Smoothing the finite differences defining the Non-local Total Variation and application in image restoration
Résumé
In the usual non-local variational models , such as the non-local total variations (NLTV), the image is regularized by minimizing an energy term that penalizes gray-levels discrepancy between some specified pairs of pixels. The pairs of pixels are interpreted as weighted edges joining nodes in a graph. The nodes correspond to the pixels of the image. The weights are ponderations. The graph structure is usually determined a-priori or, sometimes , is adjusted (most of the time empirically) during the iterative process. In this paper, we study the possibility to impose some regularity to the graph. More precisely, we study a model in which two edges whose starting nodes are near and such that the starting and ending points form a parallelogram to carry similar weights. In order to do so, we minimize a function involving a regularization term, analogous to an H 1 term, on the graph edges. Doing so, the finite differences defining the image regularity depend on their environment. They are therefore more stable. The model is also better suited to solve inverse problems for which the design of the graph is not straightforward (one can think of computerized to-. TZ is partially supported by NSFC 11271049, RGC 12302714 and RFGs of HKBU. mography problems, reconstruction from random measures, inpainting...) We provide all the details necessary for the implementation of a PALM algorithm with proved convergence. We illustrate the ability of the model to restore relevant unknown edges from the neighboring edges on an image inpainting problem. We also argue on inpainting and denoising problems that the model better recovers thin structures.
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