sparse image representation via combined transforms - Convex ...

B.2. DISCRETE ALGORITHM 153
B.2.1
Synopsis
We think of the Radon transform for an image as the result of the following five steps:
Outline
image
⇓
f(x, y)
⇓
ˆf(x, y)
⇓
ˆf(ρ, θ)
⇓
ˆf(ρ i ,θ j )
⇓
Radon transform
(1) Interpolate the discrete data to a continuous 2-D function.
(2) Do 2-D continuous time Fourier transform.
(3) Switch from Cartesian to polar coordinates.
(4) Sample at fractional frequencies according
to a grid in the polar coordinate system.
(5) Do 1-D inverse discrete Fourier transform.
In subsubsection B.2.2, we describe the sampling idea associated with steps (1), (2), (4)
and (5). In subsubsection B.2.3, we describe a fast way to sample in Frequency domain.
B.2.2
Interpolation: from Discrete to Continuous Image
We view image {I(i, j) :i =1, 2,... ,N; j =1, 2,... ,N} as a set of sampled values of
a continuous function at grid points {(i, j) :i =1, 2,... ,N; j =1, 2,... ,N}, and the
continuous function is defined as
f(x, y) =
=
∑
i=1,2,... ,N;
j=1,2,... ,N.
⎛
⎜
⎝
∑
i=1,2,... ,N;
j=1,2,... ,N.
I(i, j)ρ(x − i)ρ(y − j)
⎞
⎟
I(i, j)δ(x − i)δ(y − j) ⎠ ⋆ (ρ(x)ρ(y)),