sparse image representation via combined transforms - Convex ...

154 APPENDIX B. FAST EDGELET-LIKE TRANSFORM
where ρ is the interpolating kernel function. We assume ρ is equal to one at the origin and
zero at all the other integers:
ρ(0) = 1, and ρ(i) =0, for i =1, 2,... ,N,
notation ⋆ stands for the convolution, and function δ(·) is the Dirac function at point 0.
The 2-D Fourier transform of f(x, y) is
⎛
⎜ ˆf(ξ,η) = ⎝
∑
i=1,2,... ,N;
j=1,2,... ,N.
⎞
I(i, j)e −√−1·2πξi e −√ −1·2πηj⎟
⎠ ˆρ(ξ)ˆρ(η),
(B.3)
where ˆρ(·) is the Fourier transform of function ρ(·). Note there are two parts in ˆf(ξ,η), the
first part denoted by F (ξ,η),
F (ξ,η) =
∑
i=1,2,... ,N;
j=1,2,... ,N.
I(i, j)e −√ −1·2πξi e −√ −1·2πηj ,
is actually the 2-D Fourier transform of I. Note F (ξ,η) is a periodic function with period
one for both ξ and η: F (ξ +1,η)=F (ξ,η) andF (ξ,η +1)=F (ξ,η). If we sample ξ and η
at points 1 N , 2 N
,... ,1, then we have the discrete Fourier transform (DFT). We know there
is an O(N log N) algorithm to implement.
Function ˆρ(·) typically has finite support. In this paper, we choose the support to have
length equal to one, so that the support of ˆρ(ξ)ˆρ(η) forms a unit square. We did not choose
a support wider than one for some reason we will mention later.
From equation (B.3), ˆf(ξ,η) is the periodic function F (ξ,η) truncated by ˆρ(ξ)ˆρ(η). The
shape of ˆρ(ξ) andˆρ(η) determines the property of function ˆf(ξ,η).
Section B.6.1 gives three examples of interpolation functions and some related discussion.
In this paper, we will choose the raised cosine function as our windowing function.
B.2.3
X-interpolation: from Cartesian to Polar Coordinate
In this section, we describe how to transfer from Cartesian coordinates to polar coordinates.
As in the synopsis, we do the coordinate switch in the Fourier domain.
From equation (B.3), function ˆf(ξ,η) is just a multiplication of function F (ξ,η) witha
windowing function. If we know the function F (ξ,η), the function ˆf(ξ,η) is almost a direct