The Experts below are selected from a list of 408 Experts worldwide ranked by ideXlab platform
Valentino Magnani - One of the best experts on this subject based on the ideXlab platform.
-
On a general Coarea inequality and applications
2020Co-Authors: Valentino MagnaniAbstract:Abstract. We prove a Coarea inequality for Lipschitz maps between stratified groups. As a consequence we obtain a Sard-type theorem and the nonexistence of nontrivial Coarea Formulae between Heisenberg groups. In the case of real valued Lipschitz maps on the Heisenberg group we get a Coarea Formula using the Q − 1 spherical Hausdorff measure restricted to level sets, where Q is the homogeneous dimension of the group
-
Note on Coarea Formulae in the Heisenberg group
2016Co-Authors: Valentino MagnaniAbstract:We show a rst nontrivial example of Coarea Formula for vector-valued Lipschitz maps dened on the three dimensional Heisenberg group. In this Coarea Formula, inte-gration on level sets is performed with respect to the 2-dimensional spherical Hausdor measure, built by the Carnot-Caratheodory distance. The standard jacobian is replaced by the so called \horizontal jacobian", corresponding to the jacobian of the Pansu dif-ferential of the Lipschitz map. Joining previous results, we achieve all possible Coarea Formulae for Lipschitz maps dened on the Heisenberg group. Mathematics Subject Classication: 28A75 (22E25) Key words and phrases: Coarea Formula, Heisenberg grou
-
Publ. Mat. 48 (2004), 409{422 NOTE ON Coarea FormulaE IN THE HEISENBERG GROUP
2016Co-Authors: Valentino MagnaniAbstract:We show a rst nontrivial example of Coarea Formula for vector-valued Lipschitz maps dened on the three dimensional Heisen-berg group. In this Coarea Formula, integration on level sets is performed with respect to the 2-dimensional spherical Hausdor measure, built by the Carnot-Caratheodory distance. The stan-dard jacobian is replaced by the so called \horizontal jacobian", corresponding to the jacobian of the Pansu dierential of the Lips-chitz map. Joining previous results, we achieve all possible Coarea Formulae for Lipschitz maps dened on the Heisenberg group. 1
-
Area implies Coarea
2011Co-Authors: Valentino MagnaniAbstract:We regard one side of the Coarea Formula as a measure and compute its density by an area-type Formula. As an application, we show the first nontrivial example of Coarea Formula for vector-valued sub-Riemannian Lipschitz mappings
-
AREA IMPLIES Coarea
2010Co-Authors: Valentino MagnaniAbstract:Abstract. We regard one side of the Coarea Formula as a measure and compute its density by an area-type Formula. As an application, we show the first nontrivial example of Coarea Formula for vector-valued sub-Riemannian Lipschitz mappings. Content
Mathieu Desbrun - One of the best experts on this subject based on the ideXlab platform.
-
A Variational Approach to Eulerian Geometry Processing
2008Co-Authors: Patrick Mullen, A Mckenzie, Yiying Tong, Mathieu DesbrunAbstract:Figure 1: Eulerian Geometry: our geometry processing framework offers a fully Eulerian variational approach to (left) outward and inward surface offset (here spatially varying by height), (center) simultaneous smoothing of foliations (all isosurfaces of volumetric medical data), and (right) a conservative mass advection for incompressible fluid simulation. We present a purely Eulerian framework for geometry processing of surfaces and foliations. Contrary to current Eulerian methods used in graphics, we use conservative methods and a variational interpretation, offering a unified framework for routine surface operations such as smoothing, offsetting, and animation. Computations are performed on a fixed volumetric grid without recourse to Lagrangian techniques such as triangle meshes, particles, or path tracing. At the core of our approach is the use of the Coarea Formula to express area integrals over isosurfaces as volume integrals. This enables the simultaneous processing of multiple isosurfaces, while a single interface can be treated as the special case of a dense foliation. We show that our method is a powerful alternative to conventional geometric representations in delicate cases such as the handling of high-genus surfaces, weighted offsetting, foliation smoothing of medical datasets, and incompressible fluid animation
-
a variational approach to eulerian geometry processing
International Conference on Computer Graphics and Interactive Techniques, 2007Co-Authors: Patrick Mullen, A Mckenzie, Yiying Tong, Mathieu DesbrunAbstract:We present a purely Eulerian framework for geometry processing of surfaces and foliations. Contrary to current Eulerian methods used in graphics, we use conservative methods and a variational interpretation, offering a unified framework for routine surface operations such as smoothing, offsetting, and animation. Computations are performed on a fixed volumetric grid without recourse to Lagrangian techniques such as triangle meshes, particles, or path tracing. At the core of our approach is the use of the Coarea Formula to express area integrals over isosurfaces as volume integrals. This enables the simultaneous processing of multiple isosurfaces, while a single interface can be treated as the special case of a dense foliation. We show that our method is a powerful alternative to conventional geometric representations in delicate cases such as the handling of high-genus surfaces, weighted offsetting, foliation smoothing of medical datasets, and incompressible fluid animation.
Maria Karmanova - One of the best experts on this subject based on the ideXlab platform.
-
a Coarea Formula for smooth contact mappings of carnot caratheodory spaces
Acta Applicandae Mathematicae, 2013Co-Authors: Maria Karmanova, S VodopyanovAbstract:We prove the Coarea Formula for sufficiently smooth contact mappings of Carnot manifolds to Carnot---Caratheodory spaces. In particular, we investigate level surfaces of these mappings, and compare Riemannian and sub-Riemannian measures on them. Our main tool is the sharp asymptotic behavior of the Riemannian measure of the intersection of a tangent plane to a level surface and a sub-Riemannian ball. This calculation in particular implies that the sub-Riemannian measure of the set of characteristic points (i.e., the points at which the sub-Riemannian differential is degenerate) equals zero on almost every level set.
-
geometry of carnot caratheodory spaces differentiability Coarea and area Formulas
2009Co-Authors: Maria Karmanova, Sergey Vodop YanovAbstract:We compare geometries of two different local Lie groups in a Carnot-Caratheodory space, and obtain quantitative estimates of their difference. This result is extended to Carnot-Caratheodory spaces with C1,α-smooth basis vector fields, α ∈ [0, 1], and the dependence of the estimates on α is established. From here we obtain the similar estimates for comparing geometries of a Carnot-Caratheodory space and a local Lie group. These results base on Gromov’s Theorem on nilpotentization of vector fields for which we give new and simple proof. All the above imply basic results of the theory: Gromov type Local Approximation Theorems, and for α > 0 Rashevskiǐ-Chow Theorem and Ball-Box Theorem, etc. We apply the obtained results for proving hc-differentiability of mappings of Carnot-Caratheodory spaces with continuous horizontal derivatives. The latter is used in proving the Coarea Formula for smooth contact mappings of Carnot-Caratheodory spaces, and the area Formula for Lipschitz (with respect to sub-Riemannian metrics) mappings of Carnot-Caratheodory spaces.
-
Rectifiable sets and Coarea Formula for metric-valued mappings
Journal of Functional Analysis, 2008Co-Authors: Maria KarmanovaAbstract:We study Lipschitz mappings defined on an Hn-rectifiable metric space with values in an arbitrary metric space. We find necessary and sufficient conditions on the image and the preimage of a mapping for the validity of the Coarea Formula. As a consequence, we prove the Coarea Formula for some classes of mappings with Hk-σ-finite image. We also obtain a metric analog of the Implicit Function Theorem. All these results are extended to large classes of mappings with values in a metric space, including Sobolev mappings and BV-mappings.
Patrick Mullen - One of the best experts on this subject based on the ideXlab platform.
-
A Variational Approach to Eulerian Geometry Processing
2008Co-Authors: Patrick Mullen, A Mckenzie, Yiying Tong, Mathieu DesbrunAbstract:Figure 1: Eulerian Geometry: our geometry processing framework offers a fully Eulerian variational approach to (left) outward and inward surface offset (here spatially varying by height), (center) simultaneous smoothing of foliations (all isosurfaces of volumetric medical data), and (right) a conservative mass advection for incompressible fluid simulation. We present a purely Eulerian framework for geometry processing of surfaces and foliations. Contrary to current Eulerian methods used in graphics, we use conservative methods and a variational interpretation, offering a unified framework for routine surface operations such as smoothing, offsetting, and animation. Computations are performed on a fixed volumetric grid without recourse to Lagrangian techniques such as triangle meshes, particles, or path tracing. At the core of our approach is the use of the Coarea Formula to express area integrals over isosurfaces as volume integrals. This enables the simultaneous processing of multiple isosurfaces, while a single interface can be treated as the special case of a dense foliation. We show that our method is a powerful alternative to conventional geometric representations in delicate cases such as the handling of high-genus surfaces, weighted offsetting, foliation smoothing of medical datasets, and incompressible fluid animation
-
a variational approach to eulerian geometry processing
International Conference on Computer Graphics and Interactive Techniques, 2007Co-Authors: Patrick Mullen, A Mckenzie, Yiying Tong, Mathieu DesbrunAbstract:We present a purely Eulerian framework for geometry processing of surfaces and foliations. Contrary to current Eulerian methods used in graphics, we use conservative methods and a variational interpretation, offering a unified framework for routine surface operations such as smoothing, offsetting, and animation. Computations are performed on a fixed volumetric grid without recourse to Lagrangian techniques such as triangle meshes, particles, or path tracing. At the core of our approach is the use of the Coarea Formula to express area integrals over isosurfaces as volume integrals. This enables the simultaneous processing of multiple isosurfaces, while a single interface can be treated as the special case of a dense foliation. We show that our method is a powerful alternative to conventional geometric representations in delicate cases such as the handling of high-genus surfaces, weighted offsetting, foliation smoothing of medical datasets, and incompressible fluid animation.
Desolneux Agnès - One of the best experts on this subject based on the ideXlab platform.
-
On quantitative Laplace-type convergence results for some exponential probability measures, with two applications
2021Co-Authors: De Bortoli Valentin, Desolneux AgnèsAbstract:Laplace-type results characterize the limit of sequence of measures $(\pi_\varepsilon)_{\varepsilon >0}$ with density w.r.t the Lebesgue measure $(\mathrm{d} \pi_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto \exp[-U(x)/\varepsilon]$ when the temperature $\varepsilon>0$ converges to $0$. If a limiting distribution $\pi_0$ exists, it concentrates on the minimizers of the potential $U$. Classical results require the invertibility of the Hessian of $U$ in order to establish such asymptotics. In this work, we study the particular case of norm-like potentials $U$ and establish quantitative bounds between $\pi_\varepsilon$ and $\pi_0$ w.r.t. the Wasserstein distance of order $1$ under an invertibility condition of a generalized Jacobian. One key element of our proof is the use of geometric measure theory tools such as the Coarea Formula. We apply our results to the study of maximum entropy models (microcanonical/macrocanonical distributions) and to the convergence of the iterates of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm at low temperatures for non-convex minimization
-
On quantitative Laplace-type convergence results for some exponential probability measures, with two applications
HAL CCSD, 2021Co-Authors: De Bortoli Valentin, Desolneux AgnèsAbstract:Laplace-type results characterize the limit of sequence of measures (πε)ε>0 with density w.r.t the Lebesgue measure (dπε/dλ)(x) ∝ exp[−U (x)/ε] when the temperature ε > 0 converges to 0. If a limiting distribution π0 exists, it concentrates on the minimizers of the potential U. Classical results require the invertibility of the Hessian of U in order to establish such asymptotics. In this work, we study the particular case of norm-like potentials U and establish quantitative bounds between πε and π0 w.r.t. the Wasserstein distance of order 1 under an invertibility condition of a generalized Jacobian. One key element of our proof is the use of geometric measure theory tools such as the Coarea Formula. We apply our results to the study of maximum entropy models (microcanonical/macrocanonical distributions) and to the convergence of the iterates of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm at low temperatures for non-convex minimization