sparse image representation via combined transforms - Convex ...

164 APPENDIX B. FAST EDGELET-LIKE TRANSFORM
(a) Image
(b) Image
(c) Image
(d) Image
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(e) MSET (2−5)
(f) MSET (2−5)
(g) MSET (2−5)
(h) MSET (2−5)
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Figure B.4: Multiscale fast edgelet-like transform of artificial needle-like images.
B.5.2
Edgelet-like Transform for Some Artificial Images
Since this algorithm is designed to capture the linear features in an image, it will be interesting
to see how it works on some artificial images made by a single linear singularity.
The first row of Figure B.4 shows some needle-like images. The second row is their
multiscale fast edgelet-like transforms. To explain the images of the coefficients, we need
to explain how we arrange the output of our algorithm. Suppose we only do the transform
for the whole image (scale-0 transform). Then as in B.2.4, the input I is mapped to a
coefficient matrix with two components [E1 1,E2 1 ]. When we divide the image into 2 × 2
block images (scale-1 transform), each subimage is mapped to a coefficient sub-matrix with
two components:
[ ] [ ]
I11 I 12 E
1
→ 11 E11 2 E12 1 E12
2 .
I 21 I 22 E21 1 E21 2 E22 1 E22
2