Data Processing Methods in MRS - Advance Image Processing in Magnetic Resonance Imaging - page 413

where and Writing Equation 12.17 for each acquired
sample and using matrix notation, we obtain

α = ae k

j φ

κ = −+ e

( dj 2 π f ) n

.

k k

k k k

x

= KA (12.18)

where

κ ( , ,) fd 1 κ ( f d , ,) 1 α

11 KK 1

K = A = (12.19)

κ (, , ) fdN κ ( , f d ,) α

11 K K K

{1, 2,…, K }) are known
(or some initial guess about them is available), we can solve Equation 12.18 in
the LS sense, thus obtaining

If we assume that the nonlinear parameters ( f k ,d k , k =

K † ˆ

A = (12.20)

where K † is the pseudoinverse of K . Having estimated the A parameters, we can
get rid of it in Equation 12.16 and solve a simpler problem

|| xKKx −

†

|| (12.21)

45

1

2

3

6

− 0.08 − 0.1 − 0.12

kh − 0.14

− 0.16 − 0.18

FIGURE 12.4 Example of application of AMARES for the analysis of in vivo citrate
signal. From bottom to top: the FT spectrum of the original signal, Lorentzians (FT of
fitted sinusoids), and the residual. (From Vanhamme, L., van den Boogaart, A. and van
Huffel, S. [1997]. Improved method for accurate and efficient quantification of MRS data
with use of prior knowledge. J. Magn. Reson. 129: 35-43.)