sparse image representation via combined transforms - Convex ...

3.2. WAVELETS AND POINT SINGULARITIES 59
derived from the two-scale relationships in (3.21) and (3.25):
∫
α j k
= f(x)φ(2 j x − k)dx
(3.21)
= ∑ ∫
h(n) f(x)φ(2 j+1 x − 2k − n)dx
n∈Z
= ∑ n∈Z
h(n)α j+1
n+2k ; (3.31)
∫
β j k
= f(x)ψ(2 j x − k)dx
(3.25)
= ∑ ∫
g(n) f(x)φ(2 j+1 x − 2k − n)dx
n∈Z
= ∑ n∈Z
g(n)α j+1
n+2k . (3.32)
These two relations, (3.31) and (3.32), determine the two-scale relationship in the discrete
algorithm.
Suppose that function f resides in a subspace V j : f ∈V j , j ∈ N. By definition, there
must exist a sequence of coefficients α j = {α j k
,k ∈ Z}, such that
f = ∑ n∈Z
α j k φ(2j x − k).
The subspace V j can be divided as
V j = V 0 ⊕W 0 ⊕ ...⊕W j−1 .
We want to know the decompositions of function f in subspaces V 0 , W 0 , W 1 , ... ,and
W j−1 .Letβ l = {βk l ,k ∈ Z} denote the set of coefficients that correspond to the wavelets in
the subspace W l ,1≤ l