sparse image representation via combined transforms - Convex ...

3.4 Illustration of the discrete algorithm for forward orthonormal wavelet transform
on a finite-length discrete signal. The upper graph is for cases having
3 layers. The width of each block is proportional to the length of the corresponding
subsequence in the discrete signal. The bottom one is a symbolic
versionforgeneralcases. ............................ 61
3.5 Multiresolution analysis of a point singularity with Haar wavelets. . . . . . 62
3.6 Two-dimensional wavelet basis functions. These are 32 × 32 images. The
upper left one is a tensor product of two scaling functions. The bottom right
2 by 2 images and the (2, 2)th image are tensor products of two wavelets.
The remaining images are tensor products of a scaling function with a wavelet. 63
3.7 Idealized tiling on the time-frequency plane for (a) sampling in time domain
(Shannon), (b) Fourier transform, (c) Gabor analysis, (d) orthogonal wavelet
transform, (e) chirplet, (f) orthonormal fan basis, (g) cosine packets, and (h)
wavelet packets. . . . . .............................. 67
4.1 Quasi-sparsityanddistortioncurve. ...................... 87
4.2 Function ¯ρ (as ρ in figures) and its first and second derivatives. ¯ρ, ¯ρ ′ and ¯ρ ′′
are solid curves. The dashed curve in figure (a) is the absolute value. The
dashed curve in figure (b) is the signum function. . .............. 89
4.3 Function ¯ρ at a neighborhood of origin with different values of γ, γ =
10, 20, 40, 80, 160. The dashed line is the absolute value. . .......... 90
6.1 Four test images. . . . .............................. 121
6.2 Decompositions. ................................. 123
6.3 Decaying amplitudes of coefficients. . ...................... 124
6.4 Aglobalalgorithmvs.MatchingPursuit. ................... 126
A.1 Edgelet transform of the Chinese character “Huo”: (a) is the original; (d) is
the sorted coefficients; (b), (c), (e) and (f) are reconstructions based on the
largest 200, 400, 800, 1600 coefficients, respectively. .............. 138
A.2 Edgelet transform of the sticky image: (a) is the original; (b) is the filtered
image; (c) is the sorted coefficients; (d), (e) and (f) are the reconstructions
based on the largest 100, 300 and 500 coefficients, respectively. . . . . . . . 139
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