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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