Image Processing Reference
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Fig. 11. Standard deviation in frequency  f (red-lower curve) and amplitude  a (green upper
curve) for the case with input frequencies {0.3389, 0.3390}, unity amplitude and a 16x1024
mixing matrix.
6. Comparison with other recovery methods
In this section we compare our version of OMP with an NLS optimization step for the
sinusoid frequency and amplitude at each iteration to two common methods for CS
recovery: OMP with a linear least squares amplitude estimator at each iteration and convex
optimization based on the ell-1 norm of the sparse target vector plus the ell-2 norm of the
measurement constraint given by eq. (2). It should be noted that most of the cases presented
in the previous sections cannot be solved with OMP/LS or penalized ell-1 norm methods so
it is necessary to pick a special case to even perform the comparison. Consider a noise-free
signal that consists of 5 unity amplitude sinusoids at 5 different frequencies. We assume
N=1024 time samples and an M=30 x N=1024 complex measurement matrix made up of the
sum of random reals plus i times different random reals, both sets of reals uniformly
distributed between -1 and 1.
6.1 Baseline case OMP-NLS
We performed 100 different calculations with the frequencies chosen by a pseudo-random
number generator. In order to control the number of significant figures, we took the
frequencies from rational numbers uniformly distributed between 0 and 1 in steps of 10 -6 .
Table 2 shows the fraction of failed recoveries and the average standard deviation in the
value of the recovered frequency as a function of the oversampling ratio.
6.2 OMP with Linear Least Squares
We performed the same 100 calcuations using conventional OMP in which the NLS step is
replaced by LS as in Tropp and Gilbert (2007). Note that the number of failed recoveries is
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