1,721,083 research outputs found
The total variation flow in R^N
Full text linkIn this paper, we study the minimizing total variation flow u(t) = div(Du/\DU\) in R^N for initial data u(0) in L-loc(1)(R^N), proving an existence and uniqueness result. Then we characterize all bounded sets Omega of finite perimeter in R^2 which evolve without distortion of the boundary. In that case, no = chi(Omega) evolves as u(t, x) = (1 - lambda(Omega)t)(+) chi(Omega),, where chi(Omega) is the characteristic function of Omega, lambda(Omega) := P(Omega)/\Omega\, and P(Omega) denotes the perimeter of Omega. We give examples of such sets. The solutions are such that upsilon := lambda(Omega)chi(Omega) solves the eigenvalue problem -div(Dupsilon/\Dupsilon\) = upsilon. We construct other explicit solutions of this problem. As an application, we construct explicit solutions of the denoising problem in image processing
Explicit solutions of the eigenvalue problem - div (Du/|Du|) = u
Full text linkIn this paper we compute explicit solutions of the eigenvalue problem - div(Du/vertical bar Du vertical bar) = u in R-2, in particular explicit solutions whose truncatures are in W-loc(1,1) (R-2), and piecewise constant ones which are sums of characteristic functions of convex sets. The solutions of the above eigenvalue problem describe the asymptotic behavior of solutions of the minimizing total variation flow. As an application, we also construct explicit solutions of the denoising problem in image processing
Some remarks on uniqueness and regularity of Cheeger sets
No full textWe show that generically the subsets of R^N with finite volume have a unique Cheeger set, in the sense that there always exists a nearby set which has a unique Cheeger set. We also prove that Cheeger sets are C^(1,1), when the ambient set is C^(1,1)
Regularity for solutions of the total variation denoising problem
No full textThe main purpose of this paper is to prove a local Holder regularity result for the solutions of the total variation based denoising problem assuming that the datum is locally Holder continuous. We also prove a global estimate on the modulus of continuity of the solution in convex domains of R(N) and some extensions of this result for the total variation minimization flow
The discontinuity set of solutions of the TV denoising problem and some extensions
No full textThe main purpose of this paper is to prove that the jump discontinuity set of the solution of the total variation based denoising problem is contained in the jump set of the datum to be denoised. We also prove some extensions of this result for the total variation minimization flow, for anisotropic norms, and for some more general convex functionals, which include the minimal surface equation case and its anisotropic extensions
Crystalline mean curvature flow of convex sets
No full textWe prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in IRN. This theorem can handle the facet breaking/bending phenomena, and can be generalized to any anisotropic mean curvature flow. The method provides also a generalized geometric evolution starting from any compact convex set, existing up to the extinction time, satisfying a comparison principle, and defining a continuous semigroup in time. We prove that, when the initial set is convex, our evolution coincides with the flat phi-curvature flow in the sense of Almgren-Taylor-Wang. As a by-product, it turns out that the flat phi-curvature flow starting from a compact convex set is unique
Total Variation and Cheeger sets in Gauss space
No full textThe aim of this paper is to study the isoperimetric problem with fixed volume inside convex sets and other related geometric variational problems in the Gauss space, in both the finite and infinite dimensional case. We first study the finite dimensional case, proving the existence of a maximal Cheeger set which is convex inside any bounded convex set. We also prove the uniqueness and convexity of solutions of the isoperimetric problem with fixed volume inside any convex set. Then we extend these results in the context of the abstract Wiener space, and for that we study the total variation denoising problem in this context
A characterization of convex calibrable sets in R^N with respect to anisotropic norms
Full text linkA set is called "calibrable" if its characteristic function is an eigenvector of the subgradient of the total variation. The main purpose of this paper is to characterize the "phi-calibrability" of bounded convex sets in R(N) with respect to a norm phi (called anisotropy in the sequel) by the anisotropic mean phi-curvature of its boundary. It extends to the anisotropic and crystalline cases the known analogous results in the Euclidean case. As a by-product of our analysis we prove that any convex body C satisfying a phi-ball condition contains a convex phi-calibrable set K such that, for any V is an element of [vertical bar K vertical bar,vertical bar C vertical bar], the subset of C of volume V which minimizes the phi-perimeter is unique and convex. We also describe the anisotropic total variation flow with initial data the characteristic function of a bounded convex set
Uniqueness of the Cheeger set of a convex body
No full textWe prove that if C subset of R-N is of class C-2 and uniformly convex, the Cheeger set of C is unique. The Cheeger set of C is the set that minimizes, inside C, the ratio of perimeter over volume
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