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23
Fillingin by joint interpolation of vector fields and gray levels
 IEEE Trans. Image Processing
, 2001
"... Abstract—A variational approach for fillingin regions of missing data in digital images is introduced in this paper. The approach is based on joint interpolation of the image graylevels and gradient/isophotes directions, smoothly extending in an automatic fashion the isophote lines into the holes ..."
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Cited by 128 (22 self)
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Abstract—A variational approach for fillingin regions of missing data in digital images is introduced in this paper. The approach is based on joint interpolation of the image graylevels 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 graylevels 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—Fillingin, Gestalt principles, image gradients, image graylevels, interpolation, partial differential equations, variational approach. I.
A handbook of Γconvergence
 in “Handbook of Differential Equations – Stationary Partial Differential Equations
"... The notion of Γconvergence has become, over the more than thirty years after its introduction by Ennio De Giorgi, the commonlyrecognized notion of convergence for variational problems, and it would be difficult nowadays to think of any other ‘limit ’ than a Γlimit when talking about asymptotic an ..."
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Cited by 18 (11 self)
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The notion of Γconvergence has become, over the more than thirty years after its introduction by Ennio De Giorgi, the commonlyrecognized notion of convergence for variational problems, and it would be difficult nowadays to think of any other ‘limit ’ than a Γlimit when talking about asymptotic analysis in a general variational setting (even though special convergences may fit better specific problems, as Moscoconvergence, twoscale convergence, G and Hconvergence, etc.). This short presentation is meant as an introduction to the many applications of this theory to problems in Partial Differential Equations, both as an effective method for solving asymptotic and approximation issues and as a means of expressing results that are derived by other techniques. A complete introduction to the general theory of Γconvergence is the bynowclassical book by Gianni Dal Maso [85], while a userfriendly introduction can be found in my book ‘for beginners ’ [46], where also simplified onedimensional versions of many of the problems in this article are treated. These notes are addressed to an audience of experienced mathematicians, with some background and interest in Partial Differential Equations, and are meant to direct the reader to what I regard as the most interesting features of this theory. The style of the exposition is how I would present the subject to a colleague in a neighbouring field or to an interested PhD student: the issues that I think
Partial localization, lipid bilayers, and the elastica functional. in prep
, 2006
"... Abstract. Partial localization is the phenomenon of selfaggregation of mass into highdensity structures that are thin in one direction and extended in the others. We give a detailed study of an energy functional that arises in a simplified model for lipid bilayer membranes. We demonstrate that thi ..."
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Cited by 14 (9 self)
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Abstract. Partial localization is the phenomenon of selfaggregation of mass into highdensity structures that are thin in one direction and extended in the others. We give a detailed study of an energy functional that arises in a simplified model for lipid bilayer membranes. We demonstrate that this functional, defined on a class of twodimensional spatial mass densities, exhibits partial localization and displays the ‘solidlike ’ behavior of cell membranes. Specifically, we show that density fields of moderate energy are partially localized, i.e. resemble thin structures. Deviation from a specific uniform thickness, creation of ‘ends’, and the bending of such structures all carry an energy penalty, of different orders in terms of the thickness of the structure. These findings are made precise in a Gammaconvergence result. We prove that a rescaled version of the energy functional converges in the zerothickness limit to a functional that is defined on a class of planar curves. Finiteness of the limit enforces both optimal thickness and nonfracture; if these conditions are met, then the limit value is given by the classical Elastica (bending) energy of the curve.
Disocclusion By Joint Interpolation Of Vector Fields And Gray Levels
 SIAM Journal Multiscale Modelling and Simulation
, 2003
"... In this paper we study a variational approach for fillingin 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 pre ..."
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Cited by 12 (0 self)
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In this paper we study a variational approach for fillingin 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 graylevels 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.
Approximation by Γconvergence of a curvaturedepending functional in visual reconstruction
 Comm. Pure Appl. Math
, 2004
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Variational approximation of functionals with curvatures and related properties
 J. Convex Anal
, 1997
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Locality of the mean curvature of rectifiable varifolds
, 2008
"... The aim of this paper is to investigate whether, given two rectifiable kvarifolds in R n with locally bounded first variations and integervalued multiplicities, their generalized mean curvatures coincide H kalmost everywhere on the intersection of the supports of their weight measures. This soca ..."
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Cited by 7 (4 self)
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The aim of this paper is to investigate whether, given two rectifiable kvarifolds in R n with locally bounded first variations and integervalued multiplicities, their generalized mean curvatures coincide H kalmost everywhere on the intersection of the supports of their weight measures. This socalled locality property, which is wellknown 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 1varifolds, while for kvarifolds, 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.
Uses of elliptic approximations in computer vision
 Variational Methods for Discontinuous Structures, R.Serapioni and F.Tomarelli eds., Birkhauser
, 1996
"... 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 c ..."
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Cited by 3 (0 self)
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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
Curvature Theory of Boundary phases: the two dimensional case
 Interfaces Free Bound
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