sparse image representation via combined transforms - Convex ...

148 APPENDIX A. DIRECT EDGELET TRANSFORM
y is equal to one if and only if it is at the position corresponding to e, and zero elsewhere.
Note δ e is a generalized version of the Dirac function. Equation (A.1) becomes
∀e, 〈Tx, δ e 〉 = 〈x, T ⋆ δ e 〉. (A.2)
The left-hand side of the above equation can be written as
〈Tx, δ e 〉 = T(x, e) = ∑ α
ω(α, e)x(α).
Substituting the above into (A.2), we have
T ⋆ δ e (α) =ω(α, e).
(A.3)
This gives the formula for the adjoint edgelet transform.
Actually, in linear algebra, this is obvious: if the forward transform corresponds to the
transform matrix, then the adjoint transform corresponds to the transpose of the transform
matrix.
A.3.6
Discussion
The following observations make a fast algorithm for the edgelet transform possible:
• We can utilize inter-scale relationships. Some edgelets at coarse scales are just a linear
combination of other edgelets at a finer scale. So its coefficient is a linear combination
of other edgelet coefficients. An analogue of it is the 2-scale relationship in orthogonal
wavelets.
• We may use the special pattern of the edgelet transform. This idea is similar to the
idea in [18].
• The edgelet transform is actually the Radon transform restricted in a subsquare. It
is possible to do approximate fast Radon transform via fast Fourier transforms.
We avoid further discussion on this topic.