sparse image representation via combined transforms - Convex ...

6.3. DECOMPOSITION 121
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Figure 6.1: Four test images.
[2] They are simple but sufficient to examine whether our basic assumption is true—
that different transforms will automatically represent the corresponding features in a
sparse image decomposition.
6.3 Decomposition
A way to test whether this approach works is to see how it decomposes the image into
parts associated with included transforms. Our approach presumably provides a global
sparse representation. Because of the global sparsity, if we reconstruct part of an image by
using only coefficients associated with a certain transform, then this partial reconstruction
should be a superposition of a few atoms (from the dictionary made by representers of the
transform). So we expect to see features attributable to the associated transforms in these
partial reconstructions.
We consider decomposing an image into two parts—wavelet part and edgelet part. In
principle, we expect to see points and patches in the wavelet part and lines in the edgelet
part. Figure 6.2 shows decompositions of the four testing images. The first row is for Car.
The second, third and fourth row are for Pentagon, Overlap and Separate respectively.
In each row, from the left, the first squared sub-image is the original, the second is the
wavelet part of the image, the third is the edgelet part of the image, and the last is the
superposition of the wavelet part and the edgelet part. With appropriate parameter λ, the