Image Processing Reference
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shown in detail here, the closest frequency pairs in the signal are {0.2663, 0.2689} and
{0.7715, 0.7736}, but while signals with nearly the same frequency are difficult cases, here the
combined OMP/NLS recovers all the sinusoids to machine precision. Fig. 2 shows the initial
calculation of G ( f , y ) for a 128x1024 mixing matrix and 8192 frequency points ( N f = 8). Note
that most, but not all of the frequencies have peaks in the initial scan of G ( f , y ) . Fig. 3 shows
G ( f , r 19 ) during the 20 th iteration of the Do loop in the algorithm shown in Table 1. After
refining the frequencies by finding the minimum of R ( f ) in (10), the frequency errors are
reduced to less than 10 -16 and the amplitude errors are reduced to 4x10 -14 . Our results
compare favorably to those obtained using other recovery methods for a test problem with
20 arbitrary frequency complex sinusoids, N = 1024, and variable numbers of measurements
M (Duarte and Baraniuk, 2010).
Fig. 2. The initial calculation of G ( f , y ) for a signal with 20 input frequencies mixed with a
128 x 1024 matrix. The red dots indicate the input frequencies.
Fig. 3. The next to last calculation of G ( f , r 19 ) for a signal with 20 input frequencies mixed
with a 128x1024 matrix showing a large peak near the frequency of the only remaining
sinusoid.
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