sparse image representation via combined transforms - Convex ...

122 CHAPTER 6. SIMULATIONS
fourth sub-image should be close to the original.
We have some interesting observations:
1. Overall, we observe the wavelet parts possess features resembling points (fine scale
wavelets) and patches (coarse scale scaling functions), while the edgelet parts possess
features resembling lines. This is most obvious in Car and least obvious in Pentagon.
The reason could be that Pentagon does not contain many linear features. (Boundaries
are not lines.)
2. We observe some artifacts in the decompositions. For example, in the wavelet part of
Car, we see a lot of fine scale features that are not similar to points, but are similar to
line segments. This implies that they are made by many small wavelets. The reason
for this is the intrinsic disadvantage of the edgelet-like transform that we have used.
As in Figure B.3, the representers of our edgelet-like transform have a fixed width.
This prevents it from efficiently representing narrow features. The same phenomena
emerge in Overlap and Separate. A way to overcome it is to develop a transform
whose representers have not only various locations, lengths and orientations, but also
various widths. This will be an interesting topic of future research.
6.4 Decay of Coefficients
As we discussed in Chapter 2, one can measure sparsity of representation by studying the
decay of the coefficient amplitudes. The faster the decay of coefficient amplitudes, the
sparser the representation.
We compare our approach with two other approaches. One is an approach using merely
the 2-D DCT; the other is an approach using merely the discrete 2-D wavelet transform.
The reasons we choose to compare with these transforms:
1. The 2-D DCT, especially the one localized to 8 × 8 blocks, has been applied in an
important industry standard for image compression and transmission known as JPEG.
2. The 2-D wavelet transform is a modern alternative to 2-D DCT. In some developing
industry standards (e.g., draft standards for JPEG-2000), the 2-D wavelet transform
has been adopted as an option. It has been proven to be more efficient than 2-D DCT
in many cases.