sparse image representation via combined transforms - Convex ...

56 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES
assume that the basis {ψ(x − k) :k ∈ Z} is orthonormal:
∫
ψ(x)ψ(x − k)dx = δ(k), k ∈ Z. (3.24)
Since ψ(x) isinV 1 , there must exist a real-valued sequence {g(n) :n ∈ Z} such that
ψ(x) = ∑ n∈Z
g(n)φ(2x − n). (3.25)
Since functional subspace V 0 and functional subspace W 0 are orthogonal, we have
∫
ψ(x)φ(x − k) =0, k ∈ Z. (3.26)
From (3.20) and (3.21), we have
∑
h(n)h(n − 2k) =2δ(k), k ∈ Z. (3.27)
n∈Z
From (3.24) and (3.25), we have
∑
g(n)g(n − 2k) =2δ(k), k ∈ Z. (3.28)
n∈Z
From (3.26), (3.21) and (3.25), we have
∑
g(n)h(n − 2k) =0, k ∈ Z. (3.29)
n∈Z
Actually, if relations (3.27), (3.28) and (3.29) hold, then the sequence g can only be the
reverse of the sequence h modulated by the sequence {(−1) n : n ∈ Z}. The following
theorem provides a formal description.
Theorem 3.6 (Quadratic Mirror Filter) If two sequences {h(n) :n ∈ Z} and {g(n) :
n ∈ Z} satisfy the relations listed in (3.27), (3.28) and (3.29), then the sequence {g(n) :
n ∈ Z} is uniquely determined by the sequence {h(n) :n ∈ Z} via
∀n ∈ Z, g(n) =(−1) n h(1 + 2k − n), (3.30)
where k is a pre-determined integer.