sparse image representation via combined transforms - Convex ...

6.5. COMPARISON WITH MATCHING PURSUIT 125
approach ultimately falls well below the curve associated with the combined approach.
(This seems to imply that the wavelet-only approach gives a sparser asymptotic representation.)
Particularly, it is important to remember that amplitudes are plotted on a logarithmic
scale. A big drop at the beginning can not be shown significantly. The advantage
of combined representation at very high levels of compression is difficult to discuss
in this display. Note that the curve associated with the wavelet-only approach falls
below the curve associated with the combined approach when the order index is large.
If we study carefully the first few largest amplitudes, the curve associated with the
combined approach goes down faster. This fact is evident for Overlap. Asweknow,in
the compression or sparse representation, the decay of the first few biggest amplitudes
is the most important factor, so we think our combined approach still gives better
results, even for Pentagon and Overlap.
2. The DCT only approach always gives the least sparse coefficients. As we have mentioned,
the DCT is good for images with homogeneous components. Unfortunately,
in our examples, none of them seems to have a high proportion of homogeneous components.
This may explain why here DCT is far from optimal.
3. For more “realistic” images, as in the case of Car, the wavelet-only approach works
only as well as the DCT-only approach, but our combined approach is significantly
better. This may imply that our approach is better-suited for natural images than
existing methods. Of course more careful study and extensive experiments are required
to verify this statement.
6.5 Comparison with Matching Pursuit
Our approach is a global approach, in the sense that we minimize a global objective function.
Our method is computationally expensive. A potentially cheaper method is Matching
Pursuit (MP) [102]. MP is a greedy algorithm. In a Hilbert space, MP at every step picks
up the atom that is the most correlated with the residual at that step. But MP runs the
danger of being trapped by unfortunate choices at early steps into badly suboptimal decompositions
[27, 40, 25]. We examine the decay of the amplitudes of coefficients for both
MP and our approach.