sparse image representation via combined transforms - Convex ...

130 CHAPTER 7. FUTURE WORK
These images possess a variety of features. The results of a sparse decomposition can be
used to determine which class of features is dominant in an image, and then to tell which
transform is well-suited to processing the image.
We are going to experiment on different dictionaries. We have experimented on a dictionary
that is a combination of 2-D wavelets and edgelet-like features. Some other possible
combinations are:
•{2-D DCT + 2-D wavelets + edgelets}. This dictionary contains elements for homogeneous
image components, point singularities and linear singularities. It should
provides sparser decomposition, but will increase the computational cost.
•{2-D DCT + 2-D wavelets + curvelets}. It is similar to the idea for the previous
dictionary, but with a better-desiged curvelets set in place of the edgelets set, it may
lead to an improvement in the computational efficiency.
•{2-D DCT + curvelets}. Comparing with the previous result, we may be able to
tell how important the wavelet components are in the image, by knowing how many
wavelets we need in sparsely representing the desired image. We can explore the same
idea for 2-D DCT and curvelets, respectively.
There are some open questions. We hope to gain more insights (and hopefully answer
the questions) via more computational experiments and theoretical analysis. Some of these
open questions are:
• We want to explore the limit of our approach in finding a sparse atomic decomposition.
For example, if the desired image can be made by a few atoms from a dictionary, will
our approach find the sparsest atomic decomposition in the dictionary?
• For most of natural images (e.g., those images we have seen in our ordinary life), can
they be represented by a few components from a dictionary made by 2-D DCT, 2-D
wavelets, edgelets and curvelets? If not, what are other transforms we should bring
in (or develop)?
• We think our sparse image representation approach can be used as a tool to preprocess
a class of images. Based on the sparse representation results, we can decide which
image transform should be chosen for the specific class of images. We are going to
explore this idea. This gives a way of doing method selection and it has applications
in image search, biological image analysis, etc.