Signal Separation with A Priori Knowledge Using Sparse Representation - Advances in Heuristic Signal Processing and Applications - page 317

Fig. 14.2 Illustration of dictionary construction: ( a ) a simulated MRS spectrum with two peaks
(simulated with Gaussian functions); ( b ) normalized Gaussian basis functions in the corresponding
dictionary (the black lines represent the best approximations of the two spectral peaks in ( a ))

Figure 14.2 shows a simulated MRS spectrum with three spectrum peaks and the
corresponding basis functions in a dictionary to represent the spectrum. The black
lines in Fig. 14.2 (b) correspond to the basis functions which can best approximate
the two peaks in Fig. 14.2 (a). The objective of our method is to find the two basis
functions.
The constructed dictionary is denoted as a matrix with M basis vectors

[ L . If the sparse representation vector

of the mixed spectrum S is denoted as w , then

d 1 , d 2 ,..., d i ,..., d M ] , where M = K ×

M K

S = Dw = d i w i = D k w k , (14.14)

i = 1 k = 1

T . Theoretically, only the basis functions
which best approximate the resonances of interest correspond to nonzero repre-
sentation coefficients. In the case, where different resonances have different peak
frequencies, the basis vectors in group D k can only represent the resonance s k .
Therefore, s k =

w 1 T , w 2 T ,..., w k T ,... w K T where w =[ ]

w 1 ∧ T , w 2 ∧ T ,..., w k ∧ T ,..., w K ∧ T D k w k .Let w ∧ (

T ) denote the
estimated sparse representation of a mixed spectrum, the resonance s k can then be
estimated as s k ∧ =

[ ]

D k w k ∧ .
However, because of the presence of baseline component in an observed spec-
trum, the mixed spectrum S is unavailable. For dealing with this problem, a strategy
using a wavelet filter is exploited. In the frequency domain, baselines are commonly
assumed to be smooth and broad compared to the resonance signals. Therefore,
a wavelet filter is used to remove the smooth components of an observed spec-
trum. Because of the overlapping of the baseline and the resonances of interest, the
removed components contain not only the baseline, but also a portion of the use-
ful signal. The signal remaining after the filtering consists of only a component of
mixed resonances of interest which does not overlap with the baseline. Our idea is
to carry out the estimation of sparse representation on the remaining signal to finally
reconstruct the resonances of interest in their entirety.