sparse image representation via combined transforms - Convex ...

126 CHAPTER 6. SIMULATIONS
10 0
10 −1
greedy
10 0 greedy
10 −1
10 −2
10 −3
10 −2
global
10 −4
global
10 −3
0 200 400 600 800 1000
10 −5
0 200 400 600 800 1000
10 −1
10 −1
10 0 greedy
10 −2
10 0 greedy
10 −2
10 −3
10 −3
global
10 −4
10 −4
global
10 −5
0 200 400 600 800 1000
10 −5
0 200 400 600 800 1000
Figure 6.4: A global algorithm vs. Matching Pursuit.
Figure 6.4 shows the decay of the amplitudes of the coefficients from both of the two
approaches. In these plots, the solid lines always correspond to the MP, and these dashed
lines correspond to our (minimizing the l 1 norm) approach. From upper row to lower row,
left to right, the plots are for Car, Pentagon, Overlap and Separate.
We have the following observations:
1. In all four cases, our global approach always provides a sparser representation than
MP does: the decay of the amplitudes of coefficients for our approach is faster than for
MP. This validates our belief that a global optimization scheme should often provide
a sparser representation than one from a greedy algorithm.
2. If we adopt the idea that the decay of the amplitudes of coefficients at the beginning is
important, then our global approach shows the largest advantage in the example of a
realistic image (case Car). This is promising because it may imply that our approach
is more adaptable to real images.
3. We observe that our approach achieves a slightly greater advantage for Overlap than
for Separate. As we know, MP is good for separated features but not for overlapped
ones. This belief is confirmed here.