ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing

TL;DR

This paper introduces directional operator splitting schemes combined with implicit time-stepping to efficiently and stably solve a challenging fourth-order nonlinear PDE from image processing, specifically for denoising, deblurring, and inpainting.

Contribution

The paper develops a novel numerical method that combines directional splitting with implicit schemes to effectively solve complex fourth-order PDEs in image processing.

Findings

Stable and efficient numerical solution demonstrated.

Applicable to denoising, deblurring, and inpainting tasks.

Reduces computational cost compared to existing methods.

Abstract

We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the $H^{-1}$-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation.