Truncated Total Least Squares: A New Regularization Method for the Solution of ECG Inverse Problems

Abstract

The reconstruction of epicardial potentials (EPs) from body surface potentials (BSPs) can be characterized as an ill-posed inverse problem which generally requires a regularized numerical solution. Two kinds of errors/noise: geometric errors and measurement errors exist in the ECG inverse problem and make the solution of such problem more difficulty. In particular, geometric errors will directly affect the calculation of transfer matrix A in the linear system equation AX = B. In this paper, we have applied the truncated total least squares (TTLS) method to reconstruct EPs from BSPs. This method accounts for the noise/errors on both sides of the system equation and treats geometric errors in a new fashion. The algorithm is tested using a realistically shaped heart-lung-torso model with inhomogeneous conductivities. The h-adaptive boundary element method [h-BEM, a BEM mesh adaptation scheme which starts from preset meshes and then refines (adds/removes) grid with fixed order of interpolation function and prescribed numerical accuracy] is used for the forward modeling and the TTLS is applied for inverse solutions and its performance is also compared with conventional regularization approaches such as Tikhonov and truncated single value decomposition (TSVD) with zeroth-, first-, and second-order. The simulation results demonstrate that TTLS can obtain similar results in the situation of measurement noise only but performs better than Tikhonov and TSVD methods where geometric errors are involved, and that the zeroth-order regularization is the optimal choice for the ECG inverse problem. This investigation suggests that TTLS is able to robustly reconstruct EPs from BSPs and is a promising alternative method for the solution of ECG inverse problems.

Truncated Total Least Squares Method with a Practical Truncation Parameter Choice Scheme for Bioluminescence Tomography Inverse Problem

“…Due to the complexity of error sources, it is difficult to have an exact mathematic model to describe the system errors accurately. Hence we adopted the commonly used Gaussian white noise [28–30] and exponential noise to simulate the errors in matrix A , respectively.…”

-Norm Regularization in Volumetric Imaging of Cardiac Current Sources

“…These methods include Tikhonov regularization method [8], least squares QR (LSQR) [13], truncated total least square (TTLS) [14], Kalman filter [15], generalized minimal residual [16], and level-set [17] and statistical approaches [18]. Although incorporating the L 2-norm-based constraint handles the ill-posedness of this inverse problem and provides stability in the presence of noise, it ultimately diffuses the source reconstruction solution.…”

Convergence Results on Iteration Algorithms to Linear Systems

“…Moreover, they have proposed a novel preconditioner for the PCG method to solve (1) and a cheap iterative method such as successive overrelaxation (SOR) to further refine the solution for a desired accuracy. In 2008, Shou et al [7] have showed that the reconstruction of epicardial potentials (EPs) from body surface potentials (BSPs) can be characterized as an ill-posed inverse problem and geometric errors in the ECG inverse problem will directly affect the calculation of transfer matrix A in (1). In the field of systems and control science, Ding and Chen [8] have pointed out that Sylvester equations in systems and control especially Lyapunov equations in continuous- and discrete-time stability analysis can be converted into equivalent equations as (1).…”