sparse image representation via combined transforms - Convex ...

Appendix B
Fast Edgelet-like Transform
In this chapter, we present a fast algorithm to approximate the edgelet transform in discrete
cases. Note the result after the current transform is not exactly the result after a direct
edgelet transform as we presented in the previous chapter. They are close, in the sense that
we can still consider the coefficients after this transform are approximate integrations along
some line segments, but the line segments are not exactly the edgelets we described in the
previous chapter.
It is clear that in order to have a fast algorithm, it is necessary to modify the original
definition of edgelets. In many cases, there is a trade off between the simplicity or efficiency
of the algorithm and the loyalty to the original definition. The same is true here. In this
chapter, we show that we can change the system of the edgelets a little, so that a fast
(O(N 2 log N)) algorithm is feasible, and the transform still captures the linear features in
an image.
The current algorithm is based on three key foundations:
1. Fourier slice theorem,
2. fast Fourier transform (FFT),
3. fast X-interpolation based on fractional Fourier transform.
In the continuous case, the idea presented here has been extensively developed in
[48, 50, 52, 51, 49]. This approach is related to unpublished work on fast approximate
Radon transforms by Averbuch and Coifman, and to published work in the field of Synthetic
Aperture Radar (SAR) tomography and medical tomography. These connections to
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