Sobolev Gradients and Image Interpolation

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 Bertalmío, Bertozzi, and Sapiro [Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2001, pp. 355–362] is reformulated as a variational principle based on the minimization of a well-chosen functional by a steepest descent method using Sobolev gradients. This new theoretical framework offers an alternative to the direct solving of a high-order PDE, with the practical advantage of an easier and more flexible computer implementation. In particular, the proposed algorithm does not require any constant tuning or advanced knowledge of numerical methods for Navier–Stokes equations (slope limiters, dynamic relaxation for Poisson equation, anisotropic diffusion steps, etc.). Using a straightforward finite difference implementation, we demonstrate, through various examples for image inpainting and image interpolation, that the novel algorithm is faster than the original implementation of the Navier–Stokes model, while providing results of similar quality. This paper also provides the mathematical theory for the analysis of the algorithm. Using an evolution equation in an infinite dimensional setting, we obtain global existence and uniqueness results as well as the existence of an $\omega$-limit. This formalism is of more general interest and could be applied to other image processing models based on variational formulations.