# A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science

## TL;DR

This paper introduces second-order geometric quasilinear hyperbolic PDEs inspired by image processing applications, demonstrating their advantages over first-order methods through theoretical analysis and numerical comparisons.

## Contribution

It develops and analyzes a new class of second-order PDEs for image denoising and correction, including existence, uniqueness, and long-term behavior results.

## Key findings

- Second-order PDEs outperform first-order flows in numerical experiments.
- Existence and uniqueness of solutions are established for both PDE models.
- Analytical solutions are derived for simplified initial data cases.

## Abstract

Motivated by important applications in image processing, we study a class of second-order geometric quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems associated to gradient flows for energy decaying. In numerical computations, it turns out that the second-order methods are superior to their first-order counter-parts. We concentrate on (i) a damped second-order total variation flow for e.g., image denoising, and (ii) a damped second-order mean curvature flow for level sets of scalar functions. The latter is connected to a non-convex variational model capable of correcting displacement errors in image data (e.g. dejittering). For the former equation, we prove the existence and uniqueness of the solution and its long time behavior, and provide an analytical solution given some simple initial datum. For the latter, we draw a connection between the equation and some second-order geometric PDEs evolving the hypersurfaces, and show the existence and uniqueness of the solution for a regularized version of the equation. Finally, some numerical comparison of the solution behavior for the new equations with first-order flows are presented.