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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
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École Doctorale d’Informatique,
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5 Résumé Nou abordons dans cette
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8 TABLE DES MATIÈRES 3.5 Frequency
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10 TABLE DES MATIÈRES
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112 5. Application : Restoration of
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- Alaska Labrador Siberia Restorati
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128 Conclusion
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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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