sparse image representation via combined transforms - Convex ...
sparse image representation via combined transforms - Convex ... sparse image representation via combined transforms - Convex ...
18 CHAPTER 2. SPARSITY IN IMAGE CODING l p space if and only if θ has a finite weak l p norm. For fixed p, the statement “Θ ⊂ wl p ” simply means that every sequence in Θ has a finite weak l p norm. The critical index of a functional space Θ, denoted by p ∗ (Θ), is defined as the infimum of p such that the weak l p space includes Θ: p ∗ (Θ) = inf{p :Θ⊂ wl p }. In Section 2.2.2, we describe the linkage between critical index and optimal exponent, where the latter is a measure of efficiency of “optimal” coding in a certain functional space. A Result in the Finite Case Before we move into the asymptotic discussion, the following result provides insight into coding a finite-length sequence. The key idea is that the faster the finite sequence decays, the fewer bits required to code the sequence. Here the sparsity is measured by the decay of the sorted amplitudes, which in the 1-D case is the decay of the sorted absolute values of a finite-length sequence. In general, it is hard to define a “universal” measure of speed of decay for a finite-length sequence. We consider a special case. The idea is depicted in Figure 2.5. Suppose that two curves of sorted amplitudes of two sequences (indicated by A and B in the figure) have only one intersection. We consider the curve that is lower at the tail part (indicated by B in the figure) corresponds to a sparser sequence. (Obviously, this is a simplified version because there could be more than one intersection.) We prove that by using the most straightforward coding scheme, which is to take an identical uniform scalar quantizer at every coordinate, we need fewer bits to code B than to code A. To be more specific, without loss of generality, suppose we have two normalized nonincreasing sequences. Both of these sequences have L 2 norm equal to 1. One decays “faster” than the other, in the sense that when after a certain index, the former sorted-amplitude sequence is always below the latter one, as in Figure 2.5. (Curve A is always above curve B after the intersection.) Suppose we deploy a uniform scalar quantizer with the same quantization parameter q on every coordinate. Then the number of bits required to code the ith element θ i in the sequence θ is no more than log 2 [θ i /q]+1(where[x] is the closest integral value to x). Hence the total number of bits to code vector θ = {θ i :1≤ i ≤ N} is upper bounded by ∑ N i=1 log 2[θ i /q] +N. Since q and N are constants, we only need to
2.2. SPARSITY AND COMPRESSION 19 log(|c| ) (i) A B Figure 2.5: Intuition to compare the speeds of decay for two sequences. A sequence whose sorted-amplitudes-curve B is, as we consider, sparser than the sequence whose sortedamplitudes-curve is A. i consider ∑ N i=1 log |θ i|. The following theorem shows that a “sparser” sequence needs fewer bits to code. First suppose {x i :1≤ i ≤ N} and {y i :1≤ i ≤ N} are two non-increasing positive real-valued sequences with a fixed l 2 norm: x 1 ≥ x 2 ≥ ...≥ x N > 0, y 1 ≥ y 2 ≥ ...≥ y N > 0, ∑ N i=1 x2 i = C; ∑ N i=1 y2 i = C; where C is a constant. When ∃k, ∀j ≥ k, x j ≥ y j ,and∀j < k,x j
- Page 1 and 2: SPARSE IMAGE REPRESENTATION VIA COM
- Page 3: I certify that I have read this dis
- Page 7 and 8: To find a sparse image representati
- Page 9: Abstract We consider sparse image d
- Page 12 and 13: xii
- Page 14 and 15: 2.3 Discussion.....................
- Page 16 and 17: 6 Simulations 119 6.1 Dictionary...
- Page 18 and 19: xviii
- Page 21 and 22: List of Figures 2.1 Shannon’s sch
- Page 23 and 24: A.3 Edgelet transform of the wood g
- Page 25 and 26: Nomenclature Special sets N .......
- Page 27 and 28: List of Abbreviations BCR .........
- Page 29 and 30: Chapter 1 Introduction 1.1 Overview
- Page 31 and 32: Chapter 2 Sparsity in Image Coding
- Page 33 and 34: 2.1. IMAGE CODING 5 INFORMATION SOU
- Page 35 and 36: 2.1. IMAGE CODING 7 2.1.2 Source an
- Page 37 and 38: 2.1. IMAGE CODING 9 x ✲ T y ERROR
- Page 39 and 40: 2.1. IMAGE CODING 11 where Q stands
- Page 41 and 42: 2.2. SPARSITY AND COMPRESSION 13 Pr
- Page 43 and 44: 2.2. SPARSITY AND COMPRESSION 15 av
- Page 45: 2.2. SPARSITY AND COMPRESSION 17 wi
- Page 49 and 50: 2.3. DISCUSSION 21 tail compact. Th
- Page 51 and 52: 2.4. PROOF 23 The index l does not
- Page 53 and 54: Chapter 3 Image Transforms and Imag
- Page 55 and 56: 27 Some of the figures show the bas
- Page 57 and 58: 3.1. DCT AND HOMOGENEOUS COMPONENTS
- Page 59 and 60: 3.1. DCT AND HOMOGENEOUS COMPONENTS
- Page 61 and 62: 3.1. DCT AND HOMOGENEOUS COMPONENTS
- Page 63 and 64: 3.1. DCT AND HOMOGENEOUS COMPONENTS
- Page 65 and 66: 3.1. DCT AND HOMOGENEOUS COMPONENTS
- Page 67 and 68: 3.1. DCT AND HOMOGENEOUS COMPONENTS
- Page 69 and 70: 3.1. DCT AND HOMOGENEOUS COMPONENTS
- Page 71 and 72: 3.1. DCT AND HOMOGENEOUS COMPONENTS
- Page 73 and 74: 3.1. DCT AND HOMOGENEOUS COMPONENTS
- Page 75 and 76: 3.1. DCT AND HOMOGENEOUS COMPONENTS
- Page 77 and 78: 3.1. DCT AND HOMOGENEOUS COMPONENTS
- Page 79 and 80: 3.1. DCT AND HOMOGENEOUS COMPONENTS
- Page 81 and 82: 3.2. WAVELETS AND POINT SINGULARITI
- Page 83 and 84: 3.2. WAVELETS AND POINT SINGULARITI
- Page 85 and 86: 3.2. WAVELETS AND POINT SINGULARITI
- Page 87 and 88: 3.2. WAVELETS AND POINT SINGULARITI
- Page 89 and 90: 3.2. WAVELETS AND POINT SINGULARITI
- Page 91 and 92: 3.2. WAVELETS AND POINT SINGULARITI
- Page 93 and 94: 3.3. EDGELETS AND LINEAR SINGULARIT
- Page 95 and 96: 3.4. OTHER TRANSFORMS 67 uncertaint
2.2. SPARSITY AND COMPRESSION 19<br />
log(|c| )<br />
(i)<br />
A<br />
B<br />
Figure 2.5: Intuition to compare the speeds of decay for two sequences. A sequence whose<br />
sorted-amplitudes-curve B is, as we consider, <strong>sparse</strong>r than the sequence whose sortedamplitudes-curve<br />
is A.<br />
i<br />
consider ∑ N<br />
i=1 log |θ i|. The following theorem shows that a “<strong>sparse</strong>r” sequence needs fewer<br />
bits to code.<br />
First suppose {x i :1≤ i ≤ N} and {y i :1≤ i ≤ N} are two non-increasing positive<br />
real-valued sequences with a fixed l 2 norm:<br />
x 1 ≥ x 2 ≥ ...≥ x N > 0,<br />
y 1 ≥ y 2 ≥ ...≥ y N > 0,<br />
∑ N<br />
i=1 x2 i = C;<br />
∑ N<br />
i=1 y2 i = C;<br />
where C is a constant. When ∃k, ∀j ≥ k, x j ≥ y j ,and∀j < k,x j