sparse image representation via combined transforms - Convex ...

B.2. DISCRETE ALGORITHM 155
extension. In the following, we treat ˆf(ξ,η) as a box function (e.g., the indicator function
of the unit square).
Without loss of generality, we redefine F (ξ,η) as
F (ξ,η) =
∑
i=0,1,... ,N−1;
j=0,1,... ,N−1.
I(i +1,j+1)e −2π√ −1ξi e −2π√ −1ηj .
Note the range of the index of I is changed. This is to follow the convention in DFT. By
applying the FFT, we can get the following matrix by an O(N 2 log 2 N) algorithm:
because for 0 ≤ k, l ≤ N − 1,
F ( k N , l N )=
∑
i=0,1,... ,N−1
j=0,1,... ,N−1
This is the 2-D discrete Fourier transform.
(
F (
k
N , l N )) k=0,1,... ,N−1 ,
l=0,1,... ,N−1
I(i +1,j+1)e −2π√ −1 k N i e −2π√ −1 l N j .
For Radon transform, instead of sampling at Cartesian grid point (k/N, l/N), we need
to sample at polar grid points. We develop an X-interpolation approach, which is taking
samples at grid points in polar coordinate. So after X-interpolation, we get the following
N × 2N matrix:
where
[ (F
(
k
N − 1 2 ,s(k)+l · ∆(k) − 1 2 )) k=0,1,... ,N−1
l=0,1,... ,N−1
,...
(
F (s(k)+l · ∆(k) −
1
2 , N−k
N
− 1 2 )) k=0,1,... ,N−1
l=0,1,... ,N−1
]
, (B.4)
s(k) = k N ,
∆(k) = 1 2k
N
(1 −
N ).
(B.5)
Figure B.1 illustrates the idea of X-interpolation.
There is a fast way to compute the matrix in (B.4), using an idea from [6].