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

TL;DR

The paper introduces and analyzes two second-order damped geometric PDEs for imaging: a damped second-order TVF and a damped second-order level-set MCF. It establishes well-posedness for the TVF via Yosida approximations and develops a regularized existence theory for the level-set MCF, linking level-set evolution to hypersurface flows. Numerically, the authors present a unified symplectic-time-stepping algorithm that handles both models, showing improved efficiency over first-order flows and demonstrating denoising and dejittering capabilities with quantitative metrics. The work highlights the complementary strengths of the two approaches in preserving image features while reducing noise and displacement errors, and outlines several open theoretical questions, particularly for the degenerate hyperbolic level-set MCF.

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.

Figures (11)

• Figure 1: Trajectories of equation \ref{['eq:ode_atvf']} with respect to $\eta=10$ (blue) $\eta=1$ (red) and $\eta=0.1$ (cyan) over the time interval $[0,20]$ with homogeneous initial velocity. The vertical lines in the right figure show the discontinuity of the plotted functions.
• Figure 1: An example that distinguishes the damped second-order MCF and the damped second-order TVF. From left to right: the square image; the result of damped second-order TVF (middle) and the result of damped second-order MCF (right).
• Figure 2: Trajectories of equation \ref{['eq:ode_atvf']} with respect to initial velocities $\psi_0=0$ (left) and $\psi_0=-\eta \xi_0$ (right), respectively, for $\xi_0=255$ over the time interval $[0,300]$. Different $\eta$ values as in Figure \ref{['fig:ode_plot1']} are compared also with the first-order flow (black color).
• Figure 2: Initial value $u_0$ and initial velocity $v_0$ with respect to different eigenfunctions. From left to right, and from above to bottom: $u_0$, $u_1$ and the result from the second-order TV flow after 50, 500, 1000, 5000 iterations, respectively. Note that we choose the initial velocity $v_0=-\eta u_1$ ($\eta=10$).
• Figure 3: Trajectories of equation \ref{['eq:ode_atvf']} with respect to inhomogeneous initial velocity $\psi_0=-1.5\eta \xi_0$, $\psi_0=-\eta \xi_0$ and $\psi_0=-0.5\eta \xi_0$ for $\eta=0.1$ (left), $\eta=1$ (middle) and $\eta=10$ (right) over the time interval $[0,100]$ corresponding to $\xi_0=100$, respectively.

Theorems & Definitions (24)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Lemma 2.4
• Theorem 2.5
• Proof 1
• Theorem 2.6
• Proof 2
• Proposition 2.7
• Proof 3