sparse image representation via combined transforms - Convex ...
sparse image representation via combined transforms - Convex ... sparse image representation via combined transforms - Convex ...
60 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES Similarly, from (3.32), for any j ∈ N, the sequence β j−1 is a downsampled version of a convolution of two sequences α j and g: α j−1 =(h ∗ α j ) ↓ 2. (3.34) The algorithm for the DWT arises from a combination of equations (3.33) and (3.34). We start with α j in subspace V j , j ∈ N. From (3.33) and (3.34) we get sequences α j−1 and β j−1 corresponding to decompositions in subspaces V j−1 and W j−1 , respectively. Then we further map the sequence α j−1 into sequences α j−2 and β j−2 corresponding to decompositions in subspaces V j−2 and W j−2 . We continue this process until it stops at scale 0. The concatenation of sets α 0 , β 0 , β 1 , ... , β j−1 gives the decomposition into a union of all the subspaces V 0 , W 0 , W 1 , ... , W j−1 . The sequence α 0 corresponds to a projection of function f into subspace V 0 . For l =0, 1,... ,j− 1, the sequence β l corresponds to a projection of function f into subspace W l . The set α 0 ∪ β 0 ∪ β 1 ∪ ...∪ β j−1 is called the DWT of the sequence α j . The first graph in Figure 3.4 gives a scaled illustration of the DWT algorithm for sequence α 3 . Note that we assume a sequence has finite length. The length of each block is proportional to the length of its corresponding coefficient sequence, which is a sequence of α’s and β’s. We can see that for a finite sequence, the length of the DWT is equal to the length of the original sequence. The second graph in Figure 3.4 is a non-scaled depiction for a more generic scenario. 3.2.4 Point Singularities We will explain why the wavelet transform is good at processing point singularities. The key reason is that a single wavelet basis function is a time-localized function. In other words, the support of the basis function is finite. Sometimes we say that wavelet basis function has compact support. The wavelet basis is a system of functions made by dilations and translations of a single wavelet function, together with some scaling functions at the coarsest scale. In the discrete wavelet transform, for a signal concentrated at one point (or, equivalently, for one point singularity) at every scale because of the time localization property, there are only a few significant wavelet coefficients. For a length-N signal, the number of scales is O(log N), so for a time singularity, a DWT should give no more than O(log N) wavelet coefficients that have significant amplitudes. Compared with the length
3.2. WAVELETS AND POINT SINGULARITIES 61 α 3 β 0 α 2 β 2 α 1 β 1 ✠ ❅ L H ❅ ❅❘ ✠ α 0 ❅ L H❅ ❅❘ SIGNAL ❅ ✠ L H ❅❅❘ α 1 α 0 β 0 α j α j−1 β j−1 α j−2 β j−2 β j−3 Figure 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 version for general cases.
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3.2. WAVELETS AND POINT SINGULARITIES 61<br />
α 3<br />
β 0<br />
α 2<br />
β 2<br />
α 1 β 1<br />
✠<br />
❅<br />
L H ❅ ❅❘<br />
<br />
✠<br />
α 0<br />
❅<br />
L H❅ ❅❘<br />
SIGNAL<br />
<br />
❅<br />
✠ L<br />
H ❅❅❘<br />
<br />
α 1 <br />
<br />
<br />
α 0 β 0<br />
α<br />
<br />
j <br />
α j−1<br />
β j−1<br />
<br />
α j−2 β j−2<br />
<br />
β j−3<br />
Figure 3.4: Illustration of the discrete algorithm for forward orthonormal wavelet transform<br />
on a finite-length discrete signal. The upper graph is for cases having 3 layers. The width<br />
of each block is proportional to the length of the corresponding subsequence in the discrete<br />
signal. The bottom one is a symbolic version for general cases.
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