Image Processing Reference
In-Depth Information
about the same as the baseline OMP-NLS but the frequency error is huge by comparison.
This is the natural result of the frequency grid, which is the limit on the OMP resolution.
Timing comparisons with our software show that OMP-NLS takes about 50% longer than
conventional OMP. We have also windowed the OMP calculations in order to reduce
„spectral leakage“ and hopefully achieve better performance. Aside from the lowered
failure fraction for N f = 2, windowing OMP appears to have no statistically significant
effect.
Method\ N f
1
2
4
8
OMP with NLS
95
41
11
6
OMP
96
35
11
6
OMP with window
93
19
9
10
(a)
Method\ N f
1
2
4
8
OMP with NLS
3.9 10^-15
3.9 10^-15
3.5 10^-15
3.7 10^-15
OMP
0.000150
0.000136
0.000085
0.000060
OMP with window
0.000168
0.000141
0.000084
0.000059
(b)
Table 2. Comparing OMP with NLS to OMP and OMP with windowing for 4 values of the
overcomplete dictionary N f = 1,2,4,8. (a) failure fraction, %. (b) rms error in recovered
frequencies.
We have also compared windowed OMP to OMP/NLS in the presence of noise. Fig. 12
shows the frequency and amplitude errors,  f and  a , as a function of the noise standard
deviation  for OMP (blue) and OMP-NLS (red) for a signal composed of two sinusoids
with N = 128, M = 20 and N f = 4 averaged over 100 trials with randomly chosen input
frequencies. Note that the OMP frequency error drops to an asymptote of about 6 x 10 -4 and
the OMP amplitude error to about 0.23 for  < 0.1 while the OMP-NLS errors continue to
drop linearly proportional to  for  < 0.1.
6.3 Convex optimization
We have performed the same calculations with a penalized ell-1 norm code (Loris, 2008).
None of these calculations returns reliable estimates of frequencies off the grid. Windowing
helps recover frequencies slightly off the grid but not arbitrary frequencies. Subdividing the
frequency grid allows finer resolution in the recovery but only up to the fine frequency grid.
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