Rigorous Sensor Modeling and Triangulation for OrbView-3 - asprs

RIGOROUS SENSOR MODELING AND TRIANGULATION FOR ORBVIEW-3
ABSTRACT
Kurt J. de Venecia, Product Manager
BAE Systems
5299 DTC Boulevard , Suite 1120, Greenwood Village, CO 80111, USA
kurt.devenecia@baesystems.com
Fidel Paderes Jr., Senior Photogrammetrist
BAE Systems
10920 Technology Place, San Diego, CA 92127, USA
fidel.paderes@baesystems.com
A. Stewart Walker, Marketing Director
BAE Systems
10920 Technology Place, San Diego, CA 92127, USA
stewart.walker2@baesystems.com
BAE Systems is a leader in geospatial image processing using a wide range of image sources. Photogrammetric
processing for end-user products such as orthophotos, digital terrain models, and feature data requires rigorous
sensor modeling or highly accurate approximations to the rigorous sensor model in the form of RPC (Rational
Polynomial Coefficients) or RSM (Replacement Sensor Model). Both rigorous and approximate sensor models must
allow triangulation. In many cases triangulation is necessary to improve accuracy for derived products, but in some
instances, the input georeferencing metadata associated with the sensor may meet accuracy requirements. A sensor
model, as defined in BAE Systems’ software products SOCET SET ® and SOCET GXP ® , consists of three
components: support data input/output; ground-to-image function (or, alternatively, image-to-ground function);
triangulation. Triangulation with a sensor model requires: identification of the sensor model parameters that are
adjustable; determination of default accuracies for each adjustable parameter; error propagation model. This paper
outlines the approach for implementing the rigorous OrbView-3 sensor model, including reading support data,
implementing the image-to-ground function, and triangulation, illustrated with an OrbView-3 data set.
INTRODUCTION: REVIEW OF SENSOR MODELING
The mathematical model for a sensor describes the relationship between an image and the object space (ground
coordinates). The sensor model is the basis for the software on a photogrammetry workstation. The core component
of the model is the function to convert ground-to-image coordinates, and vice versa. Both transformations are
necessary, even though one of the transformations may simply be an iterative process of the other. The
transformation is actually a series of transformations with two chief components - the interior orientation (image to
sensor) and exterior orientation (sensor to ground). Other secondary affects are also considered in the transformation
process, including atmospheric refraction and sensor aberration.
The sensor model may take the form of rigorous functions, including collinearity equations. There are also
approximate sensor models, including Replacement Sensor Model (RSM) and Rational Polynomial Coefficients
(RPC). Software products such as BAE Systems’ SOCET SET, Leica Geosystems’ LPS, and Intergraph’s
ImageStation have sensor modeling as a core component (Wang et al., 2004; Olander and Walker, 1999).
Adjustment of the rigorous or approximate sensor model is also an important core component in a
photogrammetry workstation. The input metadata for a given satellite sensor may not be accurate enough for certain
mapping applications. Therefore, triangulation of the mathematical model is necessary. The popularity of
commercial high resolution satellite sensors has yielded many publications on the subject of satellite triangulation,
such as the contributions by Biard and Madani (2005) on DigitalGlobe QuickBird images in ImageStation and De
Venecia et al. (2005) on automated triangulation of various satellite sensors.
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Replacing the rigorous sensor model with an RPC approximation is popular mainly for its simplicity. The
January 2003 issue of Photogrammetric Engineering & Remote Sensing contained four papers discussing RPC and
RPC triangulation. An RPC model, especially when embedded in standard imagery formats such as NITF, allows
software developers to implement the RPC model once for any sensor using the standardized format. Unfortunately
RPC modeling does not lend itself to long image segments where high frequency corrections cannot be accounted
for in the standalone RPC model. The Replacement Sensor Model (McGlone, 2004) is a promising extension to the
RPC model. But, until the RSM is fully accepted by the end-user, data provider, and software developer
communities, rigorous models will continue to be used for complex modeling of airborne and satellite sensors.
ORBVIEW-3 SENSOR MODELING AND ASSOCIATED SUPPORT DATA
The OrbView-3 satellite carries a linear array pushbroom sensor, providing one meter panchromatic and four
meter multispectral imagery. GeoEye sells imagery and metadata at varying levels of processing. There are currently
eight OrbView-3 products, from BASIC imagery to Orthorectified products (ORTHO). This paper focuses on
OrbView-3 BASIC Enhanced Panchromatic Imagery, which includes enhanced telemetry metadata.
Cooperation between the data provider and the photogrammetry software supplier is essential to understanding
and implementing the rigorous sensor model for a specific sensor. In the case of OrbView-3 imagery, agreements
were signed between BAE Systems and ORBIMAGE (now GeoEye), then technical material and sample imagery
were provided by ORBIMAGE. OrbView-3 BASIC Enhanced Imagery including enhanced telemetry data was sent
to BAE Systems along with ground control and documentation describing the imagery and metadata. Clearly,
cooperation in both business and technical aspects is mutually beneficial for the data provider (GeoEye) and data
user (BAE Systems) – GeoEye wishes to sell imagery products and BAE Systems wishes to sell image exploitation
software to mapping and intelligence agencies and commercial companies.
Support data includes information such as sensor location, velocity, orientation angles, focal length, time of
acquisition, and camera calibration data. Support data are often supplied as auxiliary files to the raw imagery. In the
past imagery and metadata were typically reformatted in the photogrammetry workstation into a common format for
convenient use by the image processing software solutions such as SOCET SET, but today the image providers are
typically delivering their imagery in common formats such as NITF and TIFF. Therefore, the imagery is often used
directly for photogrammetric processing in the software. The associated image metadata is normally stored in a
binary or ASCII format. Documentation provided by ORBIMAGE details the metadata definition and format. The
SOCET SET and SOCET GXP software products use the OrbView-3 imagery and metadata directly without the
need to reformat either input data source. Since triangulation is typically required by end-users of the
photogrammetry software, a solution providing updated sensor parameters must be provided. In the case of SOCET
SET and SOCET GXP, the updated (triangulated) sensor parameters are stored in a separate ASCII support file.
The sensor model converts ground space locations to image space locations, or vice versa. The ground space
coordinates are in the form of Earth-Centered-Earth-Fixed (ECEF) coordinates in meters. The image coordinates are
sub-pixel line and sample coordinates. The transformations are used in nearly every photogrammetric process, from
simply moving the cursor on an image to creating products such as orthophotos. The OrbView-3 sensor model
documentation provided by ORBIMAGE outlines the image-to-ground transformation. Therefore, the direct imageto-ground
transformation was implemented in SOCET SET and SOCET GXP with the ground-to-image
implemented as an iterative computation of the image-to-ground. The OrbView-3 documentation outlines the
projective image-to-ground transformation as
where
G 1 A k
O
= A + M a
(1)
G
A is the unknown object space coordinate triplet in the ECEF WGS84 coordinate system in meters
O
A is the instantaneous sensor position triplet in the ECEF WGS84 coordinate system in meters
interpolated from the ephemeris metadata
k is a scalar
M is the 3x3 image-to-ground rotation matrix based on instantaneous attitude information in the form of
quaternion values from the attitude metadata
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a is the imaging vector consisting of image space coordinates and focal length transformed to meters
using interior orientation coefficients provided in the OrbView-3 metadata.
Implementation in software of a satellite sensor model including triangulation is typically an 8-10 week task,
encompassing design, code, and test prior to release in a specified software version. Technical cooperation and
communication between BAE Systems and ORBIMAGE resulted in a smooth development cycle. Complete details
regarding the transformation (equation 1) were provided by ORBIMAGE and it is provided here to allow discussion
of the triangulation for OrbView-3 implemented in SOCET SET. In addition to the direct projective transformation
model in equation 1 are secondary affects such as atmospheric correction and vehicle aberration.
ORBVIEW-3 SENSOR MODEL TRIANGULATION
The triangulation function provides a mechanism for updating the interior and exterior orientation elements for
the OrbView-3 sensor’s metadata. The multi-sensor triangulation function in SOCET SET requires:
Identification of which parameters of the sensor model are adjustable
Determination of whether the adjustment provides corrections or absolute values
Determination of the names for the adjustable parameters for a logical user interface
Determination of the default accuracies for each adjustable parameter
Determination of the perturbation values for the parameters, to compute partial derivatives numerically.
Based on the OrbView-3 documentation and engineering experience with satellite pushbroom sensors, 23
parameters were chosen for adjustment. All adjustable parameters of the OrbView-3 sensor model within SOCET
SET are corrections to the actual values provided in the image metadata. The 23 OrbView-3 sensor model
parameters consist of five corrections for interior orientation and 18 corrections for the six elements of exterior
orientation. The five interior orientation elements are non-time dependent, while the six exterior orientation
elements are quadratic functions of acquisition time. The model and triangulation assume the time duration of
acquisition is relatively short. If larger segments are to be triangulated, then additional parameters may need to be
included. The additional parameters used in other satellite pushbroom sensor models in SOCET SET include an
along-track correction to the attitude and position in the form of a finite element correction profile. Details of the
five triangulated parameters for interior orientation are
where
⎡ x
⎢
= ⎢ y
⎢
⎣ f
⎤ ⎡ x
⎥ ⎢
⎥ = ⎢y
⎥ ⎢
⎦ ⎣
δ
+ x ⎤ 1 S
δ ⎥
+ y S⎥
δ
f ⎥
⎦
δ δ
0
δ
a
δ
δ
δ
0 1
δ
a is the correction to the imaging vector in meters
δ δ
δ
x , y , and f are the corrections to the image x, y coordinates and focal length, respectively
δ δ
x0
and y0
are the triangulated offset corrections to the image x, y coordinates
δ δ
x1
and y1
are the triangulated linear corrections to the image x, y coordinates
S is the input sample coordinate for the image-to-ground transformation.
The nine triangulated parameters for exterior orientation attitude corrections are
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(2)

where
where
δ δ
ω , ϕ , and κ
δ
δ δ δ
δ 2
⎡ω
⎤ ⎡ω
⎤ ⎡ ⎤ ⎡ ⎤
0 ω1
TC
ω2TC
⎢ δ ⎥ ⎢ δ ⎥ ⎢ δ ⎥ ⎢ δ 2 ⎥
⎢ϕ
⎥ = ⎢ϕ0
⎥ + ⎢ϕ1
TC
⎥ + ⎢ϕ2
TC
⎥
⎢ δ ⎥ ⎢ δ ⎥ ⎢ δ ⎥ ⎢ δ 2 ⎥
⎣κ
⎦ ⎣κ0
⎦ ⎣κ1
TC
⎦ ⎣κ
2TC
⎦
are the corrections to the image-to-ground Euler orientation angles, omega, phi, and
kappa, respectively, whose initial values are assumed zeros
δ δ δ
ω0 , ϕ0
, and κ0
are the triangulated offset correction values for omega, phi, and kappa, respectively
δ δ δ
ω1 , ϕ1
, and κ1
are the triangulated linear (velocity) correction values for omega, phi, and kappa,
respectively
δ δ δ
ω2 , ϕ2
, and κ2
are the triangulated non-linear (acceleration) correction values for omega, phi, and
kappa, respectively
TC is the time at the given input line coordinate with origin at image center.
The exterior orientation positional corrections are
A
δ
⎡X
⎢
= ⎢Y
⎢
⎣ Z
δ
δ
δ
δ δ
δ 2
⎤ ⎡X
⎤ ⎡ ⎤ ⎡ ⎤
0 X1
Tc
X 2Tc
⎥ ⎢ δ ⎥ ⎢ δ ⎥ ⎢ δ 2 ⎥
⎥ = ⎢Y0
⎥ + ⎢Y1
Tc
⎥ + ⎢Y2
Tc
⎥
⎥ ⎢ δ ⎥ ⎢ δ ⎥ ⎢ δ 2 ⎥
⎦ ⎣ Z0
⎦ ⎣ Z1
Tc
⎦ ⎣ Z2
Tc
⎦
δ
A is the correction for the sensor position vector in ECEF WGS84 meters
δ δ
δ
X , Y , and Z
are corrections to the sensor position in ECEF WGS84 meters, respectively
δ δ
δ
X 0 , Y0
, and Z0
are the triangulated offset correction values for the sensor position
δ δ
δ
X1
, Y1
, and Z1
are the triangulated linear (velocity) correction values for the sensor position
δ δ
δ
X 2 , Y2
, and Z2
are the triangulated non-linear (acceleration) correction values for the sensor position
T C is the time at the given input line coordinate with origin at image center.
Equation 1 can be updated based on the triangulation components outlined in equations 2-4. The new image-toground
function is
G O δ 1
A = A + A + k
M
δ ( a + a )
δ
where M is the 3x3 correction rotation matrix defined by image-to-ground Euler angle corrections
δ δ δ
ω , ϕ , and κ from equation 3. The M rotation matrix is formed from the quaternion values interpolated based
on time (T ) from the attitude metadata. All other parameters are defined in equations 1-4 above.
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δ
M
(3)
(4)
(5)

ORBVIEW-3 MULTI-SENSOR TRIANGULATION EXAMPLE
Tests of the OrbView-3 sensor model and triangulation were performed with the imagery, metadata and ground
control points provided by ORBIMAGE during the development phase of the sensor model. Two additional
OrbView-3 BASIC Enhanced Imagery scenes with enhanced telemetry were subsequently sent to BAE Systems for
testing the SOCET SET Multi-Sensor Triangulation (MST) with overlapping imagery from two additional satellite
sensors – IKONOS RPC and QuickBird Basic. The dataset used for testing consists of six images from three
different sensors taken over Denver, Colorado USA. An overview of the imagery is provided in Table 1.
Table 1. Overview of the six images used for triangulation. The six images were acquired from three different
satellites and are modeled by three different sensor models in SOCET SET and SOCET GXP.
Imagery
IKONOS (2)
Pan 11-bit/band NITF
Stereo Epipolar RPC Sensor Model
Thumbnail Footprints
Ground Sample Dist: 1.0 m
Size (lines x samples): 12252x9843
12252x9843
The two images overlap by 100% and are
same pass stereo.
QuickBird (2)
Basic Panchromatic 11-bit TIFF
Physical Pushbroom Sensor Model
Ground Sample Distance: 0.7 m
Size (lines x samples): 28592x27552
29888x27552
OrbView-3 (2)
Basic Panchromatic 11-bit/pixel TIFF
Physical Pushbroom Sensor Model
Ground Sample Distance: 0.98 m
Size (lines x samples): 34406x8016
30879x8016
Some cloud cover on the bottom of one
of the images.
SOCET SET MST requires proper default settings for the accuracy of the parameters of the input imagery.
These are determined at the time the sensor model is implemented. The accuracies and adjusted parameters for each
image and/or sensor can be changed by the operator within the MST application, but for an automated process the
accuracies and parameters should be well defined. With proper accuracies for the selected parameters, the
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triangulation can be performed on multiple images and sensors using tie points only in the fully weighted least
squares bundle adjustment.
Automatic Point Measurement (APM) provides a mechanism to measure tie points automatically between
overlapping imagery regardless of the image format or sensor type. APM was performed on the six input images
using a regular grid tie point pattern. The results for the tie point distribution and number are displayed in Figure 1.
Figure 1. Tie points after SOCET SET APM and MST bundle adjustment with blunder detection. The zero ray tie
points were eliminated as part of blunder detection. The maximum number of overlaps from the input imagery is
five. The three five ray points can be seen in the center of the block, which is also the location of the high resolution
orthophoto used to check the absolute accuracy of the block. The check point orthophoto footprint is shown in black.
An important element in the MST fully weighted least squares bundle adjustment is automatic blunder
detection. Enabling this feature detected and deleted 42 points from the six image block: these points are depicted in
red in Figure 1. The resultant RMS of the tie points was just under one pixel. It was determined after analysis of the
graphical display of residuals with associated imagery that the long OrbView-3 image strips were being constrained
too tightly based on the a priori accuracy estimates for the parameters. Therefore, the accuracy values for the
exterior orientation elements for the offset and linear terms in equations 3 and 4 were increased (loosened) to allow
more adjustment for these parameters. The resultant RMS for pixel space residuals decreased to 0.86 pixels.
Check points were used to determine the absolute accuracy of the six image multi-sensor triangulation. The
check points were derived from a high resolution orthomosaic that was created from digital frame images and
LIDAR courtesy of Merrick and Company. The high resolution orthomosaic has a ground sample distance of 15
centimeters. The check points were manually measured in the satellite imagery. Each check point fell within the
five-image overlap region as shown in Figure 1. Table 2 outlines the horizontal check point differences.
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Rays Color Tie
Points
0 42
2 102
3 27
4 14
5 3

Table 2. Horizontal check points derived from a 15 cm orthomosaic, which was created from digital frame
imagery and an associated LIDAR surface. Each check point fell within the five-image overlap region (see
Figure 1). MST computed the difference between the check point original coordinates and the multi-ray
intersected check point coordinates.
Check Point
Original
Coordinates
Check Point
Computed
Coordinates
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Check Point
Differences
(meters)
Point Longitude/
Number Latitude
1 Longitude -104:59:19.917 -104:59:19.983 1.572
Latitude +39:44:23.581 +39:44:23.498 2.552
2 Longitude -104:59:34.602 -104:59:34.568 -0.795
Latitude +39:44:24.226 +39:44:24.110 3.576
3 Longitude -104:59:55.875 -104:59:55.867 -0.198
Latitude +39:44:29.071 +39:44:28.993 2.404
4 Longitude -104:59:39.400 -104:59:39.400 -0.010
Latitude +39:44:41.672 +39:44:41.661 0.347
5 Longitude -104:59:18.934 -104:59:18.884 -1.183
Latitude +39:44:52.649 +39:44:52.634 0.481
6 Longitude -104:59:10.123 -104:59:10.135 0.286
Latitude +39:44:59.406 +39:44:59.339 2.070
7 Longitude -104:58:56.380 -104:58:56.377 -0.073
Latitude +39:44:51.083 +39:44:51.048 1.069
8 Longitude -104:58:56.492 -104:58:56.429 -1.516
Latitude +39:44:35.699 +39:44:35.685 0.449
9 Longitude -104:58:52.050 -104:58:52.023 -0.637
Latitude +39:44:23.994 +39:44:23.912 2.524
10 Longitude -104:59:05.486 -104:59:05.518 0.774
Latitude +39:44:35.659 +39:44:35.647 0.379
11 Longitude -104:59:26.834 -104:59:26.857 0.535
Latitude +39:44:32.052 +39:44:31.993 1.837
12 Longitude -104:59:16.928 -104:59:16.955 0.637
Latitude
RMS
+39:44:44.524 +39:44:44.489 1.091
Total Longitude
RMS
0.848
Latitude 1.874
CONCLUSION
Development of a rigorous sensor model for a photogrammetric software solution requires cooperation between
data provider and software supplier. Understanding the physical parameters of the sensor is important in determining
how parameters will be adjusted in the triangulation process.
An RPC model, especially when embedded in standard imagery formats such as NITF, allows developers to
implement the RPC model once. Only testing is required when data providers launch and deliver subsequent image
products in these standard formats. Unfortunately RPC modeling does not lend itself to long image segments, where
high frequency corrections cannot be accounted for in the RPC model. Until a replacement sensor model is
embraced by data providers, photogrammetry software suppliers, and end-users, the rigorous sensor model will
continue to be the core component of a photogrammetry software solution.

ACKNOWLEDGEMENTS
The authors would like to thank the data providers: ORBIMAGE and Space Imaging (now GeoEye);
DigitalGlobe; and Merrick and Company.
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SOCET SET® and SOCET GXP® are registered trademarks of BAE Systems National Security Solutions Inc.
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