Vector-valued image regularization with PDEs: a common framework for different applications

In this paper, we focus on techniques for vector-valued image regularization, based on variational methods and PDEs. Starting from the study of PDE-based formalisms previously proposed in the literature for the regularization of scalar and vector-valued data, we propose a unifying expression that gathers the majority of these previous frameworks into a single generic anisotropic diffusion equation. On one hand, the resulting expression provides a simple interpretation of the regularization process in terms of local filtering with spatially adaptive Gaussian kernels. On the other hand, it naturally disassembles any regularization scheme into the smoothing process itself and the underlying geometry that drives the smoothing. Thus, we can easily specialize our generic expression into different regularization PDEs that fulfill desired smoothing behaviors, depending on the considered application: image restoration, inpainting, magnification, flow visualization, etc. Specific numerical schemes are also proposed, allowing us to implement our regularization framework with accuracy by taking the local filtering properties of the proposed equations into account. Finally, we illustrate the wide range of applications handled by our selected anisotropic diffusion equations with application results on color images.