sparse image representation via combined transforms - Convex ...

152 APPENDIX B. FAST EDGELET-LIKE TRANSFORM
From the Fourier slice theorem and the above equality, we have
∫
ˆf (s)
θ
(ρ) = R(f,ρ ′ ,θ)e −2πiρρ′ dρ ′ .
ρ ′
Taking the inverse Fourier transform, we have
∫
(s)
R(f,ρ,θ) = ˆf
θ
(ρ ′ )e 2πiρρ′ dρ ′ .
ρ ′
Thus in dimension 2, the Radon transform of f is the inverse 1-D Fourier transform of
a specially sampled (along a straight line going through the origin) 2-D Fourier transform
of the original function f. Since there are fast algorithms for the Fourier transform, we can
have fast algorithms for the Radon transform.
B.2 Discrete Algorithm
To some extent, the edgelet transform can be viewed as the Radon transform restricted to
a small square. There is a fast way to do the Radon transform. For an N by N image,
we can find an O(N 2 log N) algorithm by using the fast Fourier transform. The key idea
in developing a fast approximate edgelet transform is to do a discrete version of the Radon
transform. A discrete version of the Radon transform is the algorithm we presented in this
section.
The Radon transform was originally defined for continuous functions. The Fourier slice
theorem is based on continuous functions also. Our algorithm has to be based on discrete
data, say, a matrix. A natural way to transfer a matrix to a continuous function is to
view the matrix as a sampling from a continuous function. We can calculate the analytical
Radon transform of that continuous function, then sample it to get the corresponding
discrete result.
The first question arising is how to interpolate the data. The second question is, because
the Radon transform uses polar coordinates, and a digital image is generally sampled on a
grid in the Cartesian coordinate, how do we switch the coordinates and still preserve the
fast algorithms.
Section B.2.1 gives an outline of the algorithm. Section B.2.2 describes some issues in
interpolation. Section B.2.3 is about a fast algorithm to switch from Cartesian coordinates
to polar coordinates. Section B.2.4 presents the algorithm.