sparse image representation via combined transforms - Convex ...

54 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES
method in denoising, density estimation, and signal recovery. Wavelets and related technologies
have a lot of influence in contemporary fields like signal processing, image processing,
statistic analysis, etc. Some good books that review this literature are [20, 34, 100, 107].
It is almost impossible to cover this abundant area in a short section. In the rest of the
section, we try to summarize some key points. The ones we selected are (1) multiresolution
analysis, (2) filter bank, (3) fast discrete algorithm, and (4) optimality in processing signals
that have point singularities. Each of the following subsections is devoted to one of these
subjects.
3.2.1 Multiresolution Analysis
Multiresolution analysis (MRA) is a powerful tool, from which some orthogonal wavelets
can be derived.
It starts with a special multi-layer structure of square integrable functional space L 2 .
Let V j , j ∈ Z, denote some subspaces of L 2 . Suppose the V j ’s satisfy a special nesting
structure, which is:
...⊂V −3 ⊂V −2 ⊂V −1 ⊂V 0 ⊂V 1 ⊂V 2 ⊂ ...⊂ L 2 .
At the same time, the following two conditions are satisfied: (1) the intersection of all
⋂
the V j ’s is a null set:
j∈Z V j = ∅; (2) the union of all the V j ’s is the entire space L 2 :
⋃
j∈Z V j = L 2 .
We can further assume that the subspace V 0 is spanned by functions φ(x − k),k ∈ Z,
where φ(x − k) ∈ L 2 . By definition, any function in subspace V 0 is a linear combination of
functions φ(x − k),k ∈ Z. The function φ is called a scaling function. The set of functions
{φ(x−k) :k ∈ Z} is called a basis of space V 0 . To simplify, we only consider the orthonormal
basis, which, by definition, gives
∫
{
1, k =0,
φ(x)φ(x − k)dx = δ(k) =
0, k ≠0,
k ∈ Z. (3.20)
A very essential assumption in MRA is the 2-scale relationship, which is: for ∀j ∈ Z, if
function f(x) ∈V j , then f(2x) ∈V j+1 . Consequently, we can show that {2 j/2 φ(2 j x − k) :
k ∈ Z} is an orthonormal basis of the functional space V j . Since φ(x) ∈V 0 ⊂V 1 , there