sparse image representation via combined transforms - Convex ...

158 APPENDIX B. FAST EDGELET-LIKE TRANSFORM
B.2.4
Algorithm
Here we give the algorithm to compute the fast edgelet-like transform. Note when the
original image is N by N, our algorithm generates an N × 2N matrix. Recall that I(i, j)
denotes the image value at pixel (i, j), 1 ≤ i, j ≤ N.
Computing the discrete Radon transform via Fourier transform
A. Compute the first half of the DRT matrix (size is N by N):
1. For j =0to N − 1 step 1,
take DFT of (j + 1)th column:
[g(0,j),g(1,j),... ,g(N − 1,j)]
= DFT N ([I(1,j+1),I(2,j+1),... ,I(N,j + 1)]);
End;
2. For k =0to N − 1 step 1,
(a) pointwise multiply row
{
k +1,[g(k, 0),g(k, 1),... ,g(k, N − 1)],
(
)
}
with complex sequence e −2π√ −1 j·s(k)+∆(k) j2 2
: j =0, 1,... ,N − 1 ;
(b) convolve it with complex sequence
{
}
e π√ −1∆(k)(t) 2 : t =0, 1,... ,N − 1 ;
(c) pointwise { multiply the sequence with complex } sequence
e −2π√ −1∆(k) l2 2 : l =0, 1,... ,N − 1 ;
we get another row vector: { F ( k N ,s(k)+l · ∆(k)) : l =0, 1,... ,N − 1} ;
End;
3. For l =0to N − 1 step 1,
taper the (l + 1)th column by the raised cosine, then take inverse 1-D
DFT;
End;
B. Compute the second half of the DRT matrix (size is N by N):
1. For i =0to N − 1 step 1,
For j =0to N − 1 step 1,
modulate: g(i, j) =I(i +1,j+1)e −2π√ −1 j N ;