sparse image representation via combined transforms - Convex ...

3.2. WAVELETS AND POINT SINGULARITIES 55
must exist a real sequence {h(n) :n ∈ Z}, such that
φ(x) = ∑ n∈Z
h(n)φ(2x − n). (3.21)
Equation (3.21) is called a two-scale relationship. Based on different interpretations, or
different points of view, it is also called dilation equation, multiresolution analysis equation
or refinement equation. Let function Φ(ω) and function H(ω) be the Fourier transforms of
function φ(x) and sequence {h(n) :n ∈ Z}, respectively, so that
Φ(ω) =
∫ ∞
−∞
φ(x)e −ixω dx;
H(ω) = ∑ n∈Z
h(n)e −inω .
The two-scale relationship (3.21) can also be expressed in the Fourier domain:
Φ(ω) =H( ω 2 )Φ(ω 2 ).
Consequently, one can derive the result
Φ(ω) =Φ(0)
∞∏
j=1
H( ω ). (3.22)
2j The last equation is important for deriving compact support wavelets.
Since the functional subspace V j is a subset of the functional subspace V j+1 , or equivalently
V j ⊂V j+1 , we can find the orthogonal complement of the subspace V j in space V j+1 .
We denote the orthogonal complement by W j , so that
V j ⊕W j = V j+1 . (3.23)
Suppose {ψ(x − k) :k ∈ Z} is a basis for the subspace W 0 . The function ψ is called a
wavelet. Note that the function φ is in the functional space V 0 and the function ψ is in
its orthogonal complement. From (3.23), we say that a wavelet is designed to capture the
fluctuations; and, correspondingly, a scaling function captures the trend. We can further