Abstract—A variational approach for filling-in regions of missing data in digital images is introduced in this paper. The approach is based on joint interpolation of the image gray-levels and gradient/isophotes directions, smoothly extending in an automatic fashion the isophote lines into the holes of missing data. This interpolation is computed by solving the variational problem via its gradient descent flow, which leads to a set of coupled second order partial differential equations, one for the gray-levels and one for the gradient orientations. The process underlying this approach can be considered as an interpretation of the Gestaltist’s principle of good continuation. No limitations are imposed on the topology of the holes, and all regions of missing data can be simultaneously processed, even if they are surrounded by completely different structures. Applications of this technique include the restoration of old photographs and removal of superimposed text like dates, subtitles, or publicity. Examples of these applications are given. We conclude the paper with a number of theoretical results on the proposed variational approach and its corresponding gradient descent flow. Index Terms—Filling-in, Gestalt principles, image gradients, image gray-levels, interpolation, partial differential equations, variational approach. I.

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In this paper we study a variational approach for filling-in regions of missing data in 2D and 3D digital images. Applications of this technique include the restoration of old photographs and removal of superimposed text like dates, subtitles, or publicity, or the zooming of images. The approach presented here, initially introduced in [12], is based on a joint interpolation of the image gray-levels and gradient/isophotes directions, smoothly extending the isophote lines into the holes of missing data. The process underlying this approach can be considered as an interpretation of the Gestaltist's principle of good continuation. We study the existence of minimizers of our functional and its approximation by smoother functionals. Then we present the numerical algorithm used to minimize it and display some numerical experiments. Key words. Disocclusion, Elastica, BV functions, Interpolation, Variational approach, #- convergence AMS subject classifications. 68U10, 35A15, 65D05, 49J99, 47H06, 1.

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One of the problems in Computer Vision is recovery of object shapes from noisy images. Associated with this problem is the question of what is a shape and how is it to be represented. Since answers to these questions have to be ultimately tailored to the uses one has in mind, one has to bring into consideration potential applications and with it, the question of practical

The aim of this paper is to investigate whether, given two rectifiable k-varifolds in R n with locally bounded first variations and integer-valued multiplicities, their generalized mean curvatures coincide H k-almost everywhere on the intersection of the supports of their weight measures. This so-called locality property, which is well-known for classical C 2 surfaces, is far from being obvious in the context of varifolds. We prove that the locality property holds true for integral 1-varifolds, while for k-varifolds, k> 1, we are able to prove that it is verified under some additional assumptions (local inclusion of the supports and locally constant multiplicity on their intersection). We also discuss a couple of applications in elasticity and computer vision.

A variational approach for filling-in regions of missing data in gray-level and color images is introduced in this paper. The approach is based on joint interpolation of the image gray-levels and gradient/isophotes directions, smoothly extending in an automatic fashion the isophote lines into the holes of missing data. This interpolation is computed solving the variational problem via its gradient descent flow, which leads to a set of coupled second order partial differential equations, one for the gray-levels and one for the gradient orientations. The process underlying this approach can be considered as an interpretation of the Gestaltist's principle of good continuation. No limitations are imposed on the topology of the holes, and all regions of missing data can be simultaneously processed, even if they are surrounded by completely different structures. Applications of this technique include the restoration of old photographs and removal of superimposed text like dates, subtitles, o...

We discuss possible algorithms for interpolating data given in a set of curves and#or points in the plane. We propose a set of basic assumptions to be satis#ed by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial di#erential equations. The Absolute Minimal Lipschitz Extension model #AMLE# is singled out and studied in more detail. We show experiments suggesting a possible application, the restoration of images with poor dynamic range. We also analyse the problem of unsmooth interpolation and showhowit permits a subsidiary variational method. 1