Total-Variation Minimization on Unstructured Volumetric Mesh: Biophysical Applications on Reconstruction of 3D Ischemic Myocardium

This paper describes the development and application of a new approach to total-variation (TV) minimization for reconstruction problems on geometrically-complex and unstructured volumetric mesh. The driving application of this study is the reconstruction of 3D ischemic regions in the heart from noninvasive body-surface potential data, where the use of a TV-prior can be expected to promote the reconstruction of two piecewise smooth regions of healthy and ischemic electrical properties with localized gradient in between. Compared to TV minimization on regular grids of pixels/voxels, the complex unstructured volumetric mesh of the heart poses unique challenges including the impact of mesh resolutions on the TV-prior and the difficulty of gradient calculation. In this paper, we introduce a variational TV-prior and, when combined with the iteratively re-weighted least-square concept, a new algorithm to TV minimization that is computationally efficient and robust to the discretization resolution. In a large set of simulation studies as well as two initial real-data studies, we show that the use of the proposed TV prior outperforms L2-based penalties in reconstruct ischemic regions, and it shows higher robustness and efficiency compared to the commonly used discrete TV prior. We also investigate the performance of the proposed TV-prior in combination with a L2- versus L1-based data fidelity term. The proposed method can extend TV-minimization to a border range of applications that involves physical domains of complex shape and unstructured volumetric mesh.