Facet-breaking for three-dimensional crystals evolving by mean curvature

Bellettini, Giovanni; Novaga, Matteo; Paolini, Maurizio

doi:10.4171/ifb/3

Cited by 51 publications

(67 citation statements)

References 20 publications

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“…5, one observes the celebrated "facet-breaking" phenomenon: a L-shaped facet of the original shape breaks into two rectangular facets which evolve at different speed, as predicted, and observed, in Bellettini et al (1999), Paolini and Pasquarelli (2000), see Fig. 6.…”

Section: Surface Evolution Using Parametric Maximum Flowsmentioning

confidence: 99%

On Total Variation Minimization and Surface Evolution Using Parametric Maximum Flows

Chambolle

Darbon²

2009

Int J Comput Vis

147

183

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In a recent paper Boykov et al. (LNCS, Vol. 3953, pp. 409-422, 2006) propose an approach for computing curve and surface evolution using a variational approach and the geo-cuts method of Boykov and Kolmogorov (International conference on computer vision, pp. [26][27][28][29][30][31][32][33] 2003). We recall in this paper how this is related to well-known approaches for mean curvature motion, introduced by Alm-

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Section: Surface Evolution Using Parametric Maximum Flowsmentioning

confidence: 99%

On Total Variation Minimization and Surface Evolution Using Parametric Maximum Flows

Chambolle

Darbon²

2009

Int J Comput Vis

147

183

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“…In surface evolving problems a facet may not stay as a facet even if σ ≡ 0 see for example [5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

A Comparison Principle for Singular Diffusion Equations with Spatially Inhomogeneous Driving Force for Graphs

Giga

Rybka³

2013

Arch Rational Mech Anal

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We introduce the notions of viscosity super-and subsolutions suitable for singular diffusion equations of non-divergence type with a general spatially inhomogeneous driving term. In particular, the viscosity super-and subsolutions support facets and allow a possible facet bending. We prove a comparison principle by a modified doubling variables technique. Finally, we present examples of viscosity solutions. Our results apply to a general crystalline curvature flow with a spatially inhomogeneous driving term for a graph-like curve.

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“…Different from one dimensional problem, our approach may not correspond to a solution of an original problem with a piecewise constant initial data. Such a difficulty is also observed in the unconstrained problem of crystalline flow [4] and [8], for instance.…”

Section: Introductionmentioning

confidence: 89%

Global Solvability of Constrained Singular Diffusion Equation Associated with Essential Variation

Giga

Kuroda

Yamazaki

2006

Free Boundary Problems

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Abstract. We consider a gradient flow system of total variation with constraint. Our system is often used in the color image processing to remove a noise from picture. In particular, we want to preserve the sharp edges of picture and color chromaticity. Therefore, the values of solutions to our model is constrained in some fixed compact Riemannian manifold. By this reason, it is very difficult to analyze such a problem, mathematically. The main object of this paper is to show the global solvability of a constrained singular diffusion equation associated with total variation. In fact, by using abstract convergence theory of convex functions, we shall prove the existence of solutions to our models with piecewise constant initial and boundary data.

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Facet-breaking for three-dimensional crystals evolving by mean curvature

Cited by 51 publications

References 20 publications

On Total Variation Minimization and Surface Evolution Using Parametric Maximum Flows

On Total Variation Minimization and Surface Evolution Using Parametric Maximum Flows

A Comparison Principle for Singular Diffusion Equations with Spatially Inhomogeneous Driving Force for Graphs

Global Solvability of Constrained Singular Diffusion Equation Associated with Essential Variation

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