Image Processing Reference
In-Depth Information
where U 1 =  x ( f 1 ). The argmax of G ( f , r 1 ) now yields a first estimate of the frequency of the
second sinusoid, f 2 . Next improve the estimates of both f 1 and f 2 by again solving the NLS
problem by minimizing the functional R ( f ) over f = { f 1 , f 2 }. Note that this overwrites the
previous estimate of the first frequency f 1 . The amplitudes a 1 and a 2 are recalculated using
(8) with U 2 given by
U 2 = [ Φ x ( f 1 ), Φ x ( f 2 ) ]
(15)
for the latest values of f 1 and f 2 , Finally, in this iteration estimates of the first two sinusoids
are removed from y :
r 2 = y - A ( U 2 ) U 2 . (16)
If K , the total number of sinusoids present in the signal, is known, this process is repeated K
times until f K and a K are obtained. In the absence of noise, the sum of these sinusoids solves
(5) exactly and r K = 0.
Inputs:
CS Mixing Matrix Φ
Measured data y
Maximum number of sinusoids K or threshold T
f min , f max
Oversampling ratio for dictionary N f
Initialize
U = [ ]
r 0 = y
K T = K
W = (ΦΦ H ) -1
f = ( f max -f min )/( N N f )
Do i = 1 to K
f i = Argmax G ( f , r i-1 ) over { f min , f min + f ,… f max - f , f max }
{ f 1 , f 2 ,… f i } = Argmin[ R ( f ) with initial value f = { f 1 , f 2 , …, f i } ]
U = x ( f 1 ), Φ x ( f 2 )… Φ x ( f i )}
r i = y - A ( U ) U
If r i H Wr i < T:
K T = i
Break
End If
End Do
Output of Do:
K T
{ f 1 , f 2 , … f KT }
{ a 1 , a 2 ,… a KT }= A [{ Φ x ( f 1 ), Φ x ( f 2 )… Φ x ( f K ) }]
Table 1. OMP/NLS Algorithm.
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