Discrete total variation in multiple spatial dimensions and its applications

Abstract

Total variation plays an important role in the analysis of stability and convergence of numerical solutions for one-dimensional scalar conservation laws. However, extending this approach to two or more spatial dimensions presents a formidable challenge. Existing literature indicates that total variation diminishing solutions for two-dimensional hyperbolic equations are limited to at most first-order accuracy.

The presented research contributes to overcoming the challenges associated with extending total variation to higher dimensions, particularly in the context of hyperbolic conservation laws. By addressing the limitations of conventional discrete total variation definitions, we seek answers to critical questions associated with the total variation diminishing property of solutions of scalar conservation laws in multiple spatial dimensions. We adopt a more accurate dual discrete definition of total variation, recently proposed in Condat, L. (2017) Discrete total variation: New definition and minimization, SIAM Journal on Imaging Sciences, 10(3), 1258-1290, for measuring the total variation of grid-based functions. Dual total variation can be computed as a solution to a constrained optimization problem. We propose a set of conditions on the coefficients of a general five-point scheme so that the numerical solution is total variation diminishing in the dual discrete sense and validate that through numerical experiments.

Apart from the contributions to the analysis of numerical methods for two-dimensional scalar conservation laws, we develop an algorithm to efficiently compute the dual discrete total variation and develop an imaging method, based on this algorithm. We study its performance in computed tomography image reconstruction and compare it with the state-of-the-art total variation minimization-based imaging methods.