sparse image representation via combined transforms - Convex ...

150 APPENDIX B. FAST EDGELET-LIKE TRANSFORM
published work came to light only in the final stages of editing this thesis. The first Matlab
version of the algorithm being presented was coded by David Donoho. The author made
several modifications to the algorithm, and also some analysis is presented at the end of
this chapter.
The rest of the chapter is organized as following: In Section B.1, we introduce the
Fourier slice Theorem and the continuous Radon transform. Note both are for continuous
functions. Section B.2 describes in detail the main algorithm—an algorithm for fast edgeletlike
transform. The tools necessary to derive the adjoint of this transform are presented
in Section B.3. Some discussion about miscellaneous properties of the fast edgelet-like
transform are in Section B.4, including storage, computational complexity, effective region,
and ill-conditioning. Finally, we present some examples in Section B.5.
B.1 Transforms for 2-D Continuous Functions
B.1.1
Fourier Slice Theorem
The Fourier slice theorem is the key for us to utilize the fast Fourier transform to implement
the fast Radon transform. The basic idea is that for a 2-D continuous function, if we do a
2-D Fourier transform of it, then sampling along a straight line traversing the origin, the
result is the same as the result of projecting the original function onto the same straight
line then taking 1-D Fourier transform.
We will just sketch the idea of a proof. Suppose f(x, y) is a continuous function in 2-D,
where (x, y) ∈ R 2 . We use f to denote the continuous interpolation of the image I, andf
to denote a general 2-D function. Let ˆf(ξ,η) denote its 2-D Fourier transform. Then we
have
∫
ˆf(ξ,η) =
y
∫
x
f(x, y)e −2π√ −1xξ e −2π√ −1yη dxdy.
Taking polar coordinates in both the original domain and the Fourier domain, we have
(B.1)
ξ = ρ cos θ, η = ρ sin θ, x = ρ ′ cos θ ′ , y = ρ ′ sin θ ′ .
Let
(s) ˆf
θ
(ρ) stand for the sampling of the function ˆf along the line {(ρ cos θ, ρ sin θ) :ρ ∈