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510
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 122 (20 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.
Semiconcave Functions, HamiltonJacobi Equations, and Optimal Control
, 2003
"... otion of semiconvexity. The answer is that this choice better fits minimization, as opposed to maximization, and this is the formulation we have adopted for the optimization problems of interest here. Interest in semiconcave functions was initially motivated by research on nonlinear partial di#eren ..."
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Cited by 52 (2 self)
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otion of semiconvexity. The answer is that this choice better fits minimization, as opposed to maximization, and this is the formulation we have adopted for the optimization problems of interest here. Interest in semiconcave functions was initially motivated by research on nonlinear partial di#erential equations. In fact, it was exactly in classes of semiconcave functions that the first global existence and uniqueness results were obtained for HamiltonJacobiBellman equations, see Douglis [67] and Kruzhkov [97, 98, 100]. Afterwards, more powerful uniqueness theo M. Dibdin, Blood rain, Faber and Faber, London, 1999. iii iv ries, such as viscosity solutions and minimax solutions, were developed. Nevertheless, semiconcavity maintains its importance even in modern PDE theory, as the maximal type of regularity that can be expected for certain nonlinear problems. As such, it has been investigated in the modern textbooks on HamiltonJacobi equations by Lions [107], Bardi and Capuzzo D
A MULTISCALE IMAGE REPRESENTATION USING HIERARCHICAL (BV, L²) DECOMPOSITIONS
 MULTISCALE MODEL. SIMUL.
, 2004
"... We propose a new multiscale image decomposition which offers a hierarchical, adaptive representation for the different features in general images. The starting point is a variational decomposition of an image, f = u0 { + v0, where [u0,v0] is the minimizer of a Jfunctional, J(f, λ0; X, Y) = infu+v= ..."
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Cited by 50 (9 self)
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We propose a new multiscale image decomposition which offers a hierarchical, adaptive representation for the different features in general images. The starting point is a variational decomposition of an image, f = u0 { + v0, where [u0,v0] is the minimizer of a Jfunctional, J(f, λ0; X, Y) = infu+v=f ‖u‖X + λ0‖v ‖ p} Y. Such minimizers are standard tools for image manipulations
Minimizing total variation flow
 Differential and Integral Equations
, 2001
"... (Submitted by: Jerry Goldstein) Abstract. We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect ..."
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Cited by 43 (6 self)
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(Submitted by: Jerry Goldstein) Abstract. We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect, and the solution converges to the spatial average of the initial datum as t →∞. We also prove that local maxima strictly decrease with time; in particular, flat zones immediately decrease their level. We display some numerical experiments illustrating these facts. 1. Introduction. Let Ω be a bounded set in R N with Lipschitzcontinuous boundary ∂Ω. We are interested in the problem ∂u Du = div(
Total generalized variation
 SIAM Journal on Imaging Sciences
"... The novel concept of total generalized variation of a function u is introduced and some of its essential properties are proved. Differently from the bounded variation seminorm, the new concept involves higher order derivatives of u. Numerical examples illustrate the high quality of this functional ..."
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Cited by 35 (10 self)
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The novel concept of total generalized variation of a function u is introduced and some of its essential properties are proved. Differently from the bounded variation seminorm, the new concept involves higher order derivatives of u. Numerical examples illustrate the high quality of this functional as a regularization term for mathematical imaging problems. In particular this functional selectively regularizes on different regularity levels and, as a side effect, does not lead to a staircasing effect.
A level set formulation for Willmore flow
 INTERFACES FREE BOUNDARIES
, 2004
"... A level set formulation of Willmore flow is derived using the gradient flow perspective. Starting from single embedded surfaces and the corresponding gradient flow, the metric is generalized to sets of level set surfaces using the identification of normal velocities and variations of the level set f ..."
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Cited by 28 (7 self)
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A level set formulation of Willmore flow is derived using the gradient flow perspective. Starting from single embedded surfaces and the corresponding gradient flow, the metric is generalized to sets of level set surfaces using the identification of normal velocities and variations of the level set function in time via the level set equation. The approach in particular allows to identify the natural dependent quantities of the derived variational formulation. Furthermore, spatial and temporal discretization are discussed and some numerical simulations are presented.
Existence and uniqueness for dislocation dynamics with nonnegative velocity
, 2004
"... We study the problem of large time existence of solutions for a mathematical model describing dislocation dynamics in crystals. The mathematical model is a geometric and non local eikonal equation which does not preserve the inclusion. Under the assumption that the dislocation line is expanding, we ..."
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Cited by 25 (15 self)
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We study the problem of large time existence of solutions for a mathematical model describing dislocation dynamics in crystals. The mathematical model is a geometric and non local eikonal equation which does not preserve the inclusion. Under the assumption that the dislocation line is expanding, we prove existence and uniqueness of the solution in the framework of discontinuous viscosity solutions. We also show that this solution satisfies some variational properties, which allows to prove that the energy associated to the dislocation dynamics is non increasing. AMS Classification: 35F25, 35D05. Keywords: Dislocation dynamics, eikonal equation, HamiltonJacobi equations,
Variational Restoration and Edge Detection for Color Images
 Journal of Mathematical Imaging and Vision
, 2003
"... Abstract. We propose and analyze extensions of the MumfordShah functional for color images. Our main motivation is the concept of images as surfaces. We also review most of the relevant theoretical background and computer vision literature. Keywords: color, MumfordShah functional, segmentation, va ..."
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Cited by 23 (1 self)
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Abstract. We propose and analyze extensions of the MumfordShah functional for color images. Our main motivation is the concept of images as surfaces. We also review most of the relevant theoretical background and computer vision literature. Keywords: color, MumfordShah functional, segmentation, variational methods.