ∞ p→∞
1 + g2 ∥
2
∥
L p
(4.46)
As pointed out in [Meyer, 2002], the residual v = f − u in the original ROF model, can
be expressed as the divergence of a vector field ⃗g ∈ L ∞ (Ω) since :
v = − 1 ( ) ∇u
2λ div |∇u|
Vese and Osher then consider the space of generalized functions :
(4.47)
G p (Ω) = {v = div(⃗g), ⃗g =(g 1 ,g 2 ), g 1 ,g 2 ∈ L p (Ω)} (4.48)
induced by the norm :
‖v‖ Gp(Ω) =
inf
v=div(⃗g), g 1 ,g 2 ∈L p (Ω)
√
∥ g1 2 + g2 2∥ (4.49)
As an approximation to Yves Meyer model, Vese and Osher propose the following decomposition
:
inf
(u,⃗g)∈BV (Ω)×L p (Ω) 2 |u| BV (Ω) + λ‖f − (u + div(⃗g))‖ 2 L 2 (Ω) + µ ‖|⃗g|‖ L p (Ω) (4.50)
which approximates the original decomposition model of Yves Meyer when λ →∞and
p →∞. The minimization problem () also writes as :
∫
inf
|∇u|dxdy + λ |f − u − ∂ x g 1 − ∂ y g 2 |
(u,(g 1 ,g 2 ))∈BV (Ω)×L p (Ω)
∫Ω
2 dxdy
2 Ω
(∫ √
+ µ
(
Ω
g 2 1 + g2 2 )p dxdy
) 1
p
(4.51)
where λ and µ are tuning parameters. The first term in () forces u to lie in the space
BV (Ω). The second term insures that the noisy image can be approximated as f ≈
u + div(⃗g). The last term is a penalty on the G p norm of v which regulates the amount of
texture extracted in the component v. The minimization of Vese-Osher functional requires
the computation of its partial derivative with respect to u, g 1 and g 2 which leads to three

85
coupled Euler-Lagrange equations :
u = f − ∂g 1
∂x − ∂g 2
∂y + 1 ( ) ∇u
2λ div |∇u|
) ( p−2 ∥∥∥∥ √
) 1−p (
µg 1
(√g1 2 + g2 2 g1 2 + g2 ∂(u − f)
2∥ =2λ
p
∂x
) ( p−2 ∥∥∥∥ √
) 1−p (
µg 2
(√g1 2 + g2 2 g1 2 + g2 ∂(u − f)
2∥ =2λ
p
∂y
)
+ ∂2 g 1
∂ 2 x + ∂2 g 2
∂x∂y
)
+ ∂2 g 1
∂x∂y + ∂2 g 2
∂ 2 y
(4.52)
In [Vese and Osher, 2002], Vese and Osher observed similar results for 1 ≤ p ≤ 10. When
p ≫ 10, only texture associated with small scale details can be extracted. As suggested
by Vese and Osher, we consider the case p = 1 for computational speed and the previous
equations reduce to :
u = f − ∂g 1
∂x − ∂g 2
∂y + 1 ( ) ∇u
2λ div |∇u|
( )
g 1
∂(u − f)
µ √ =2λ
+ ∂2 g 1
g
2
1 + g2
2 ∂x ∂ 2 x + ∂2 g 2
∂x∂y
( )
g 2
∂(u − f)
µ √ =2λ
+ ∂2 g 1
g
2
1 + g2
2 ∂y ∂x∂y + ∂2 g 2
∂ 2 y
(4.53)
Neumann conditions are used at the boundary of the image domain Ω. Let us denote
by (n x ,n y ) the normal to the boundary ∂Ω of the image domain, Neumann conditions
translate to :
∇u
|∇u| (n x,n y ) = 0
(
)
f − u − ∂g 1
∂x − ∂g 2
∂y
n x =0
(
)
f − u − ∂g 1
∂x − ∂g 2
∂y
n y =0
(4.54)
The three coupled equations derived from Vese-Osher model are discretized and linearized
as in section 4.2. Mixed derivatives of g 1 (and g 2 ) are approximated with :
∂ 2 g 1,i,j
∂x∂y = 1
4h 2 (g 1,i+1,j+1 + g 1,i−1,j−1 − g 1,i+1,j−1 − g 1,i−1,j+1 ) (4.55)

86 4. A Variational approach for the destriping issue
Using the same notations for C 1 , C 2 , C 3 , C 4 as in section 4.2, estimates for u, g 1 and g 2
are obtained via the fixed point iterative scheme :
(
) (
u n+1
1
i,j
=
1+ 1
f i,j − gn 1,i+1,j − gn 1,i−1,j
(C
2λh 2 1 + C 2 + C 3 + C 4 )
2h
− gn 2,i,j+1 − gn 2,i,j−1
+ 1
)
2h 2λh 2 (C 1u n i+1,j + C 2 u n i−1,j + C 3 u n i,j+1 + C 4 u n i,j−1)
⎛
g1,i,j n+1 = ⎝
4λ
h 2 µ
⎞
(
2λ
u
n
√
⎠ i+1,j − u n i−1,j
g1,i,j n 2 + g2,i,j n 2 2h
− f i+1,j − f i−1,j
2h
+ gn 1,i+1,j − gn 1,i−1,j
h 2 + 1
4h 2 (g 2,i+1,j+1 + g 2,i−1,j−1 − g 2,i+1,j−1 − g 2,i−1,j+1 )
⎞
(
⎝
2λ
u
n
√
⎠ i,j+1 − u n i,j−1
− f i,j+1 − f i,j−1
g1,i,j n 2 + g2,i,j n 2 2h
2h
⎛
g2,i,j n+1 =
4λ
h 2 µ
+ gn 2,i,j+1 − gn 2,i,j−1
h 2 + 1
4h 2 (g 1,i+1,j+1 + g 1,i−1,j−1 − g 1,i+1,j−1 − g 1,i−1,j+1 )
4.3.3 Osher-Solé-Vese’s Model
)
)
(4.56)
The model proposed by Osher, Solé and Vese in [Osher et al., 2002], provides another
practical approximation of (4.45). If the texture component can be written as v = f − u =
div(⃗g) with ⃗g ∈ L ∞ (Ω) 2 then the Hodge decomposition of ⃗g leads to :
⃗g = ∇P + ⃗ Q (4.57)
where ⃗ Q is a divergence free vector. Considering the divergence of the previous equation,
the texture component f − u can be written as :
f − u = div(⃗g) =div(∇P + ⃗ Q)=div(∇P )=△P (4.58)
From the previous equation, P can be expressed as P = △ −1 (f − u). Osher et al. then
suggest to replace the L ∞ -norm by the L 2 -norm of |⃗g|. Neglecting Q ⃗ in the Hodge decomposition
of ⃗g, Osher et al. consider the simple minimization problem :
∫ ∫
inf
u∈BV (Ω)
Ω
|∇u| + λ
inf
u∈BV (Ω)
Ω
Ω
|∇(△ −1 )(f − u)| 2 dx dy (4.59)
Using the norm in the space H −1 defined by ‖v‖ H −1 = ∫ Ω |∇(△−1 )(v)| 2 dx dy, the previous
minimization problem can be simplified to :
∫
|∇u| + λ‖f − u‖ 2 H−1 (4.60)

87
The minimization of (4.60) is obtained with the Euler-Lagrange equation :
( ) ∇u
2λ △ −1 (f − u) =div
|∇u|
which takes a pratical form after application of the Laplacian operator :
( ( )) ∇u
2λ(f − u) =△ div
|∇u|
(4.61)
(4.62)
Equation (4.62) can be solved using a fixed step gradient descent. The estimation of u at
iteration n + 1 is given by :
( ( ))) ∇u
u n+1 = u n − ∆ t
(2λ(f − u n n
) −△ div
|∇u n |
(4.63)
u 0 = f
It is interesting to point out that Osher-Solé-Vese model is very similar to the particular
case of Vese-Osher model where p = 2 and λ = ∞. In fact, the space G p (Ω) coincides
with the Sobolev space W −1,p (Ω) which is the dual of W 1,p′
0 (Ω) where p and p ′ satisfie the
relationship 1 p + 1 p
= 1. If p = 2, then the texture component v lies in G ′ 2 (Ω) = W −1,2 (Ω) =
H −1 (Ω). The decomposition model of Vese and Osher can then be written as :
inf |u| BV (Ω) + λ‖f − (u + v)‖ 2 L 2 (Ω) + µ ‖v‖ H −1 (Ω) (4.64)
(u,v)∈BV (Ω)×H −1 (Ω)
4.3.4 Other u + v models
More image decomposition models were introduced following [Vese and Osher, 2002]
and [Osher et al., 2002]. In [Aujol et al., 2005], the following energy is considered :
(
)
TV(u)+λ‖f − (u + v)‖ 2 L 2 (Ω)
(4.65)
inf
(u,v)∈BV (Ω)×G µ(Ω)
where the space G µ is defined as :
G µ = {v ∈ G(Ω)/‖v‖ G ≤ µ)} (4.66)
The minimization of (4.66) is obtained with the projection algorithm of Chambolle.
[Daubechies and Teschke, 2005] introduced an image decomposition model based on a
wavelet framework as :
inf
(u,v)∈B 1 1 (L1 (Ω))×H −1 (Ω)
(
)
2α|u| B 1
1 (L 1 (Ω)) + ‖f − (u + v)‖2 L 2 (Ω) + ‖‖2 H −1 (Ω)
(4.67)
The space BV (Ω) used in most image decomposition models is replaced by a space suited
for wavelet coefficients, namely the Besov space B 1 1 (Ω).

88 4. A Variational approach for the destriping issue
We also refer to the variational model initially proposed for 1D signals in [] and later
explored by [Nikolova, 2004]. The proposed functional replaces the L 2 norm of the original
ROF model by the L 1 norm as :
inf
(u,v)∈BV (Ω) 1 (Ω)
(
)
TV(u)+λ‖f − (u + v)‖ 1 L 1 (Ω)
(4.68)
4.3.5 Experimental results and discussion
The original ROF model, Vese-Osher (VO) and Osher-Solé-Vese (OSV) decomposition
models have been applied to Terra MODIS images from band 30 and 33, in an attempt to
extract stripe noise in the texture component v. The u + v decompositions are illustrated
in figures 4.4 and 4.5. Traditionally, the lagrange multiplier λ is selected so that the L 2
norm of v is the same for every decomposition model. This strategy aims at visually
evaluating the ability of decomposition models to discriminate texture and noise from the
image main content and is well suited when no requirements are imposed on the cartoon
component u. In our case however, the primary goal is to isolate the striping on v while
preserving an acceptable distortion in u as it constitute an estimate of the destriped image.
Consequently, we proceed by selecting a value of λ that ensures approximatively the same
ID index for u for every decomposition model.
Figures 4.4 and 4.5 underscore a better ability of (VO) and (OSV) models compared to the
ROF model in terms of texture discrimation. In fact, it can be seen that the v component
of the ROF model also contains smooth structures related to ocean and clouds, and not
visible with VO and OSV decompositions. Nevertheless, despite a strong regularization
(ID=0.7), striping is still visible in the u-component of all three decomposition models.
Remarquably, we point out that the u-component derived from ROF model contains less
stripes than that obtained with (VO) and (OSV) models. This observation is in agreement
with the conlusion drawn from the wavelet analysis of section 3.7. Indeed, as a result of its
high intensity (in terms of gradient values), striping tends to be considered as an image
discontinuity and is therefore better preserved in the cartoon component u of texture
discriminating variational models.
For all three models, the presence of residual stripes despite oversmoothing, is a limitation
related to the contradictive compromise between the terms of the energy functionals. In
fact, in addition to the true image u being attached simply to the noisy image f (regardless
the norm used), the commonly used TV-norm reinforces the anisotropic preservation of
structures and does not distinguish strong gradient values related to striping from those
corresponding to edges. We recall that the initial motivation behind the use of variational
decomposition models is the textured aspect of striping due to its unidirectionality. This
feature however, is not accounted for neither in the fidelity term nor in the regularizing
term.
We shall see in the following sections how the introduction of directional information in
variational models offers a new perpective for the removal of stripe noise.

Figure 4.4 – Decomposition of the image from Terra MODIS band 30 as u (Left) +
v (Right). λ is selected so that ID(u) is the same for the three models. From top to
bottom, TV model (NR=4.56, ID=0.739), VO model (NR=4.23, ID=0.735) and OSV
model (NR=2.47, ID=0.735)
89

90 4. A Variational approach for the destriping issue
Figure 4.5 – Decomposition of the image from Terra MODIS band 33 as a cartoon u
(Left) + v (Right). λ is selected so that ID(u) is the same for the three models. From
top to bottom, TV model (ID=0.447), VO model (ID=0.472) and OSV model (ID=0.461)

91
4.4 Destriping via gradient field integration
This far, we introduced variational methods based on PDE’s and energy functional minimization,
with image denoising applications in mind. These techniques are actually used
extensively in computer vision/graphics, to solve over-constrained geometric problems.
It is typically the case of Photometric Stereo Methods (PSM) and Shape From Shading
(SFS) applications where the goal is to recover a depth map (or an image) by integrating
a gradient field with discrete values.
Let us denote by G =(G x ,G y ) a bidimensional gradient vector defined in a subspace
Ω of R 2 . The problem of gradient field integration consists in determining a function u
whose gradient ∇u is close to G. Two classes of techniques can be used to tackle this
ill-posed problem and estimate the true image u. Local integration techniques [Coleman
et al., 1982], [Healey and Jain, 1984] and [Wu and Li, 1988] rely on a curve integration :
∫
u(x, y) =u(x 0 ,y 0 )+ G x dx + G y dy (4.69)
where γ is the integration path from a starting point (x 0 ,y 0 ) to pixel (x, y) ∈ Ω. Local integration
techniques recover an image u starting with an initial height and then propagating
height values according to the neighborhood gradient values. Local integration techniques
are dependent on the data accuracy and can propagate error values when dealing with
noisy gradient fields. Global integration methods [Horn and Brooks, 1986], [], [Frankot
and Chellappa, 1988] and [Horn, 1990] can be formulated in a variational framework as
the minimization of an energy functional :
∫
E(u) =
Ω
γ
( ) ∂u 2 ( ) ∂u 2
∂x − G x +
∂y − G y dx dy (4.70)
Unlike denoising and decomposition variational models described this far, both terms in the
energy functional (4.70) are fidelity terms that measure the L 2 norm difference between the
gradient field components of u and the observed gradient field G. For instance, the energy
functional does not include any lagrange multiplier λ. The Euler-Lagrange equations of
the previous functionals leads to :
∂
∂x
( ∂u
∂x − G x
which can be simplified into the following Poisson equation :
)
+ ∂ ( ) ∂u
∂x ∂y − G y = 0 (4.71)
∇ 2 u = ∇.G (4.72)
The previous equation can be solved using a fast marching method [Ho et al., 2006] or
the streaming multigrid method proposed in [Kazhdan and Hoppe, 2008]. Among several
other techniques, we draw a particular attention to the well-known Frankot-Chellapa

92 4. A Variational approach for the destriping issue
(FC) algorithm [Frankot and Chellappa, 1988], which provides a solution by considering
the Fourier transform of the gradient field G components. Let us denote by (ξ x ,ξ y ) the
frequency components as in [Bracewell, 1986] and recall the differentiation properties of
the Fourier transform :
F
⎧⎪ ( ∂u
( ∂x)
) = j.ξx F(u)
⎨ F ∂u
∂y
= j.ξ y F(u)
( )
F ∂ 2 u
= ξ 2 (4.73)
xF(u)
⎪ ⎩
∂ 2 x
F
(
∂ 2 u
∂ 2 y
)
= ξyF(u)
2
where j is the imaginary unit √ −1. The fourier transform of the Poisson equation (4.72)
can be written as :
(
ξ
2
x + ξy
2 )
F(u) =−jξx F(G x ) − jξ y F(G y ) (4.74)
Denoting u F , G F x and G F y the fourier transform of u, G x and G y , the Frankot-Chellapa
reconstruction algorithm gives :
u F = −jξ xG F x − jξ y G F y
ξ 2 x + ξ 2 y
(4.75)
As we will see, gradient field integration problems and their gradient-based variational
formulation offer an interesting perspective on the destriping issue. In fact, compared to
other restoration-oriented variational models, the energy functional (4.70) clearly separates
the information related to vertical gradient and horizontal gradient. Going further
in this gradient-based reasoning, we make the following remark :
If the stripe noise is additive, it mostly affects the vertical gradient of the striped image
Let us the consider the following image formation model :
I s = I + n (4.76)
where I is the stripe free true image and n is the stripe noise. The linear operator K is
considered to be the identity. Let us assume that the unidirectional signature of the stripe
noise n translates on its horizontal gradient as :
∂n(x, y)
∂x
≈ 0 (4.77)
The partial derivatives along the x and y-axis of the image formation model (4.76) :
∂I s
∂x = ∂I
∂x + ∂n
∂x
∂I s
∂y = ∂I
∂y + ∂n
∂y
(4.78)

93
Figure 4.6 – (Left) Horizontal and (Right) vertical gradient of the image from Terra
MODIS band 30. The stripe noise is isolated in the vertical gradient
can be simplified under the assumption (4.77) into :
∂I s
∂x ≈ ∂I
∂x
∂I s
∂y = ∂I
∂y + ∂n
∂y
(4.79)
Since the stripe noise only affects the horizontal gradient of the image (figure 4.6), we
propose a variational model where the directional information is distributed separately on
the fidelity term and the regularizing term. In addition, the regularization is limited to the
direction of the stripe noise. Under these requirements, we minimize the following energy :
∫
E(u) =
Ω
∂(u − I s )
∥ ∂x
∥
2
∫
dx dy + λ
Ω
∂(u − H ⊗ I s )
∥ ∂y
∥
2
dx dy (4.80)
where H is a low-pass filter applied to the noisy image I s only to approximate the true
image cross-track profile. In fact, if the regularizing term in (4.80) is replaced with ‖ ∂u
∂y ‖2 ,
lines of the estimated solution u will have zero mean. Frankot-Chellapa algorithm presented
previously can be used to minimize the previous energy functional which Euler-Lagrange
equation is :
∂ 2 (
u
∂ 2 x − ∂2 I s ∂ 2 )
∂ 2 x + λ u
∂ 2 y − ∂2 (H ⊗ I s )
∂ 2 =0 (4.81)
y
Using the differienciation properties of the fourier transform, the previous equation can
be expressed as :
ξ 2 x(u F − I F s )+λξ 2 y(u F − H F .I F s )=0 (4.82)

94 4. A Variational approach for the destriping issue
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
u
u
v
v
Figure 4.7 – Frequency response of the filter ˜H. As the value of λ decreases, only frequencies
close to the vertical axis of the fourier domain are reduced.
where u F , Is
F and H F are the fourier transform of u, I s and H. An estimate of the stripe
free true image u is obtained with a simple inverse fourier transform :
)
u = F −1 (
(ξ
2
x + λξ 2 yH F ).I F s
ξ 2 x + λξ 2 y
(4.83)
Remark : If we ignore the term H ⊗I s in (4.80) and denote ˜H a filter defined in the fourier
domain as :
ξx
˜H 2 =
ξx 2 + λξy
2 (4.84)
destriping via the variational model (4.80) is equivalent to filtering with ˜H. The fourier
response of the filter ˜H is illustrated in figure 4.7.
The destriping variational model (4.80) is applied to images extracted from Terra
MODIS band 27, 30 and 33 and the results are illustrated in figures 4.8 and 4.9. In the case
of homogeneous data, as in band 27, we observe a complete removal of the striping effect
without any visible blur in the destriped result. We point out that the overall blur, i.e the
image distortion index defined in section 3.8, depends on the lagrange multiplier λ, whch
choice is discussed further in section 4.6. Although the results on band 27 are satisfactory,
the destriping model does not perform as well on images containing strong discontinuities.
Figure 4.9, clearly shows local blurring artifacts along sharp edges at ocean-land and oceanclouds
transitions. This major limitation is, analogically to Tikhonov regularization, due
to the L 2 norm used in (4.80) which is not able to preserve sharp structures.
4.5 A unidirectional variational destriping model
From a variational perspective, we first tackled the striping issue with ROF total variation
regularization model. We also explored image decomposition models for their ability

95
Figure 4.8 – (Left) Noisy image from Terra MODIS band 27 (Right) Image destriped
with the variational model (3.76) showing no evidence of stripes or distortion (ID=0.989)
Figure 4.9 – (Left) Noisy image from Terra MODIS band 30 (b) Image destriped with the
variational model (3.76). Although stripes have been completely removed, local blurring
artifacts appear along the image discontinuities.
to discriminate texture. The attempt to isolate the stripe noise in the texture component
v fails as the total variation norm in the fidelity term provides oversmoothed results.
Gradient field integration algorithms based on a variational formulation are suited for the
issue of destriping because of the separation between vertical and horizontal information
in the energy functional to be minimized. The destriping variational model derived in the
previous section performs well on homogeneous areas but introduces local blur artifacts

96 4. A Variational approach for the destriping issue
along strong edges. In this section, we overcome this drawback and introduce a new variational
model that satisfies the requirements of optimal destriping.
In chapter 3, we attributed the limitations of standard destriping techniques to the simplistic
assumptions made on the stripe noise. Equalization based methods (moment matching,
histogram matching and OFOV techniques) assume that the stripe noise can be modelled
with a gain and offset between the detectors of the sensor. The presence of strong nonlinearities
on MODIS detectors responses, discredits this assumption. A simple and more
realistic assumption lies in the directional aspect of stripe noise. For instance, if we denote
by n the stripe noise, it is resonable to assume that :
∣ ∀(x, y) ∈ Ω,
∂n(x, y)
∣∣∣ ∣ ∂x ∣ ≪ ∂n(x, y)
∂y ∣ (4.85)
The integration of the previous inequality over the entire image domain Ω results in :
∫
∫
∂n(x, y)
∣ ∂x ∣ ≪ ∂n(x, y)
∣ ∂y ∣ (4.86)
Ω
which translates a realistic characteristic of stripe noise when written as :
Ω
TV x (n) ≪ TV y (n) (4.87)
where TV x and TV y are respectively horizontal and vertical variations of the noise n.
The L 1 norm being an edge-preserving norm can be introduced in a variational model
where the fidelity term takes into account the stripe free horizontal information while
the regularization term only smoothes the result along the vertical axis. We propose a
destriping variational model based on the minimization of the following energy :
E(u) =TV x (u − I s )+λT V y (u) (4.88)
We anticipate the non-differentiability of this functional at points where ∂u
= 0 by introducing a small parameter ɛ in (4.88) which becomes :
∂u
∂y
∂x = ∂Is
∂x
and
∫
E ɛ (u) =
Ω
√ (∂(u ) − Is ) 2 ∫
+ ɛ 2 + λ
∂x
Ω
√ (∂u ) 2
+ ɛ 2 (4.89)
The model (4.89) will be refered to hereafter as the Unidirectional Variational Destriping
Model (UVDM). Its Euler-Lagrange equation is given by :
− ∂
∂x
⎛
⎜
⎝
∂(u−I s)
∂x
√ (
∂(u−Is)
∂x
⎞
⎛
⎟
) 2 ⎠ − λ ∂ ⎜
∂y ⎝
+ ɛ 2
√ (
∂u
∂y
∂u
∂y
∂y
⎞
⎟
) 2 ⎠ =0 (4.90)
+ ɛ 2

97
Using the same notations as in sections 4.2, equation (4.90) can be discretized as :
(
) (
)
D +x (u i,j − f i,j )
D +y u i,j
D −x √ + λD −y √ =0
(D+x (u i,j − f i,j )) 2 + ɛ 2 (D+y u i,j ) 2 + ɛ 2
(
)
(u i+1,j − u i,j − f i+1,j + f i,j )
√ − (u i,j − u i−1,j − f i,j + f i−1,j )
√
(D+x (u i,j − f i,j )) 2 + ɛ 2 (D−x (u i,j − f i,j )) 2 + ɛ 2
+λ
(
)
u i,j+1 − u i,j
√
(D+y u i,j ) 2 + ɛ − u i,j − u i,j−1
√ =0
2 (D−y u i,j ) 2 + ɛ 2
(4.91)
We introduce the following linearization :
⎛
⎞
⎝ (un i+1,j − un+1 i,j
− f i+1,j + f i,j )
√
− (un+1 i,j
− u n i−1,j − f i,j + f i−1,j )
√
⎠
(D +x (u n i,j − f i,j)) 2 + ɛ 2 (D −x (u n i,j − f i,j)) 2 + ɛ 2
⎛
⎞
+λ ⎝
un i,j+1 − un+1 i,j
√
− un+1 i,j
− u n i,j−1
√
⎠ =0
(D +y u n i,j )2 + ɛ 2 (D −y u n i,j )2 + ɛ 2
(4.92)
If we denote :
C 1 =
C 2 =
C 3 =
C 4 =
1
√
(D +x (u n i,j − f i,j)) 2 + ɛ 2
1
√
(D −x (u n i,j − f i,j)) 2 + ɛ 2
(4.93)
1
√
(D +y u n i,j )2 + ɛ 2
1
√
(D −y u n i,j )2 + ɛ 2
The destriped image is obtained with a fixed point iterative scheme :
u n+1
i,j
= C 1(u n i+1,j − f i+1,j + f i,j )+C 2 (u n i−1,j + f i,j − f i−1,j )+λC 3 u n i,j+1 + λC 4u n i,j−1
C 1 + C 2 + λC 3 + λC 4
(4.94)
4.6 Optimal regularization
The limitations of standard destriping techniques discussed at the end of chapter 3,
were used as a starting point to establish the requirements of optimal destriping. One of

98 4. A Variational approach for the destriping issue
Figure 4.10 – (Left) Noisy image from Terra MODIS band 30 (Center) Destriped
with histogram matching (IMAPP) showing residual stripes (Right) Destriped with the
UVDM illustrating complete removal of striping without any bluring or ringing artifacts
Figure 4.11 – (Left) Noisy image from Terra MODIS band 33 (Center) Destriped
with histogram matching (IMAPP) showing residual stripes in spite of noisy detectors
being replaced with neighbors (Right) Image destriped with the UVDM showing efficient
removal of radom striping without bluring or ringing artifacts
these requirements dictates minimum distortion in the destriped image. The ID index in
clearly dependent on the choice of the lagrange multiplier λ. The coefficient λ regulates
the smoothness of the solution and its impact on the restored image can be interpreted as
follows : A large value of λ may result in an oversmoothed solution, visually similar to what
could be achieved with a low-pass filter, or a hard thresholding of wavelet coefficients in
the lower scales. On the other hand, if λ is too small, the result might still contain noise. In
fact, if we considerer the ROF model, its minimizer converges to the original noisy image
f when λ → 0. In the case of the UVDM, when λ → 0, the solution converges to that of
the following optimization problem :
√
∫ (∂(u ) − Is ) 2
E ɛ (u) =
+ ɛ
∂x
2 (4.95)
Ω

99
which is not unique. Indeed, the solution of (4.95) is the set {u ∈ BV (Ω)/u = f +
C, ∫ ∂C
Ω ∂x
=0} which corresponds to the set of images that differs from f on a constant
per line. Due to the variations of stripe noise along the x-direction, any solution of (4.95)
will still display residual stripes. We are then lead to the question of how to determine a
value of λ that removes all the stripe noise while maintainin the image distortion index
close to 1.
4.6.1 Tadmor-Nezzar-Vese (TNV) hierarchical decomposition
In [Tadmor et al., 2004], the authors introduce a multiscale image representation based
on a hierarchical adaptive decomposition. The ROF model is iteratively solved to generate
a sequence of solutions which sum converges to the original image. We propose a
modest adaptation of this strategy to our variational destriping model. Let us consider
the following decomposition for a striped image I s :
[u 0 ,v 0 ]= argmin
(u,v)/u+v=I s
TV x (v)+λ 0 TV y (u)
(4.96)
If the inital value λ 0 is not too small, then u 0 can be considered as a cartoon approximation
of the true stripe-free image I, while the component v 0 mainly contains stripe noise.
Following Tadmor et al.’s remark that a texture at a scale λ contains edges at a refined
scale λ 2 , the noisy component v 0 can also be decomposed using the same variational model :
[u 1 ,v 1 ]= argmin
(u,v)/u+v=v 0
TV x (v)+ λ 0
2 TV y(u) (4.97)
The previous decomposition can be seen as a dyadic refinement to the approximation u 0
and can be iterated as follows :
[u k ,v k ]= argmin
(u,v)/u+v=v k−1
TV x (v)+ λ 0
2 k TV y(u) (4.98)
leading to a simple multilayered representation of the original noisy image I s :
I s = u 0 + v 0
= u 0 + u 1 + v 1
= u 0 + u 1 + u 2 + v 2
(4.99)
= ...
= u 0 + u 1 + u 2 + u 3 + ... + u k + v k
The previous expansion translates the hierarchical decomposition of I s :
j=k
∑
lim u j = I s (4.100)
k→∞
j=0

100 4. A Variational approach for the destriping issue
Figure 4.12 – Optimal destriping using Tadmor et al. hierarchical decomposition approach.
From left to right and top to bottom, noisy image from Terra band 30 and successive
destriped results for iterations 1 to 7. The value λ 0 = 10 have been selected to ensure
initial oversmoothing
This iterated process provides a multiscale representation of I s where the successive terms
u k extract details of the noise-free image I related to the scale λ 0 2 k , while sharper details
associated with stripe and included in the finer scale λ 0 2 k+1 are isolated in the term v k . The
original TNV hierachical decomposition was initially proposed as a multiscale framework
in the context of ROF regularization but does not aim at removing noise from the images.
In the later case, a stopping criteria is required so that the term ∑ j=k
j=0 u j does not contain
stripe details. This is discussed in section 4.6.3. TNV hierarchical decomposition is applied
to a striped image extracted from Terra MODIS band 30 and illustrated in figure 4.12.
The stripe noise extracted at each iteration is displayed in figure 4.13
4.6.2 Osher et al. iterative regularization method
In the orignal ROF model, the term f − u can still contain fine edges and textures if
the lagrange multiplier λ is not choosen carefully. To minimize the removal of structures
from the estimate solution, [Osher et al., 2005] introduced an iterative refinement of the
ROF model and proposed a generalization for other variational models based on the use of
Bregman distances. In the case of ROF regularization, the iterative methodology follows
three steps :
Step 1 : Solve the original ROF model :
{∫ ∫
u 1 =
inf
u∈BV (Ω)
|∇u| + λ
Ω
Ω
(f − u) 2 }
(4.101)

101
Figure 4.13 – From left to right and top to bottom, extracted stripe noise using Tadmor
et al. hierarchical decomposition approach for iterations 1 to 8.
This leads to a decomposition of the image f as f = u 1 + v 1 where v 1 is the estimated
noisy component.
Step 2 : Inject the noise estimated in the first step v 1 in the fidelity term and solve :
{∫ ∫
u 2 =
inf
u∈BV (Ω)
|∇u| + λ
Ω
Ω
(f + v 1 − u) 2 }
(4.102)
This correction step provides a decomposition of the form f + v 1 = u 2 + v 2
Step 3 : Solve :
{∫ ∫
}
u k+1 = inf |∇u| + λ (f + v k − u) 2
u∈BV (Ω) Ω
Ω
(4.103)
where v k is obtained after k iterations as v k = v k−1 + f − u k . This iterated methodology
is extended by Osher et al. to other inverse problems by considering a general variational
model of the form :
inf {J(u)+H(u, f)} (4.104)
u
where J(u) is a convex non negative regularizing functional and H(u, f) the fidelity term.
The general case consists of iteratively solving :
u k+1 = inf
u
{J(u)+H(u, f)− < u, p k >} (4.105)
where < ., . > is the duality product and p k is a subgradient of J at u k . The adaptation to
the UVDM is straightforward. Replacing J(u) with λT V y (u) and H(u, f) with TV x (u−f),

102 4. A Variational approach for the destriping issue
the iterative procedure can be written as :
u k+1 = inf
u
{
TV x (u − f)+λT V y (u) −
∫
Ω
}
u∂J(u k )
(4.106)
In practice, we consider a smoothed version of TV y , and the subdifferential ∂J can be
replaced by the gradient of J(u) with respect to u. Equation (4.106) then becomes :
⎛ ⎛
⎞⎞
∫
u k+1 = inf TV x (u − f)+λT V y (u) − u ⎝λ ∂ ∂u k
⎝
∂y
√ ⎠⎠ (4.107)
u
Ω ∂y
( ∂u k
∂y )2 + ɛ
4.6.3 Stopping criteria
Whether using TVN decomposition or Osher et al. methodology, there exists an iteration
k, where the estimate û k is the closest to the true sripe free image. In both cases,
if k →∞, û k converges to the noisy image I s . In [Osher et al., 2005] the discrepancy
principle is used as a stopping rule. Assuming that the noise level δ is known, the iterative
procedure is stopped as soon as the residual term ‖û k − I s ‖ reaches a value of the same
order as δ. In our case, the stripe noise level is not known and we have to relie on another
approach. The unidirectionality of stripe noise can be exploited, again, to define a reliable
stopping criteria. The variational model (4.84), was designed in order to constrain the
regularization only to the direction of striping. Nevertheless, if the lagrange multiplier λ is
exessively high, the estimated solution will also be smoothed in the horizontal direction.
If we recall that the unidirectionality of striping translates as the horizontal gradients of
the noisy image I s and the true image I being of the same order, λ has to be chosen so
that :
∫
∂(û − f)
∣ ∂x ∣ ≤ ɛ (4.108)
Ω
where ɛ is a tolerance parameter that regulates the amount of distortion introduced in
û. In pratice, the difficulties related to the determination of ɛ can be easily overcome. In
fact, an optimal destriping is expected to preserve the ensemble averaged power spectrum
down the lines of the noisy image because striping only affects one direction. This means
that the spectral distribution of the information averaged accross the swath should be
approximatively the same for the striped image and the estimated true scene. Conveniently,
such measure is provided by the Image distortion index (ID) ; We recall that the ID reflects
a spectral fidelity between the destriped and original signals in the direction orthogonal to
striping. As TVN decomposition and Osher et al. iterative procedures result in a sequence
of solutions {u k } that converges to an image û = I s + C with TV x (C) = 0, we have :
lim
k→∞ ID(u k)=1 (4.109)
A stopping criteria can then be established using a threshold value, ID thres = 0.95.
This threshold was determined heuristically to ensure simultaneously complete removal

103
Image distortion
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
1 2 3 4 5 6 7 8
Iteration
Image distortion
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
1 2 3 4 5 6 7 8 9 10 11 12
Iteration
Figure 4.14 – Image distortion index as a function of number of iterations starting with
λ 0 = 10 and using (Left) Tadmor et al. dyadic hierarchical decomposition (Right) Osher
et al. iterative method
of stripes and minimum distortion. If the threshold value is very close to 1, the estimated
solution might include residual striping due to non linearities or random stripes. In fact,
the case ID = 1 corresponds to λ = 0.
4.6.4 Experimental results and discussion
The UVDM was applied to the entire data set used in this study. Experimental comparison
of TVN and Osher et al. procedure shows that TVN provides a faster convergence
to the solution with an ID close to 1. It also offers more flexibility to reduce the number of
iterations which is dependent on the initial choice of λ 0 . This can be achieved using tryadic
or higher order n updates of the lagrange multiplier λ as λ k+1 = λ 0 /n k . The application of
TVN hierarchical decomposition on the image from Terra MODIS band 30 shows the gradual
extraction of details through the iterations without retrieval of striping (figure 4.12).
Figure 4.13 illustrates how the extracted striping noise is progressively decorrelated from
the original signal and isolated from the true scene structures.
Figures 4.10 and 4.11 show the destriping results obtained with NASA’s IMAPP software
and the UVDM on images from Terra MODIS band 30 and 33. We point out that the
strategy adopted in the IMAPP software to remove non linear effects and random stripes
is to replace corresponding detectors with neighboors. Despite the distortion introduced by
such procedure, residual stripes are still persistent in the resotred image. Complementary
analysis of cross-track profiles (figure 4.15) and ensemble averaged power spectrum down
the columns (figure 4.16), underscores the performances of the UVM. Cross-track profiles
are properly smoothed without erasing sharp fluctuations related to transitions between
ocean and land. Futhermore, examination of column power spectrums obtain with the
UVM show that 1) spectral peaks associated with periodic stripes are completely removed
2) rapid fluctuations in the high frequency range due to random stripes are canceled 3)
the overall shape of the column spectrum is preserved. These qualitative observations are

104 4. A Variational approach for the destriping issue
supported by quantitative measurements reported in tables 4.1 and 4.2.
Although the UVM was derived intuitively by combining edge-preserving models and gradient
field integration approaches, similar results can be achieved with slightly different
variational models. These are based on the minimization of the following energy functionals
:
E 1 (u) =TV x (u − f)+λT V (u) (4.110)
∫ ( ) ∂ (u − f) 2
E 2 (u) =
+ λT V (u) (4.111)
Ω ∂x
∫ ( ) ∂ (u − f) 2
E 3 (u) =
+ λT V y (u) (4.112)
∂x
Ω
In the model (4.112), the L 1 -norm on the x-gradient is replaced by the L 2 -norm. This
modification does not cause bluring artifacts as the edges are still preserved by the regularizing
term which includes the L 1 -norm of the y-gradient. Nevertheless, the original
UVDM is more suited to isolate striping due to the L 1 -norm being weaker than the standard
L 2 -norm (L 2 (Ω) ∈ L 1 (Ω)). For u-components holding the same ID index and derived
from (4.112) and the UVDM, the noisy component v extracted with the UVDM contains
more stripe-related texture. This feature comes in handy in the TVN hierarchical decomposition
where the destriped image is obtained by progressively extracting striping from
oversmoothed u estimates. The L 1 -norm in the fidelity term then ensures a minimal number
of iterations compared to the L 2 -norm.
The model (4.110) replaces the regularizing term of the UVM model with the commonly
used TV-norm which provides similar results than the UVM but increases the computational
cost required for the minimization of its energy functional. This drawback, together
with the use of L 2 -norm in the fidelity term are both present in the variational model
(4.111).
From a denoising perspective, the UVDM and models (4.110), (4.111), (4.112) satisfy the
destriping requirements imposed in chapter 3, namely complete stripe removal without
blur/ringing artifacts. In fact, both TVN and/or Osher et al. iterative procedures ensure
that the ID index of the optimally destriped image is close to 1. This emphazises the
importance of a suitable fidelity term which, for all destriping models described above,
takes into account the unidirectional property of striping.

105
17500
23500
Mean value
16500
15500
Mean value
23000
22500
14500
0 100 200 300 400 500
Line Number
22000
0 100 200 300 400 500
Line Number
17500
23500
Mean value
16500
15500
Mean value
23000
22500
14500
0 100 200 300 400 500
Line Number
22000
0 100 200 300 400 500
Line Number
17500
23500
Mean value
16500
15500
Mean value
23000
22500
14500
0 100 200 300 400 500
Line Number
22000
0 100 200 300 400 500
Line Number
Figure 4.15 – Cross-Track profiles for Terra MODIS band 30 (Left) and band 33 (Right).
From top to bottom : Original image, histogram matching with IMAPP and proposed
UVDM

106 4. A Variational approach for the destriping issue
10
8
Power spectrum
9
8
7
6
5
4
3
Power spectrum
7
6
5
4
3
2
0 0.1 0.2 0.3 0.4 0.5
Normalized frequency
2
0 0.1 0.2 0.3 0.4 0.5
Normalized frequency
10
8
Power spectrum
9
8
7
6
5
4
3
Power spectrum
7
6
5
4
3
2
0 0.1 0.2 0.3 0.4 0.5
Normalized frequency
2
0 0.1 0.2 0.3 0.4 0.5
Normalized frequency
10
8
Power spectrum
9
8
7
6
5
4
3
Power spectrum
7
6
5
4
3
2
0 0.1 0.2 0.3 0.4 0.5
Normalized frequency
2
0 0.1 0.2 0.3 0.4 0.5
Normalized frequency
Figure 4.16 – Column power spectrum for Terra MODIS band 27 (Left) and band 33
(Right). From top to bottom : Original image, histogram matching with IMAPP and
proposed UVDM

Table 4.1 – Noise Reduction (NR), Image distortion (ID), radiometric Improvement Factors IF 1 and
IF 2 for the Terra MODIS band 27, 30 and 33
– Terra MODIS band 27 Terra MODIS band 30 Terra MODIS band 33
index NR ID IF 1 IF 2 NR ID IF 1 IF 2 NR ID IF 1 IF 2
moment matching 45.8895 0.6305 7.3085 11.5732 9.4280 0.9963 6.2656 6.6818 – 0.9995 0.1637 0.1724
histogram (IMAPP) 101.0629 0.9459 8.5463 20.1003 15.3316 0.8816 5.7455 12.8977 – 0.9773 4.8320 11.1566
IFOV 100.5751 0.9692 5.8563 13.7338 5.4195 0.8991 4.6190 5.7684 – 0.9388 -0.1046 0.5466
Frequency filtering 1784 0.9868 9.8866 21.5316 25.5615 0.9884 7.3092 14.8566 – – – –
Wavelet thresholding 6194 0.9039 9.9038 38.4647 54.1577 0.8956 6.34871 18.7944 – 0.8765 4.9008 17.1591
Proposed 2602 0.9881 9.7695 42.3641 15.6980 0.9897 11.8861 22.4904 – 0.9877 8.3956 21.3488
Table 4.2 – Noise Reduction (NR), Image distortion (ID), radiometric Improvement Factors IF 1 and
IF 2 for the Aqua MODIS band 27, 30 and 36
– Aqua MODIS band 27 Aqua MODIS band 30 Aqua MODIS band 36
index NR ID IF 1 IF 2 NR ID IF 1 IF 2 NR ID IF 1 IF 2
moment matching 12.643 0.984 10.4393 12.0686 4.2363 0.9991 6.4180 12.6573 1.90 0.9997 3.1844 2.3252
histogram (IMAPP) 33.8517 0.9828 11.5966 18.1798 4.4847 0.9839 5.5991 12.6135 1.9593 0.9957 3.2537 2.4242
IFOV 1 12.1310 0.9890 9.3335 11.9212 1.8756 0.9646 -3.6762 8.9016 1.1396 0.8627 -1.8263 2.3097
frequency filtering 510.5828 0.9854 10.3039 24.8529 7.8478 0.9874 6.35 17.8977 3.2048 0.9868 0.7912 10.6393
wavelet thresholding 1284 0.9176 9.7060 25.6599 14.8403 0.9198 6.2033 18.1075 5.6922 0.8790 -0.1253 11.6181
Proposed 282.1545 0.9788 12.0818 36.1859 3.8588 0.9985 10.3925 14.4371 2.3648 0.9868 10.4755 15.8115
107

108 4. A Variational approach for the destriping issue
Figure 4.17 – Destriping results with the UVDM on the MODIS data set (TL) Terra
band 27 (TR) UVMD on Terra band 27 (CL) Terra band 30 (CR) UVMD on Terra
band 30 (BL) Terra band 33 (BR) UVMD on Terra band 33

Figure 4.18 – Destriping results with the UVDM on the MODIS data set (TL) Aqua
band 27 (TR) UVMD on Aqua band 27 (CL) Aqua band 30 (CR) UVMD on Aqua
band 30 (BL) Aqua band 36 (BR) UVMD on Aqua band 36
109

110 4. A Variational approach for the destriping issue

111
Chapitre 5
Application : Restoration of Aqua
MODIS Band 6
5.1 Context
In light of the destriping results reported in the chapter 3, it is clear that the quality
of Aqua MODIS data is higher than that collected by Terra MODIS and have been given
preference for many remote sensing applications including ocean colour product generation.
This is mainly due to a better pre-launch calibration and limited non linear effects of
the individual detectors response. Nevertheless, a major issue have been reported on one
of Aqua MODIS bands. 15 of the 20 detectors of Aqua MODIS band 6 (data available at
a resolution of 500 m) are either non functional or noisy. Non functional detectors do not
measure any information and the resulting missing lines, in addition to functional detectors
mis-calibrations, induces a sharp striping pattern accross the entire swath (figure 5.1).
Due to the placement of band 6 (1.6281.652 µm) in the electromagnetic spectrum, many
important applications are affected by this degradation. Two aerosol products, M and
A-aerosols and the corresponding aerosol optical depth are derived over the ocean by the
CERES Science Team using band 6 and band 1 [Tanre et al., 1997], [Ignatov et al., 2005].
Forest biomass estimation and canopy water stress also relie on SWIR range measurements
[Fensholt and Sandholt, 2003]. The relationship between biomass and MODIS band 6 reflectance
was characterized in [Baccini et al., 2004]. More importantly, in the context of
climate change, the missing and noisy scans of Aqua MODIS band 6 is a serious problem
for MODIS snow products. Snow cover information have gained attention of many scientific
studies due to its impact on climate change. The spatial distribution and temporal
evolution of snow provides a valuable information on the amount of snowmelt, used as
input for water management applications and hydrological cycle studies. In addition, the
Earth’s albedo, amount of solar energy reflected back into the atmosphere, is partially regulated
the snow cover. Depending on their fractional cover, snow and ice high reflectivity
acts as a shield against sun radiation thus reducing the Earth surface temperature. Most

112 5. Application : Restoration of Aqua MODIS Band 6
Figure 5.1 – (Left) Image from Aqua MODIS band 6 showing 4 non functional detectors
(Right) Image from Aqua band 6 after removal of missing lines : functional detectors are
contaminated with stripe noise
approaches related to the analysis of snow cover use the Normalized Difference Snow Index
(NDSI). This index exploits snow high reflectivity in the visible (0.5-0.7 µm) and its low
reflectance in the SWIR (1-4 µm). Defined as a spectral band ratio, the NDSI allows a
robust separation of snow from clouds and also reduces the influence of atmospheric effects
and viewing geometry [Salomonson and Appel]. On MODIS, the NDSI can be expressed
as the difference of reflectances ρ measured in the visible band 4 (0.555 µm) and a SWIR
band such as band 6 divided by the sum of the two reflectances :
NDSI 16 = ρ 4 − ρ 6
ρ 4 + ρ 6
(5.1)
The evaluation of the NDSI index on the Aqua sensor is problematic due to the non
functional detectors. Nevertheless, to ensure complementary observations to Terra MODIS
band 6 and provide continuous monitoring and mapping of snow coverage, scientists have
relied on Aqua MODIS band 7 (2.105-2.155 µm). This is a reasonnable alternative, given
that snow has similar reflectance properties over bands 6 and 7, which happen to display a
high correlation over land surfaces. As a result, Aqua-based snow coverage is determined
using a secondary NDSI defined as :
NDSI 17 = ρ 4 − ρ 7
ρ 4 + ρ 7
(5.2)
It was however pointed out in [Hall et al.], that the reflectance of snow is slightly weaker
in band 7 compared to band 6. Consequently, values of NDSI 17 are higher than NDSI 16
and the estimation of snow coverage is compromised. Given the importance of retrieving

113
Figure 5.2 – Location of the Terra MODIS scenes selected over snow-covered regions
for the study of Aqua MODIS band 6 restoration (Left) Alaska, May 5, 2001 (Center)
Labrador, Canada, November 7, 200 (Right) Siberia, Russia, May 24, 2001
accurate quantitative variables associated with snow and ice components, and used in
climate change numerical models, an alternative option to NDSI 17 is mandatory. To this
purpose, restoration techniques for Aqua MODIS band 6 have been recently proposed.
5.2 Existing restoration techniques
5.2.1 Global interpolation
[L. L. Wang and Nianzeng] were the first to investigate the relationship between bands
6 and 7 and demonstrated the feasability of restoring Aqua MODIS band 6 missing data.
The authors first remarqued that the difference of snow reflectance between Terra MODIS
bands 6 and 7 is very close to the same difference on Aqua MODIS. Using observations
consisting of Terra MODIS level 1B calibrated and geolocated radiances at TOA, polynomial
regression was used to quantify the analytical relationship between Terra MODIS
bands 6 and 7. Quantitative analysis conducted by the authors over snow covered areas,
shows that TOA reflectances in Terra MODIS band 6/7, and NDSI 16 /NDSI 17 were highly
correlated with correlation coefficients of 0.9821 and 0.9777 respectively. Linear, quadratic,
cubic, and fourth-degree polynomials were derived from Terra bands 6/7 scatter plots in
order to be applied to Aqua MODIS. Wang et al. suggest restoring Aqua MODIS band 6
using cubic and quadratic polynomes of the corresponding band 7 reflectances :
ρ 6 =1.6032ρ 3 7 − 1.9458ρ 2 7 +1.7948ρ 7 +0.012396
ρ 6 = −0.70472ρ 2 7 +1.5369ρ 7 +0.025409
(5.3)
These analytical relationships between bands 6 and 7 were derived fom Terra measurements
over snow covered regions. It obviously do not account for land surface cover types,
spectral characteristics, scanning geometry. Application of this approach to restore band 6

114 5. Application : Restoration of Aqua MODIS Band 6
missing data over vegetation, clouds, desert or oceanic surfaces require further refinement
to distinguish surface cover types. In addition, it is assumed that the polynomial relation
established on Terra data is transposable to Aqua. This assumption is all the more sensitive
to unknown calibration differences between Terra and Aqua MODIS, and the striping
noise described in the previous chapters. Furthermore, the analytical relation is derived
without prior pre-processing of bands 7 and 6 for stripe noise removal.
5.2.2 Local interpolation
More recently, Rakwatin2009 proposed a restoration procedure for Aqua MODIS band
6 that consists in three step :1) the determination of non functioning detectors ; 2) the
correction of periodic stripes for functioning detectors using histogram matching ; 3) the
estimation of missing pixels via local cubic polynomial regression between band 6 and band
7. Similarly to the approach proposed in [L. L. Wang and Nianzeng], the high correlation
between band 6 and 7 reflectances is quantified using polynomial regression. However, the
fitting is computed locally to account for land cover types. The restoring algorithm can
be summarized with the following :
1) For a dead pixel x in band 6, a initial rectangular window of size 15 × 3 is centered
at x. The minimum and maximum values of the window in band 7, respectively ρ min
7
and ρ max
7 and their location x min , x max are determined.
2) If ρ min
7 ≤ ρ 7 (x) ≤ ρ max
7 , a local cubic polynomial function is calculated from the
values ρ 7 (x min ), ρ 7 (x max ), ρ 6 (x min ) and ρ 6 (x max ) and used to estimated the values of
ρ 6 (x)
3) If ρ 7 (x) < ρ min
7 , ρ 7 (x) > ρ max
7 or if pixels x min or x max in band 6 are also dead,
the size of the analizing window is increased until these criteria are met.
4) Step 1, 2 and 3 are repeated for every dead pixel of band 6
This local cubic interpolation procedure is illustrateed in figure 5.3.
5.3 Proposed approach
We propose here a simple methodology to estimate the value of Aqua MODIS band 6
missing pixels. The approach is based on a concept widely used in the field of hyperspectral
image classification, spectral similarity. Data collected from MODIS can be perceived as a
cube, where the third dimension represents the signal’s wavelenght and can also be used to
extract useful information. In hyperspectral remote sensing, the determination of surface
composition requires the analysis of its reflectance spectrum and comparison with known
field spectra via spectral matching techniques [Kruse et al.]. Although conditionned by the
number of available spectral bands, this reasoning also applies to multispectral imagery.

115
ρ 6 (x max )
MODIS Band 6
ρ 6 (x)
ρ 6 (x min )
ρ 7
min
ρ 7 (x)
MODIS Band 7
ρ 7
max
Figure 5.3 – Restoration procedure for Aqua MODIS band 6 proposed in (Rakwatin et
al., 2009) and based on a local cubic interpolation
For the issue of Aqua MODIS band 6 restoration, missing pixels can be estimated using
spectrally similar pixels from functionning detectors. To this purpose, let us first recall
the definition of few spectral similarity measures commonly used in hyperspectral image
classification.
5.3.1 Spectral similarity
We denote by ρ(x) the reflectance of a given pixel x, which we consider as a vector in
a n-dimensional space as :
ρ(x) =(ρ 1 (x),ρ 2 (x), ..., ρ n (x)) T (5.4)
Each component of the vector ρ(x) in (5.4) corresponds to the reflectance of pixel x in a
given spectral band.
5.3.1.1 Spectral Correlation Measure
The Spectral Correlation Measure (SCM) was defined in [der Meero and Bakker] and
is computed for two pixels x and y as :
n ∑ n
1
SCM(x, y) =
ρ(x)ρ(y) − ∑ n
1 ρ(x) ∑ n
√
1
∑ ρ(y)
[n n
1 ρ(x)2 − ( ∑ n
1 ρ(x))2 ][n ∑ n
1 ρ(y)2 − ( ∑ n
(5.5)
1 ρ(y))2 ]
where n is the number of overlapping spectral bands. The SCM measures the correlation
between the two vectors ρ(x) and ρ(y) and takes into account the mean value and variance
of the overall spectral shape. The values of SCM are contained in the interval [-1,1].

116 5. Application : Restoration of Aqua MODIS Band 6
5.3.1.2 Spectral Angle Measure
The Spectral Angle Measure (SAM) is defined in [Kruse et al.] as the following angle :
( ∑n
1
SAM(x, y) = arcos
ρ(x) ∑ )
n
1
√∑ ρ(y)
n
1 ρ2 (x) ∑ n
(5.6)
1 ρ2 (y)
The SAM is not very sensitive to pixel reflectance values and as such tends to reduce
differences due to atmospheric effects or viewing geometry.
5.3.1.3 Euclidian Distance Measure
The Euclidian Distance Measure (EDM) between pixels x and y is computed as :
∑
EDM(x, y) = √ n (ρ(x) − ρ(y)) 2 (5.7)
Unlike the SCM and SAM, the EDM depends on the reflectance differences between pixels
x and y.
5.3.1.4 Spectral Information Divergence Measure
The Spectral Information Divergence Measure (SIDM) is a stochastic index that measures
the distance between the probability distribution of spectrums associated with pixels
x and y :
SIDM(x, y) =D(x||y)+D(y||x) (5.8)
D(x||y) is the relative entropy of y with respect to x computed as :
1
D(x||y) =
n∑
p i (x)D i (ρ i (x)||ρ i (y)) =
i=1
n∑
p i (x)(I(ρ i (x)) − I(ρ i (y))) (5.9)
i=1
where
p i (x) =
ρ i (x)
∑ n
j=1 ρ j(x) and I(ρ i(x)) = −log p i (x) (5.10)
The measure I(ρ i (x)) is the self-information of x in the spectral band i.
5.3.2 Spectral inpainting
It is clear from the limitations discussed in [L. L. Wang and Nianzeng] that a reliable
restoration of Aqua MODIS band 6 requires accurate distinction of land cover types which
can not be achieved using only the correlation between band 6 and 7. To compute robust
estimate of missing pixels values from band 6, we suggest to rely solely on spectral information
available in other bands where all the detectors are knnown to be functional. Dead

117
pixels can then be restored with a spectral-based non local approach.
Non local or neighborhood filters have been first introduced in the context of image processing
for denoising applications [Yaroslavsky and Eden, 1996]. In the general case, we
consider a noisy image f defined in a bounded domain Ω of R 2 . The denoised value of a
pixel x is obtained as an average of pixels that have a value close to f(x). Such radiometric
neighborhood is used in the Yarolavsky non local filter and the denoised value of a pixel x
is given by :
Y NF h,r (f(x)) = 1 ∫
f(y)e − |f(y)−f(x)|2
h
C(x)
2 (5.11)
B r(x)
where B r (x) is a ball of radius r centered at pixel x, C(x) is a normalization factor
and h is a filtering parameter that controls the decay of the weighting coefficients. More
recently, the high redunduncy observed in natural images was used for texture synthesis
in [Efros and Leung, 1999]. Efros and Leung algorithm exploits non local self similarities
and is particularly suited for inpainting applications. This very principle is used in the
Non Local Means denoising algorithm introduced in [Buades et al., 2005]. With analogy
to the Yaroslavsky filter, the denoised value of a pixel x is obtained as a weighted average
of pixels with a similar local configuration as :
NL(f(x)) = 1
C(x)
∫
Ω
f(y)w(x, y)dy (5.12)
where w(x, y) is a weighting between pixels x and y obtained by comparing intensity
patches of windows centered at x and y as :
( ∫
G a (z)|f(x + z) − f(y + z)| 2 )
w(x, y) =exp −
dz
(5.13)
Ω
where G a is a gaussian function with standard deviation a.
The concept of neighborhood used in nonlocal filters for image processing applications is
a valuable tool for the restoration of Aqua MODIS band 6. Indeed, the value of a dead
pixel can be determined from the set of pixels with similar spectral properties. For a given
Spectral Similarity Measure (SSM), we define the spectral neighborhood of a pixel x as :
h 2
S(x) ={y ∈ Ω|SSM(x, y)

118 5. Application : Restoration of Aqua MODIS Band 6
Reflectance
0.7
0.6
0.5
0.4
0.3
0.2
Land
Ice over land
Sea ice
Coastal waters
Deep ocean
Cloud
Reflectance
0.7
0.6
0.5
0.4
0.3
0.2
Ice 1
Ice 2
Ice 3
Ice 4
Ice 5
0.1
0.1
0
3 4 5 6 7
MODIS Spectral band
0
3 4 5 6 7
MODIS Spectral band
Figure 5.4 – Illustration of spectral similarity (Left) Reflectance spectrum of pixels
selected over areas of different geophysical nature (Right) Reflectance spectrum of pixels
all selected over a homogeneous snow-covered area
band 6. For example, the EDM of two pixels x and y is given by :
∑
EDM(x, y) = √
7 (ρ i (x) − ρ i (y)) 2 (5.16)
i=3,i≠6
Additionally, for a missing pixel x from band 6, we define its spectral neighborhood as :
S(x) ={y ∈ Ω|SSM(x, y)

119
Figure 5.5 – Restoration of Terra Modis Band 6 with synthetic non functional detectors.
From left to right : Original image (Alaska), restored with local interpolation, restored with
gobal interpolation and restored with spectral inpainting. Striping is a clear indication of
poor estimation of missing pixels value
Figure 5.6 – From left to right : Original image (Labrador), restored with local interpolation,
restored with gobal interpolation and restored with spectral inpainting.
Figure 5.7 – From left to right : Original image (Siberia), restored with local interpolation,
restored with gobal interpolation and restored with spectral inpainting.
5.4 Experimental results
5.4.1 Validation of spectral inpainting using NDSI measurements
Three Terra MODIS images have been selected to evaluate the restoration of Aqua
MODIS band 6 obtained with [L. L. Wang and Nianzeng], [Rakwatin et al.] and the pro-

120 5. Application : Restoration of Aqua MODIS Band 6
1
1
0.8
0.8
Simulated NDSI
0.6
0.4
0.2
0
−0.2
Simulated NDSI
0.6
0.4
0.2
0
−0.2
−0.4
−0.4
−0.6
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
Measured NDSI
−0.6
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
Measured NDSI
Simulated NDSI
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
Measured NDSI
Figure 5.8 – Scatter plots of simulated and measured NDSI from Terra MODIS band 6
(Labrador) (Left) Local interpolation (Right) Global interpolation (Bottom) Spectral
inpainting
posed approache refered to hereafter as spectral inpainting. The locations of these scenes
coincide with those used in the work of [Salomonson and Appel] where MODIS data were
used to estimate fractional snow cover. We recall that detectors of Terra MODIS band
6 are all functional and therefore, restoring algorithms described above are applied on
images where measurements from 4 detectors have been set to zero to simulate non functional
detectors. The true and estimated images for Terra band 6 are then combined with
band 4 data in order to compute the NDSI index. The resulting measured and simulated
NDSI indexes are then compared in terms of absolute mean difference, root mean square
error and correlation coefficients to evaluate the efficiency of each approach. Results are
reported in table .
Let us mention few notes on the computational aspect of the spectral inpainting method.
In practice, for a missing pixel x from band 6, a window of limited size (instead of the
entire image domain Ω) is used to determine the spectral similarity between its pixels and
pixel x. Since the proposed approach does not relie on spatial information, standard periodic
or symmetric extensions of boundaries cannot be used. Instead, the search window can

e square or rectangular depending on the position of the pixel with respect to the image
borders. Alternatively, the image can be fragmented into blocks of K × K pixels and the
algorithm is applied separately for each block with a search window composed of the entire
domain of the subimage. The choice of K then depends on the available computational
power. Results illustrated in this study were obtained with K = 200.
Another pratical aspect of the spectral inpainting method lies in the selection of the parameter
N in equation (5.20). The choice N = 1 holds a physical meaning and ensures
radiometric coherence as it corresponds to the case where a missing pixel from band 6 is
given the same value as the most spectrally similar available pixel in the same band. Furthermore,
all spectral bands (3, 4, 5, 6 and 7) have been destriped prior to the restoration
procedure and, therefore all measurements used in the spectral inpainting can be assumed
to be noise free.
The spectral similarity measures defined in the previous section have all been tested except
the Spectral Correlation Measures which requires overlapping spectral bands.
The two existing techniques for the restoration of Aqua MODIS band 6 are entirely based
on the correlation of measurements between bands 6 and 7. The global polynomial regression
proposed in [L. L. Wang and Nianzeng] only applies to snow covered regions and do
not make any distinction in land cover types. Estimation of band 6 missing lines via equations
(5.3) results in strong striping in the restored image even over snow covered areas.
Improved results are obtained when the global polynomial relationship between band 6
and 7 is estimated using data from the current swath.
Another limitation of the global interpolation method proposed by Wang et al., is attached
to the assumption that the relationship between bands 6 and 7 of Terra MODIS can be
directly transposed to Aqua MODIS. In pratice, this assumption do not hold because of
calibration differences between Terra and Aqua MODIS instruments.
The local interpolation procedure described in [Rakwatin et al.] is a refinement of the
global interpolation. To account for land cover types, the estimation of missing pixels is
derived from a polynomial regression between bands 6 and 7, restricted to a local windows.
Results achieved with this technique are often disapointing, as confirmed by the
strong striping visible in the restored band 6. This can be attributed to the incoherent
mixing of spatial and spectral information. Indeed, the value of a dead pixel from band
6 is determined from minimum and maximum values of band 7 in a local window, which
actually correspond to spectrally distant pixels. With such a procedure, the spatial locality
does not garantee any distinction in land cover types. On the contrary, the approach
of [Rakwatin et al.] amounts to a spatio-spectral interpolation where pixels used for the
restoration are spectrally distant and spatially close.
The spectral inpainting method described above enables a robust estimation of missing
lines as it is only based on spectral information. As illustrated in figures 5.5, 5.6 and 5.7
restoration with spectral inpainting provides images visually identical to the original band
6, and unlike global and local interpolation does not display any stripes. Scatter plots (figure
) and quantitative analysis reported in table indicate reliable results obtained
with the spectral inpainting for all three images of Alaska, Labrador and Siberia.
121

122 5. Application : Restoration of Aqua MODIS Band 6
5.4.2 Assessing the impact of stripe noise
The UVDM described in the previous chapter was applied to all bands used by the
spectral inpainting technique. Nevertheless, the striping on Terra MODIS bands 3 to 7
is extremely weak and its impact on the retrieval of NDSI index is not substantial. We
propose here to quantitatively evaluate the necessity of a robust destriping methodology
prior to the generation of high level products. To this purpose, the UVDM was used to
extract a stripe noise from Terra MODIS emissive band 27. This band was specifically
chosen beacause its detectors suffer from strong nonlinear effects that standard destriping
techniques fail to correct. The stripe noise from band 27 is then injected in bands 5, 6 and
7 with different intensity levels so that the Peak Signal-to-Noise ratio (PSNR) of images
from each of these bands is equal to 5, 10, 15, 20, 25, 30 and 35 dB. The spectral inpainting
method is then applied to restore band 6 missing lines with and without preliminary
destriping. The spectral similarity measure used for this experiment is the EDM as it has
shown to provide better results compared to SAM and SDI. In addition to the UVDM,
histogram matching is also used to illustrate the impact of residual stripes on the retrieval
of NDSI indexes.
Simulated and measured NDSI values have been compared for each level of striping using
three scenarios :
- Images from bands 5, 6 and 7 are not destriped prior to spectral inpainting
- Images from bands 5, 6 and 7 are destriped with the histogram matching prior to spectral
inpainting
- Images from bands 5, 6 and 7 are destriped using the UVDM prior to spectral inpainting
Absolute mean difference (AMD), Root Mean Square Errors (RMSE) and Correlation
(CORR) coefficients for the three scenarios are reported in table . Many interesting
observations can be made from these results (also plotted in figure 5.10) to underscore the
benefits of a reliable destriping algorithm.
It can be seen that for a reasonable amount of stripe noise (PSNR≥35 dB), the histogram
matching techniques does not provide any improvement in the estimation of the NDSI
index. In case of very strong striping (PSNR=5dB), the application of the UVDM before
the restoration of band 6 results in NDSI values as accurate as those obtained with the
histogram matching when the stripe noise corresponds to a PSNR=25dB. In addition, the
correlation coefficient obtained with the UVDM, remains above 0.99 for all tested levels of
striping. The extreme case of PSNR=5dB, illustrated in figure 5.9 shows how the UVDM
enables the preservation of the structures contained in the original image.

Figure 5.9 – (TL) Original image from Terra MODIS band 6 (TR) Extreme striping on
band 6 (PSNR=5dB) prior to synthetic non functional detectors. Bands 5 and 7 are contaminated
with a similar striping (BL) Restoration with spectral inpainting after destriping
with histogram matching (BR) Restoration with spectral inpainting after destriping with
UVDM
123

– Alaska Labrador Siberia
Restoration AMD RMSE CORR AMD RMSE CORR AMD RMSE CORR
Wang et al. 0.0087 9.0000 0.9944 0.0031 12.8145 0.9901 4.3822e-004 5.3724 0.9987
Rakwatin et al. 0.0120 12.4084 0.9893 0.0188 16.2538 0.9859 0.0230 11.9939 0.9950
Proposed SAM 5.5582e-004 6.4964 0.9969 1.7451e-004 6.9496 0.9971 0.0011 7.4547 0.9974
SDIM 6.5480e-004 6.5729 0.9968 3.3377e-004 6.5852 0.9974 0.0016 5.4037 0.9986
EDM 1.7882e-004 3.0273 0.9993 6.5103e-005 4.3314 0.9989 0.0021 2.3593 0.9998
Table 5.1 – Absolute mean difference (AMD), Root Mean Square Error (RMSE) and Correlation
(CORR) between simulated and measured NDSI values using different restoration techniques for
Aqua MODIS band 6
– Without destriping Histogram matching UVDM
Stripe noise level AMD RMSE CORR AMD RMSE CORR AMD RMSE CORR
PSNR=5dB 1.1632 7.2247e+004 0.0054 0.0136 36.5744 0.9407 0.0014 10.9336 0.9944
PSNR=10dB 0.3251 4.5093e+003 0.0081 0.0078 24.8579 0.9731 0.0020 7.4031 0.9975
PSNR=15dB 0.0572 81.8125 0.7653 0.0055 17.4133 0.9878 0.0017 5.7920 0.9985
PSNR=20dB 0.0157 39.0451 0.9335 0.0046 12.3446 0.9950 0.0012 4.9865 0.9989
PSNR=25dB 0.0057 22.0486 0.9777 0.0042 9.7847 0.9976 8.1850e-004 3.5721 0.9994
PSNR=30dB 0.0030 13.5149 0.9915 0.0039 8.4876 0.9986 5.0663e-004 3.1400 0.9995
PSNR=35dB 0.0021 7.6345 0.9973 0.0036 7.6683 0.9991 3.8126e-004 2.4931 0.9997
Table 5.2 – Absolute mean difference (AMD), Root Mean Square Error (RMSE) and Correlation
(CORR) between simulated and measured NDSI values using spectral inpainting with EDM and different
levels of stripe noise in bands 5, 6 and 7 prior to restoration
124 5. Application : Restoration of Aqua MODIS Band 6

125
40
1
40
1
35
30
0.99
35
30
0.99
RMSE
25
20
15
RMSE
CORR
0.98
0.97
CORR
RMSE
25
20
15
RMSE
CORR
0.98
0.97
CORR
10
5
0.96
10
5
0.96
0
0.95
5 10 15 20 25 30 35
Level of stripe noise (dB)
0
0.95
5 10 15 20 25 30 35
Level of stripe noise (dB)
Figure 5.10 – RMSE and CORR values between simulated and measured NDSI values
obtained for the Alaska image when different levels of stripe noise are injected in bands
5, 6 and 7. Band 6 is restored using the spectral inpainting method with the EDM. Both
histogram matching (Left) and the UVDM (Right) are used to remove stripe noise prior
to the restoration. In presence of extreme striping (PSNR=5dB) the UVDM allows an
estimation of the NDSI index as accurate as the one provided by the histogram matching
technique for a stripe noise level corresponding to a PSNR=25dB

126 5. Application : Restoration of Aqua MODIS Band 6

127
Conclusion
Contribution
Perspectives and furtur work

128 Conclusion

129
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