sparse image representation via combined transforms - Convex ...

140 APPENDIX A. DIRECT EDGELET TRANSFORM
(a) Wood Grain Image
(b) Largest−10000 coeff. (log)
(c) Largest−20000 coeff.
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(d) Edgelet trans. coeff.(best−1500000/1839104)
(e) Largest−40000 coeff.
(f) Largest−80000 coeff.
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5 10 15
x 10 5
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Figure A.3: Edgelet transform of the wood grain image: (a) is the original; (d) is the sorted
coefficients; (b), (c), (e) and (f) are reconstructions based on the largest 1 × 10 4 ,2× 10 4 ,
4 × 10 4 ,8× 10 4 coefficients, respectively.
so when the image is a gray-scale image, I(i, j) is the gray scale at point (i, j). A 2-D
function f is defined in the following way: for i, j =1, 2,... ,N,ifx and y fall in the
cell ((i − 1)/N, i/N ] × ((j − 1)/N, j/N ], then f(x, y) =I(i, j). Suppose e is an edgelet as
described in Section A.1. The edgelet transform of I corresponding to e is the integration
of the function f divided by the length of e:
E(I,e)=
∫
e f
length(e) .
Note this integration is a weighted sum of I(i, j) where the edgelet e intersects the square
((i − 1)/N, i/N ] × ((j − 1)/N, j/N ]. The weight of I(i, j) depends on the fraction of e in
the cell ((i − 1)/N, i/N ] × ((j − 1)/N, j/N ].