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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
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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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12 1. Introduction A common issue f
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14 1. Introduction
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116 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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