Sobolev gradients and image interpolation

TL;DR

This paper introduces a novel image inpainting algorithm using Sobolev gradients combined with the Navier-Stokes model, offering a simpler alternative to high-order PDE solutions with demonstrated effectiveness.

Contribution

It reformulates Bertalmio et al's model as a variational problem solved via Sobolev gradient flow, avoiding complex numerical schemes and providing theoretical analysis of the algorithm.

Findings

Effective smoothing and preconditioning properties in image inpainting

Global existence and uniqueness of the proposed algorithm

Successful demonstration through various examples

Abstract

We present here a new image inpainting algorithm based on the Sobolev gradient method in conjunction with the Navier-Stokes model. The original model of Bertalmio et al is reformulated as a variational principle based on the minimization of a well chosen functional by a steepest descent method. This provides an alternative of the direct solving of a high-order partial differential equation and, consequently, allows to avoid complicated numerical schemes (min-mod limiters or anisotropic diffusion). We theoretically analyze our algorithm in an infinite dimensional setting using an evolution equation and obtain global existence and uniqueness results as well as the existence of an $\omega$-limit. Using a finite difference implementation, we demonstrate using various examples that the Sobolev gradient flow, due to its smoothing and preconditioning properties, is an effective tool for use in the image inpainting problem.