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Electrical activity of the heart 101
Figure 4.4 Laplacian electrodes are used to measure the thoracic electrical potential produced by
an intracardiac dipole source (Fig. 4.4).
Technically, inverse solutions are obtained through the minimization of
2 2 p
x α 5 arg min :Cx2b: 1 α :Lx: ; ð4:5Þ
p
2
x
|fflfflfflfflffl{zfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
penalty term
least sqaures approx:
where the penalty term provides a priori information on the solution. Should L 5 I
and p 5 2 then Eq. (4.5) is the Tikhonov regularization of order zero, yielding
T 2 21 T
x Tik 5 C C1α I C b: ð4:6Þ
When L 5 D (the gradient operator) and p 51, Eq. (4.5) yields the total variation
(TV) of the potential x (Rudin et al., 1992; Mukherjee et al., 2016)
T
2
x TV 5 C C1α D W x D 21 C b; ð4:7Þ
T
T
where W x is the weight diagonal matrix
!
1 1
W x 5 diag p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð4:8Þ
2 Dx 1 β
2
½ i
Successive approximation (fixed point iterations)
x ðk11Þ 5 C C1α D W ðkÞD 21 C y; ð4:9Þ
T
T
T
2
x
may be used to solve for x.
