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136 APPENDIX A. DIRECT EDGELET TRANSFORM
i, j =1, 2,... ,N. It’s natural to assume that a vertex mentioned in [E2] must be located
at a pixel. The cardinality of an edgelet system has O(N 2 log 2 N). More details are given
in Section A.3.2.
An edgel is a line segment connecting a pair of pixels in an image. Note if we take all the
possible edgels in an N × N image, we have O(N 4 ) of them. Moreover, for any edgel, it’s
proven in [50] that it takes at most O(log 2 N) edgelets to approximate it within a distance
1/N + δ, where δ is a constant.
The coefficients of the edgelet transform are simply the integration of the 2-D function
along these edgelets.
There is a fast algorithm to compute an approximate edgelet transform [50]. For an
N × N image, the complexity of the fast algorithm is O(N 2 log 2 N). The fast edgelet
transform will be the topic of the next chapter.
This transform has been implemented in C and it is callable via a Matlab MEX function.
It can serve as a benchmark for testing other transforms, which are designed to capture
linear features in images.
A.2 Examples
Before we present details, let’s first look at some examples. The key idea of developing this
transform is hoping that if the original image is made by a few needle-like components,
then this transform will give a small number of significant coefficients, and the rest of the
coefficients will be relatively small. Moreover, if we apply the adjoint transform to the coefficients
selected by keeping only these with significant amplitudes, then the reconstructed
image should be close to the original.
To test the above idea, we select four images:
[Huo] a Chinese character, 64 × 64;
[Sticky] a sticky figure, 128 × 128;
[WoodGrain] an image of wood grain, 512 × 512;
[Lenna] the lenna image, 512 × 512.
[Huo] was selected because it is made by a few lines, so it is an ideal testing image.
[Sticky] has a patch in the head. We apply an edge filter to this image before we apply the