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16
Applications of parametric maxflow in computer vision
"... 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 ..."
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Cited by 39 (7 self)
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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 highknowledge 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.
An introduction to total variation for image analysis
 in Theoretical Foundations and Numerical Methods for Sparse Recovery, De Gruyter
, 2010
"... These notes address various theoretical and practical topics related to Total Variationbased image reconstruction. They focuse first on some theoretical results on functions which minimize the total variation, and in a second part, describe a few standard and less standard algorithms to minimize th ..."
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Cited by 16 (2 self)
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These notes address various theoretical and practical topics related to Total Variationbased image reconstruction. They focuse first on some theoretical results on functions which minimize the total variation, and in a second part, describe a few standard and less standard algorithms to minimize the total variation in a finitedifferences setting, with a series of applications from simple denoising to stereo, or deconvolution issues, and even more exotic uses like the minimization of minimal partition problems.
Uniqueness of the Cheeger set of a convex body
, 2007
"... 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. ..."
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Cited by 11 (3 self)
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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.
The rst eigenvalue of the Laplacian, isoperimetric constants, and the max ow mincut theorem
 Arch. Math
, 2006
"... Abstract. We show how ’test ’ vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical M ..."
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Cited by 8 (0 self)
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Abstract. We show how ’test ’ vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical Max Flow Min Cut Theorem for networks implies that Cheeger’s constant may be obtained precisely from such vector fields. Finally, we apply these ideas to reprove a known lower bound for Cheeger’s constant in terms of the inradius of a plane domain. 1.
The TV ? L1 model : a geometric point of view
 SIAM Journal on Multiscale Modeling and Simulation
, 2009
"... The aim of this paper is to investigate the geometrical behavior of the TVL1 model used in image processing, by making use of the notion of Cheeger sets. This mathematical concept was recently related to the celebrated RudinOsherFatemi image restoration model, yielding important advances in both f ..."
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Cited by 6 (0 self)
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The aim of this paper is to investigate the geometrical behavior of the TVL1 model used in image processing, by making use of the notion of Cheeger sets. This mathematical concept was recently related to the celebrated RudinOsherFatemi image restoration model, yielding important advances in both fields. We provide the reader with a geometrical characterization of the TVL1 model. We show that, in the convex case, exact solutions of the TVL1 problem are given by an opening followed by a simple test over the ratio perimeter/area. Shapes remain or suddenly vanish depending on this test. As a result of our theoritical study, we suggest a new and efficient numerical scheme to apply the model to digital images. As a byproduct, we justify the use of the TVL1 model for image decomposition, by establishing a connection between the model and morphological granulometry. Eventually, we propose an extension of TVL1 into an adaptive framework, in which we derive some theoretical results.
On the selection of maximal Cheeger sets
 Differential and Integral Equations
, 2007
"... 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 wellknown fact. Uniqueness is a more delicate issue and is not t ..."
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Cited by 6 (2 self)
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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 wellknown 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 pLaplacian 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.