Signal Separation with A Priori Knowledge Using Sparse Representation - Advances in Heuristic Signal Processing and Applications - page 315

Fig. 14.1 35 ms) of normal human brain tissue ( a )whichis

composed of several resonances associated to different metabolites of interest, a baseline compo-
nent and a noise ( b )

Observed 1H MRS spectra (TE =

a baseline and a noise, as shown in Fig. 14.1 . It is necessary to recover these res-
onances for accurately quantitating the corresponding metabolites. In our previous
work, we have proposed a method using sparse representation and wavelet filter
for separating different components in observed MRS spectra. Here, we introduce
this method for specifying signal separation with a priori knowledge using sparse
representation.

14.3.1 Signal Models

Generally, an observed MRS spectrum x can be modeled as

K

x = S + B + e = s k + B + e , (14.11)

k = 1

where S represents the mixed spectrum of interest which is the linear combination
of several resonances s k (k 1 ,...,K) , B a baseline contribution, e a Gaussian

noise. Each resonance can be modeled as a lineshape. A Lorentzian lineshape in Eq.
( 14.12 ), or a Gaussian lineshape in Eq. ( 14.13 ), or a combination of Lorentzian and
Gaussian lineshapes is usually used:

=

a k

L k (f) =

2 ,

(14.12)

1 +[ (f − f k )/d k ]