SCIRun Forward/Inverse ECG Toolkit - Scientific Computing and ...

tor of measured potentials (see figure 4.1-1,3 ). In this case the module will calculate the
standard Tikhonov regularization with l2 regularization:
x = argmin x ‖Ax − y‖ 2 + λ‖x‖ 2 (4.1)
Optionally, the algorithm allows to use other constraints in both the solution and the
measurements. In the module, this matrices are respectively called solution constraint
matrix (R) and residual constraint matrix (L) (see figure 4.1-2,4). In the general case the
formula would result:
x = argmin x ‖Ax − Ly‖ 2 + λ‖Rx‖ 2 (4.2)
The module is also prepared to compute the solution more efficiently depending in
whether the problem is underdetermined or overdetermined. In both cases the underlying
algorithm will be the Gaussian elimination, but the equations to solve will differ. This will
give for the underdetermined case:
(ALL T A T + λRR T )x ′ = y (4.3)
and for the overdetermined case:
x = LL T A T x ′ (4.4)
(A T L T LA + λR T R)x = A T L T Ly (4.5)
In both cases, the algorithm could be solved by using a linear operator G. This operator
requires some inverses and might result in an inefficient implementation. For this reason it
is not used as is, but it might be obtained by the user as one of the outputs of the module.
In the case of the overdetermined case the operator is:
G = (A T P T P A + λ 2 LL T ) −1 A T P T P (4.6)
and for the underdetermined case:
G = W W T A T (AW W T A T + λ 2 CC T ) −1 . (4.7)
28 Chapter 4