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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
Page 1: École Doctorale d’Informatique,
Page 5: 5 Résumé Nou abordons dans cette
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- Alaska Labrador Siberia Restorati
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130 BIBLIOGRAPHIE A. Chambolle. An
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132 BIBLIOGRAPHIE E. Hochberg, S. A
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134 BIBLIOGRAPHIE P. Perona and J.
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