sparse image representation via combined transforms - Convex ...

Chapter 6
Simulations
Section 6.1 describes the dictionary that we use. Section 6.2 describes our testing images.
Section 6.3 discusses the decompositions based on our approach and its implication. Section
6.4 discusses the decay of amplitudes of coefficients and how it reflects the sparsity
in representation. Section 6.5 reports a comparison with Matching Pursuit. Section 6.6
summarizes the computing time. Section 6.7 describes the forthcoming software package
that is used for this project. Finally, Section 6.8 talks about some related efforts.
6.1 Dictionary
The dictionary we choose is a combination of an orthonormal 2-D wavelet basis and a set
of edgelet-like features.
2-D wavelets are tensor products of two 1-D wavelets. We choose a type of 1-D wavelets
that have a minimum size support for a given number of vanishing moments but are as
symmetrical as possible. This class of 1-D wavelets is called “Symmlets” in WaveLab [42].
We choose the Symmlets with 8 vanishing moments and size of the support being 16. An
illustration of some of these 2-D wavelets is in Figure 3.6.
Our “edgelet dictionary” is in fact a collection of edgelet features. See the discussion
of Sections 3.3.1–3.3.3. In Appendix B, we define a collection of linear functionals ˜λ e [x]
operating on x belonging to the space of N × N images. These linear functionals are
associated with the evaluation of an approximate Radon transform as described in Appendix
B. In effect, the Riesz representers of these linear functionals, { ˜ψ e (k 1 ,k 2 ):0≤ k 1 ,k 2