By Jean-Luc Starck, Fionn Murtagh, Jalal M. Fadili
This ebook provides the state-of-the-art in sparse and multiscale photograph and sign processing, protecting linear multiscale transforms, equivalent to wavelet, ridgelet, or curvelet transforms, and non-linear multiscale transforms in keeping with the median and mathematical morphology operators. fresh innovations of sparsity and morphological range are defined and exploited for numerous difficulties similar to denoising, inverse challenge regularization, sparse sign decomposition, blind resource separation, and compressed sensing. This ebook weds idea and perform in studying purposes in components reminiscent of astronomy, biology, physics, electronic media, and forensics. a last bankruptcy explores a paradigm shift in sign processing, exhibiting that earlier limits to details sampling and extraction should be conquer in very major methods. Matlab and IDL code accompany those tools and functions to breed the experiments and illustrate the reasoning and technique of the study to be had for obtain on the linked website.
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Additional resources for Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity
Example text
In the reconstruction or synthesis side, the coefficients are up-sampled by inserting a 0 between each sample and then convolved with the dual filters h˜ and g; ˜ the resulting coefficients are summed, and the result is multiplied by 2. The procedure is iterated up to the smallest scale, as depicted in the lower part of Fig. 5. Compared to the CWT, we have far fewer scales because we consider only dyadic scales, that is, scales a j that are a power of 2 of the initial scale a0 (a j = 2 j a0 ). Therefore, for a discrete signal X [n] with N samples, one would typically use J = log2 N scales; the indexing here is such that j = 1 corresponds to the finest scale (high frequencies).
3. a = a + a . ), the discretized CWT of X. If discrete computations assume periodic boundary conditions, the discrete convolution ψ¯ a X can be performed in the Fourier domain: ψ¯ a X = IFFT(FFT(ψ¯ a )FFT(X)), where FFT and IFFT denote the fast Fourier transform and its inverse, respectively. In this case, if the number of voices per octave is set to 12, it is easy to check that the computation of the CWT requires O(12N log2 N)2 operations (Rioul and Duhamel 1992). If the convolution is done in the original (sample) 21 22 The Wavelet Transform domain, we can choose other ways to deal with the boundaries.
The inclusion of the wavelet transform in JPEG 2000, the recent still-picture compression standard, testifies to this lasting and significant impact. 7 (bottom right) shows the decompressed image for a compression ratio of 40, and as can be seen, the result is near-perfect. 6 NONDYADIC RESOLUTION FACTOR Feauveau (1990) introduced quincunx analysis, based on Adelson’s work (Adelson et al. 1987). This analysis √ is not dyadic and allows an image decomposition with a resolution factor equal to 2. The advantage is that only one nonseparable wavelet is needed.