Image Processing Reference
In-Depth Information
4.3 Signal composed of 2 sinusoids with large dynamic range
For signals composed of 2 well separated frequencies but widely different amplitudes in the
absence of noise, we recover the amplitude and frequency of the 2 sinusoids when
a
1
= 1
and
a
2
is as small as 10
-14
with an 8x1024 mixing matrix. For this case the amplitude and
frequency of the large signal are recovered to machine precision while the frequency and
amplitude error of the weak signal are 3x10
-4
and 1%, respectively. Naturally, such
performance is not found in the presence of noise as discussed below.
4.4 Signal composed of 2 sinusoids with closely spaced frequencies
We have also input a signal with 2 very closely spaced frequencies and unity amplitudes.
For frequencies {0.3389, 0.3390} we recover the frequencies to machine precision with a
16x1024 mixing matrix. Smaller values of
M
for the mixing matrix yield one root half way
between the two frequencies. For frequencies {0.33899,0.33900} mixed with 16x1024 and
32x1024 matrices the OMP part of our algorithm yields a signal with one frequency at
0.338995 and an amplitude of 1.9996. Attempts to find a second frequency yield a badly
conditioned matrix for
U
H
WU
and the inversion required to find the 2
nd
amplitude in eq.
(11) fails. For a 64x1024 mixing matrix OMP finds two separated estimates of the frequencies
and this allows NLS determination of both frequencies to an accuracy of a few parts in 10
5
.
These results are in contrast to those obtained using the “spectral compressive sensing”
algorithms that use “a signal model that inhibits closely spaced sinusoids” (Duarte and
Baraniuk, 2010).
4.5 Dependence on dimensions of the mixing matrix
We have investigated the requirements on
M
, the small dimension of the measurement
matrix, to recover a signal composed of a small number of sinusoids using the OMP-NLS
algorithm. Fig. 4 shows the fraction of failed recoveries as a function of M for a problem in
which the signal is composed of 1,3,5, or 7 sinusoids and
N
= 128. For each value of
K
we
performed 1000 trials so a failure fraction of 0.1 corresponds to 100 failures. The
conventional relation between
K
,
M
, and
N
for recovery is given by
M
=
C K
log(
N/K
)
(Baraniuk, 2007; Candes and Wakin, 2008). From Fig. 4 we see that the curves for
K
= 3,5 and
7 are equispaced and correspond to
C
~ 1.5.
We have also investigated several different types of the measurement matrix as displayed in
Fig. 5. The three curves correspond to three different measurement matrices. For the blue
curve the mixing matrix is generated from the sum of random integers drawn from {-1,0,1}
plus i times different random integers drawn from {-1,0,1}; for the red curve, complex
numbers with the real and imaginary parts given by reals uniformly distributed between -1
and 1 and i times uniformly distributed reals; for the magenta curve, the mixing matrix is
generated from randomly chosen -1's and 1's. The magenta curve for a real mixing matrix
made from 1's and -1's is inferior to the blue and red curves for the two complex mixing
matrices. The differences between the red and blue curves in Fig. 5 appear to be random
fluctuations and are in agreement with other CS results that Gaussian and Bernoulli
measurement matrices perform equally well (Baraniuk, 2007; Candes and Wakin, 2008). Fig.
6 compares calculations with the weighting matrix given by eq. (7) to calculations with the
weighting matrix set to the identity matrix. One can see that the green curve with the
weighting matrix set to the identity matrix is significantly worse in the important region of
less than 1% failure.
