Vector-valued image regularization with PDE's: a common framework for different applications

Abstract

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.

4. Rotation invariance in exemplar-based image inpainting

“…In [CS01], a regularization term, taking into account the total variation, was proposed. Later this idea was extended by Tschumperlé et al who presented a framework for arbitrary vector-valued regularization techniques [TD05]. The PDE-based methods have been extended for application to video-sequences [CCB03], for application on ancient frescoes [BFMS08] and to encounter higher-order PDE's [Sc09, upto order 4].…”

Recovering Missing Data on Satellite Images

“…Data assimilation is then applied as explained in Section 3. Bertalmio et al [4] and Tschumperlé et al [14] are also used on the same data. Results are displayed on Fig.…”

“…A second class of methods concerns the so-called "inpainting" approaches, which make use of oriented diffusion processes. Using the local orientation of image gradient, it becomes possible to close interrupted lines [6], and even, to recover large regions by diffusing the image texture in the direction of the image gradient [4,5,7,11,14]. However, these methods are either spatial or space-time techniques, with time only considered as an additional dimension: they do not use any knowledge on the underlying dynamics that is visualized by the image sequence.…”

Self-Invertible 2D Log-Gabor Wavelets

“…Most currently used denoising methods are based on anisotropic diffusion (Tschumperlé and Deriche, 2005;Sroubek and Flusser, 2003;Hamza et al, 2002) or wavelet thresholding techniques (Donoho, 1995;Coifman and Donoho, 1995;Portilla et al, 2003). Wavelet or multiresolution image denoising applications usually proceed in three stages: first a transformation, then a thresholding operation and finally the inverse transform for reconstructing the image.…”