sparse image representation via combined transforms - Convex ...

A.3. DETAILS 139
(a) Stick image
(b) Filtered stick image
(c) Edgelet trans. coeff.(largest−10000/362496)
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(d) Edgelets assoc. with largest−100 coeff.
(e) Largest−300 coeff.
(f) Largest−500 coeff.
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Figure A.2: Edgelet transform of the sticky image: (a) is the original; (b) is the filtered
image; (c) is the sorted coefficients; (d), (e) and (f) are the reconstructions based on the
largest 100, 300 and 500 coefficients, respectively.
explains how to compute a single edgelet coefficient. Section A.3.5 shows how to do the
adjoint transform.
A.3.1
Definition
In this subsection, we define what a direct edgelet transform is.
For continuous functions, the edgelet transform is defined in [50]. But in modern computing
practice, an image is digitalized: it is a discrete function, or a matrix. There are
many possibilities to realize the edgelet transform for a matrix, due to many ways to interpolate
a matrix as a continuous 2-D function.
We explain a way of interpolation. Suppose the image is a squared image having N
rows and N columns. The total number of pixels is N 2 . We assume N is a power of
2, N =2 n . Let the matrix I = I(i, j) 1≤i,j≤N ∈ R N×N correspond to the digital image,