sparse image representation via combined transforms - Convex ...

A.2. EXAMPLES 137
edgelet transform. [WoodGrain] is a natural image, but it has significant linear features in
it. [Lenna] is a standard testing image in image processing. As for [Sticky], we apply an
edge filter to [Lenna] before doing the edgelet transform. The images in the above list are
roughly ranked by the abundance (of course this could be subjective) of linear features.
Figure A.1, A.2, A.3 and A.4 show the numerical results. From the reconstructions
based on partial coefficients—Figure A.1 (b), (c), (e) and (f), Figure A.2 (d), (e) and (f),
Figure A.3 (b), (c), (e) and (f), Figure A.4 (d), (e) and (f)—we see that the significant
edgelet coefficients capture the linear features of the images. The following table shows the
percentages of the coefficients being used in the reconstructions. Note all the percentages
Recon. 1 Recon. 2 Recon. 3 Recon. 4
[Huo] 0.75 % 1.50 % 2.99 % 5.98 %
[Sticky] 0.03 % 0.08 % 0.14 % NA
[WoodGrain] 0.54 % 1.09 % 2.17 % 4.35 %
[Lenna] 0.23 % 0.47 % 0.93 % NA
Table A.1: Percentages of edgelet coefficients being used in the reconstructions.
in the above table are small, say, less than 6%. The larger the percentage is, the better the
reconstruction captures the linear features in the images.
A.2.1
Edge Filter
Here we design an edge filter. Let A represent the original image. Define 2-D filters D 1 , D 2
and D 3 as
( )
( )
( )
− +
− −
− +
D 1 =
, D 2 =
, D 3 =
.
− +
+ +
+ −
Let “⋆” denote 2-D convolution. The notation [·]·2 means square each element of the matrix
in the brackets. Let “+” be an elementwise addition operator. The edge filtered image of
A is defined as
A E =[A⋆D 1 ]·2 +[A⋆D 2 ]·2 +[A⋆D 3 ]·2 .
As we have explained, for the [Sticky] and [Lenna] image, we first apply an edge filter.