Results 1  10
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320
Computable elastic distances between shapes
 SIAM J. of Applied Math
, 1998
"... Abstract. We define distances between geometric curves by the square root of the minimal energy required to transform one curve into the other. The energy is formally defined from a left invariant Riemannian distance on an infinite dimensional group acting on the curves, which can be explicitly comp ..."
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Cited by 120 (19 self)
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Abstract. We define distances between geometric curves by the square root of the minimal energy required to transform one curve into the other. The energy is formally defined from a left invariant Riemannian distance on an infinite dimensional group acting on the curves, which can be explicitly computed. The obtained distance boils down to a variational problem for which an optimal matching between the curves has to be computed. An analysis of the distance when the curves are polygonal leads to a numerical procedure for the solution of the variational problem, which can efficiently be implemented, as illustrated by experiments.
Error Estimate for Approximate Solutions of a Nonlinear ConvectionDiffusion Problem
, 2002
"... This paper proves the estimate ku" uk L 1 (Q T ) C" 1=5 , where, for all " > 0, u" is the weak solution of (u" ) t + div(q f(u" )) ('(u")+"u") = 0 with initial and boundary conditions, u is the entropy weak solution of u t + div(qf(u)) (&apos ..."
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Cited by 37 (13 self)
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This paper proves the estimate ku" uk L 1 (Q T ) C" 1=5 , where, for all " > 0, u" is the weak solution of (u" ) t + div(q f(u" )) ('(u")+"u") = 0 with initial and boundary conditions, u is the entropy weak solution of u t + div(qf(u)) ('(u)) = 0 with the same initial and boundary conditions, and C > 0 does not depend on ". The domain
Convergence of a Finite Volume Scheme for Nonlinear Degenerate Parabolic Equations.
, 2002
"... One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation u t + div(qf(u)) '(u) = 0 by a piecewise constant function uD using a discretization D in space and time and a finite volume scheme. The convergence of uD to u is shown as the size of the space and time st ..."
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Cited by 36 (14 self)
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One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation u t + div(qf(u)) '(u) = 0 by a piecewise constant function uD using a discretization D in space and time and a finite volume scheme. The convergence of uD to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on uD are used to prove the convergence, up to a subsequence, of uD to a measure valued entropy solution (called here an entropy process solution). A result of uniqueness of the entropy process solution is proved, yielding the strong convergence of uD to u. Some numerical results on a model equation are shown.
Global Solutions of some Chemotaxis and Angiogenesis Systems in high space dimensions
, 2003
"... We consider two simple conservative systems of parabolicelliptic and parabolicdegenerate type arising in modeling chemotaxis and angiogenesis. Both systems share the same property that when the L d 2 norm of initial data is small enough, where d ≥ 2 is the space dimension, then there is a global ( ..."
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Cited by 34 (5 self)
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We consider two simple conservative systems of parabolicelliptic and parabolicdegenerate type arising in modeling chemotaxis and angiogenesis. Both systems share the same property that when the L d 2 norm of initial data is small enough, where d ≥ 2 is the space dimension, then there is a global (in time) weak solution that stays in all the Lp spaces with max{1; d 2 − 1} ≤ p < ∞. This result is already known for the parabolicelliptic system of chemotaxis, moreover blowup can occur in finite time for large initial data and Dirac concentrations can occur. For the parabolicdegenerate system of angiogenesis in two dimensions, we also prove that weak solutions (which are equiintegrable in L¹) exist even for large initial data. But breakdown of regularity or propagation of smoothness is an open problem.
On traces for functional spaces related to Maxwell's Equations  Part I: An integration...
, 1999
"... Part I. The aim of this paper is to study the tangential trace and tangential components of fields which belong to the space H(curl ;\Omega\Gamma2 when\Omega is a polyhedron with Lipschitz continuous boundary. The appropriate functional setting is developed in order to suitably define these traces ..."
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Cited by 32 (2 self)
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Part I. The aim of this paper is to study the tangential trace and tangential components of fields which belong to the space H(curl ;\Omega\Gamma2 when\Omega is a polyhedron with Lipschitz continuous boundary. The appropriate functional setting is developed in order to suitably define these traces on the whole boundary and on a part of it (for partially vanishing fields and general ones.) In both cases it is possible to define ad hoc dualities among tangential trace and tangential components and the validity of two related integration by parts formulae is provided. Part II. Hodge decompositions of tangential vector fields defined on piecewise regular manifolds are provided. The first step is the study of L 2 tangential fields and then the attention is focused on some particular Sobolev spaces of order \Gamma1=2. In order to reach this goal, it is required to properly define the first order differential operators and to study their properties. When the manifold \Gamma is the boundar...
Auxiliary variables and twostep iterative algorithms in computer vision problems
 J. Math. Imag. Vision
, 1995
"... Abstract. We present a new mathematical formulation of some curve and surface reconstmctien algorithms by the introduction of auxiliary variables. For deformable models and templates, the extraction of a shape is obtained through the minimization of an energy composed of an internal regularization t ..."
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Cited by 31 (6 self)
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Abstract. We present a new mathematical formulation of some curve and surface reconstmctien algorithms by the introduction of auxiliary variables. For deformable models and templates, the extraction of a shape is obtained through the minimization of an energy composed of an internal regularization term (not necessary in the case of parametric models) and an external attraction potential. Twostep iterative algorithms have been often used where, at each iteration, the model is first locally deformed according to the potential data attraction and then globally smoothed (or fitted in the parametric case). We show how these approaches can be interpreted as the introduction of auxiliary variables and the minimization of a twovariables energy. The first variable corresponds to the original model we are looking for, while the second variable represents an auxiliary shape close to the first one. This permits to transform an implicit data constraint defined by a non convex potential into an explicit convex reconstruction problem. This approach is much simpler since each iteration is composed of two simple to solve steps. Our formulation permits a more precise setting of parameters in the iterative scheme to ensure convergence to a minimum. We show some mathematical properties and results on this new auxiliary problem, in particular when the potential is a function of the distance to the closest feature point. We then illustrate our approach for some deformable models and templates.
Exponential decay for the fragmentation or celldivision equation
 J. Diff. Eq
, 2003
"... We consider a classical integrodifferential equation that arises in various applications as a model for celldivision or fragmentation. In biology, it describes the evolution of the density of cells that grow and divide. We prove the existence of a stable steady dynamics (first positive eigenvector ..."
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Cited by 25 (7 self)
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We consider a classical integrodifferential equation that arises in various applications as a model for celldivision or fragmentation. In biology, it describes the evolution of the density of cells that grow and divide. We prove the existence of a stable steady dynamics (first positive eigenvector) under general assumptions in the variable coefficients case. We also prove the exponential convergence, for large times, of solutions toward such a steady state.
A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws
 IEEE Transactions on Automatic Control
, 2007
"... Abstract—We present a strict Lyapunov function for hyperbolic systems of conservation laws that can be diagonalized with Riemann invariants. The time derivative of this Lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions. It is shown that the ..."
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Cited by 23 (8 self)
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Abstract—We present a strict Lyapunov function for hyperbolic systems of conservation laws that can be diagonalized with Riemann invariants. The time derivative of this Lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions. It is shown that the derived boundary control allows to guarantee the local convergence of the state towards a desired set point. Furthermore, the control can be implemented as a feedback of the state only measured at the boundaries. The control design method is illustrated with an hydraulic application, namely the level and flow regulation in an horizontal open channel. Index Terms—Boundary control, conservation laws, hyperbolic systems, Lyapunov function, partial differential equations. I.
Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 21 (7 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
LOCAL GEOMETRY OF DEFORMABLE TEMPLATES
, 2005
"... In this paper, we discuss a geometrical model of a space of deformable images or shapes, in which infinitesimal variations are combinations of elastic deformations (warping) and of photometric variations. Geodesics in this space are related to velocitybased image warping methods, which have proved ..."
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Cited by 21 (6 self)
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In this paper, we discuss a geometrical model of a space of deformable images or shapes, in which infinitesimal variations are combinations of elastic deformations (warping) and of photometric variations. Geodesics in this space are related to velocitybased image warping methods, which have proved to yield efficient and robust estimations of diffeomorphisms in the case of large deformation. Here, we provide a rigorous and general construction of this infinite dimensional “shape manifold ” on which we place a Riemannian metric. We then obtain the geodesic equations, for which we show the existence and uniqueness of solutions for all times. We finally use this to provide a geometrically founded linear approximation of the deformations of shapes in the neighborhood of a given template.