SIAM Journal on Imaging Sciences, Volume 14, Issue 2, Page 645-688, January 2021.
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 nonconvex 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 comparisons of the solution behavior for the new equations with first-order flows are presented.

中文翻译：

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一类二阶几何拟线性双曲型PDE及其在成像中的应用

SIAM影像科学杂志，第14卷，第2期，第645-688页，2021年1月。
基于图像处理中的重要应用，我们研究了一类二阶几何拟线性双曲偏微分方程（PDE）。这是受最近开发的与能量衰减的梯度流相关的二阶阻尼系统的启发。在数值计算中，事实证明，二阶方法优于其一阶对应部分。我们专注于（i）阻尼二阶总变化流（例如，图像去噪）和（ii）阻尼二阶平均曲率流，用于标量函数的水平集。后者连接到能够校正图像数据中的位移误差（例如，抖动）的非凸变率模型。对于前一个方程式 我们证明了该解决方案的存在性和唯一性及其长时间的行为，并给出了一些简单的初始基准就提供了一种解析解决方案。对于后者，我们在方程与一些演化超曲面的二阶几何PDE之间建立了联系，并显示了方程正则化版本解的存在性和唯一性。最后，给出了具有一阶流的新方程解行为的一些数值比较。