The maximum flow algorithm for minimizing energy functions of binary variables has become a standard tool in computer vision. In many cases, unary costs of the energy depend linearly on parameter λ. In this paper we study vision applications for which it is important to solve the maxflow problem for different λ’s. An example is a weighting between data and regularization terms in image segmentation or stereo: it is desirable to vary it both during training (to learn λ from ground truth data) and testing (to select best λ using high-knowledge constraints, e.g. user input). We review algorithmic aspects of this parametric maximum flow problem previously unknown in vision, such as the ability to compute all breakpoints of λ and corresponding optimal configurations in finite time. These results allow, in particular, to minimize the ratio of some geometric functionals, such as flux of a vector field over length (or area). Previously, such functionals were tackled with shortest path techniques applicable only in 2D. We give theoretical improvements for “PDE cuts ” [5]. We present experimental results for image segmentation, 3D reconstruction, and the cosegmentation problem. 1.

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We prove that if C ⊂ IR N is a an open bounded convex set, then there is only one Cheeger set inside C and it is convex. A Cheeger set of C is a set which minimizes the ratio perimeter over volume among all subsets of C.

Given a bounded open subset Ω of Rd and two positive weight functions f and g, the Cheeger sets of Ω are the subdomains C of finite perimeter of Ω that maximize the ratio ∫ / ∫ C f(x) dx ∂∗C g(x) dHd−1. Existence of Cheeger sets is a well-known fact. Uniqueness is a more delicate issue and is not true in general (although it holds when Ω is convex and f ≡ g ≡ 1 as recently proved in [4]). However, there always exists a unique maximal (in the sense of inclusion) Cheeger set and this paper addresses the issue of how to determine this maximal set. We show that in general the approximation by the p-Laplacian does not provide, as p → 1, a selection criterion for determining the maximal Cheeger set. On the contrary, a different perturbation scheme, based on the constrained maximization of Ω f(u − εΦ(u)) dx for a strictly convex function Φ, gives, as ε → 0, the desired maximal set.

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No geometric approach for general overdetermined elliptic problems with nonconstant source

problems with possibly degenerate ellipticity,

Abstract. Starting from the quantitative isoperimetric inequality [21, 17], we prove a sharp quantitative version of the Cheeger inequality. A Cheeger set E for an open subset Ω ⊂ Rn, n ≥ 2, is any minimizer of the variational problem

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The 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.