sparse image representation via combined transforms - Convex ...

166 APPENDIX B. FAST EDGELET-LIKE TRANSFORM
image, divided by two, and divided by four). The upper-left image is the original. The
first row gives the columnwise maximum of the absolute values of the coefficients. We see
that there are strong patterns in the columnwise maximums; in particular, it becomes large
along the direction that is the direction of most of the needle-like features in the image.
The second row are the coefficient matrices at different scales.
It takes about 40 seconds to carry out this transform on an SGI Onyx workstation.
(a) Barbara image
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Coeff. Matrix (scale=5)
Coeff. Matrix (scale=4)
Coeff. Matrix (scale=3)
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Coeff. Matrix (scale=2)
Coeff. Matrix (scale=1)
Coeff. Matrix (scale=0)
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Figure B.6: Fast edgelet-like transform of wood grain image.
Another example is the Barbara image. Again, the upper-left image in Figure B.6 is the
original image. The remaining images are the coefficient matrices corresponding to scale 0
through 5. The dark area corresponds to the significant coefficients. We observe that when
the scale is increased, we have more significant coefficients. See the coefficient matrix at
scale equal to 5. This implies that when we divide the image into small squares, the linear
features become more dramatic, hence it becomes easier for a monoscale fast edgelet-like
transform to capture them. Further discussion is beyond the scope of this chapter; we will
leave it as future research.