sparse image representation via combined transforms - Convex ...

Appendix A
Direct Edgelet Transform
This chapter is about the implementation of edgelet transform described in [50]. We give
a review of edgelets in Section A.1, then some examples in Section A.2, and finally some
details in Section A.3.
A.1 Introduction
The edgelet [50] transform was developed to represent needle-like features in images. Edgelets
are 2-D objects taking various scales, locations and orientations. If we consider an image
as a function on a unit square [0, 1] × [0, 1], an edgelet system is constructed as follows:
[E1] Partition the unit square into dyadic sub-squares, so that the sides of these squares
take values at 1/2, 1/4, 1/8,....
[E2] On each dyadic subsquare, put equally-spaced vertices on the boundary, starting from
corners. We require each side equally partitioned by these vertices, and we generally
assume that there are dyadic and integral number of vertices on each side, so the
distance between two neighbor vertices should be a dyadic value too.
[E3] For a line segment that connects two vertices as in [E2], if it does not coincide with a
boundary, then it is called an edgelet.
[E4] The edgelet system is a collection of all the edgelets as in [E3].
An N × N digital image can be considered as a sample of a continuous 2-D function in
a unit square. The value of each pixel is equal to the sample value at points (i/N, j/N),
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