Results 1  10
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64
Computing Optical Flow via Variational Techniques
 SIAM Journal on Applied Mathematics
, 1999
"... Defined as the apparent motion in a sequence of images, the optical flow is very important in the Computer Vision community where its accurate estimation is strongly needed for many applications. It is one of the most studied problem in Computer Vision. In spite of this, not much theoretical analysi ..."
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Cited by 46 (5 self)
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Defined as the apparent motion in a sequence of images, the optical flow is very important in the Computer Vision community where its accurate estimation is strongly needed for many applications. It is one of the most studied problem in Computer Vision. In spite of this, not much theoretical analysis has been done. In this article, we first present a review of existing variational methods. Then, we will propose an extended model that will be rigorously justified on the space of functions of bounded variations. Finally, we present an algorithm whose convergence will be carefully demonstrated. Some results showing the capabilities of this method will end that work.
Image Sequence Analysis via Partial Differential Equations
, 1999
"... This article deals with the problem of restoring and motion segmenting noisy image sequences with a static background. Usually, motion segmentation and image restoration are considered separately in image sequence restoration. Moreover, motion segmentation is often noise sensitive. In this article, ..."
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Cited by 44 (3 self)
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This article deals with the problem of restoring and motion segmenting noisy image sequences with a static background. Usually, motion segmentation and image restoration are considered separately in image sequence restoration. Moreover, motion segmentation is often noise sensitive. In this article, the motion segmentation and the image restoration parts are performed in a coupled way, allowing the motion segmentation part to positively influence the restoration part and viceversa. This is the key of our approach that allows to deal simultaneously with the problem of restoration and motion segmentation. To this end, we propose a theoretically justified optimization problem that permits to take into account both requirements. The model is theoretically justified. Existence and unicity are proved in the space of bounded variations. A suitable numerical scheme based on half quadratic minimization is then proposed and its convergence and stability demonstrated. Experimental results obtaine...
Restoration of color images by vector valued BV functions and variational calculus
 SIAM J. Appl. Math
, 2006
"... Abstract. We analyze a variational problem for the recovery of vector valued functions and we compute its numerical solution. The data of the problem are a small set of complete samples of the vector valued function and a significant incomplete information where the former are missing. The incomplet ..."
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Cited by 20 (11 self)
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Abstract. We analyze a variational problem for the recovery of vector valued functions and we compute its numerical solution. The data of the problem are a small set of complete samples of the vector valued function and a significant incomplete information where the former are missing. The incomplete information is assumed as the result of a distortion, with values in a lower dimensional manifold. For the recovery of the function we minimize a functional which is formed by the discrepancy with respect to the data and total variation regularization constraints. We show existence of minimizers in the space of vector valued BV functions. For the computation of minimizers we provide a stable and efficient method. First we approximate the functional by coercive functionals on W 1,2 in terms of Γconvergence. Then we realize approximations of minimizers of the latter functionals by an iterative procedure to solve the PDE system of the corresponding EulerLagrange equations. The numerical implementation comes naturally by finite element discretization. We apply the algorithm to the restoration of color images from a limited color information and gray levels where the colors are missing. The numerical experiments show that this scheme is very fast and robust. The reconstruction capabilities of the model are shown, also from very limited (randomly distributed) color data. Several examples are included from the real restoration problem of the A. Mantegna’s art frescoes in Italy.
Adaptive Finite Element Methods for Elliptic Equations With NonSmooth Coefficients
, 2000
"... We consider a secondorder elliptic equation with discontinuous or anisotropic coefficients in a bounded two or three dimensional domain, and its finiteelement discretization. The aim of this paper is to prove some a priori and a posteriori error estimates in an appropriate norm, which are independ ..."
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Cited by 20 (1 self)
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We consider a secondorder elliptic equation with discontinuous or anisotropic coefficients in a bounded two or three dimensional domain, and its finiteelement discretization. The aim of this paper is to prove some a priori and a posteriori error estimates in an appropriate norm, which are independent of the variation of the coefficients.
An optimal variance estimate in stochastic homogenization of discrete elliptic equations
 Ann. Probab
, 2011
"... We consider a discrete elliptic equation on the ddimensional lattice Zd with random coefficients A of the simplest type: they are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the so ..."
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Cited by 17 (10 self)
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We consider a discrete elliptic equation on the ddimensional lattice Zd with random coefficients A of the simplest type: they are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric “homogenized ” matrix Ahom = ahom Id is characterized by ξ · Ahomξ =〈(ξ +∇φ) · A(ξ +∇φ) 〉 for any direction ξ ∈ Rd, where the random field φ (the “corrector”) is the unique solution of −∇ ∗ · A(ξ +∇φ) = 0 such that φ(0) = 0, ∇φ is stationary and 〈∇φ〉=0, 〈· 〉 denoting the ensemble average (or expectation). It is known (“by ergodicity”) that the above ensemble average of the energy density E = (ξ +∇φ) · A(ξ +∇φ), which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of E on length scales L satisfies the optimal estimate, that is, var [ ∑ EηL] � L−d, where the averaging function [i.e., ∑ ηL = 1, supp(ηL) ⊂{x≤L}] has to be smooth in the sense that
Enhanced Electrical Impedance Tomography via the MumfordShah Functional
 ESAIM: Control, Optimization and Calculus of Variations
, 2001
"... We consider the problem of electrical impedance tomography where conductivity distribution in a domain is to be reconstructed from boundary measurements of voltage and currents. It is wellknown that this problem is highly illposed. In this work, we propose the use of the MumfordShah functional, dev ..."
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Cited by 14 (0 self)
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We consider the problem of electrical impedance tomography where conductivity distribution in a domain is to be reconstructed from boundary measurements of voltage and currents. It is wellknown that this problem is highly illposed. In this work, we propose the use of the MumfordShah functional, developed for segmentation and denoising of images, as a regularization. After establishing existence properties of the resulting variational problem, we proceed by demonstrating the approach in several numerical examples. Our results indicate that this is an eective approach for overcoming the illposedness. Moreover, it has the capability of enhancing the reconstruction while at the same time segmenting the conductivity image. 1 Introduction and formulation of the problem The purpose of this work is to demonstrate that the MumfordShah functional from image processing can be used eectively to regularize the classical problem of electrical impedance tomography. In electrical impedance tomogr...
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 13 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
A Finite Volume Scheme for a Noncoercive Elliptic Equation With Measure Data
, 2003
"... We show here the convergence of the finite volume approximate solutions of a convectiondiffusion equation to a weak solution, without the usual coercitivity assumption on the elliptic operator and with weak regularity assumptions on the data. Numerical experiments are performed to obtain some rates ..."
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Cited by 12 (8 self)
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We show here the convergence of the finite volume approximate solutions of a convectiondiffusion equation to a weak solution, without the usual coercitivity assumption on the elliptic operator and with weak regularity assumptions on the data. Numerical experiments are performed to obtain some rates of convergence in two and three space dimensions.
Bounds of Riesz transforms on L p spaces for second order elliptic operators
 Ann. Inst. Fourier
"... Abstract. Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on R n or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p> 2, a necessary and sufficient condition is obtained for the boundedness of the Rie ..."
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Cited by 10 (1 self)
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Abstract. Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on R n or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p> 2, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇(L) −1/2 on the L p space. As an application, for 1 < p < 3 + ε, we establish the L p boundedness of Riesz transforms on Lipschitz domains for operators with V MO coefficients. The range of p is sharp. The closely related boundedness of ∇(L) −1/2 on weighted L 2 spaces is also studied. 1.
A Finite Element Model for the TimeDependent Joule Heating Problem
 Math. Comp
, 1995
"... . We study a spatially semidiscrete and a completely discrete finite element model for a nonlinear system consisting of an elliptic and a parabolic partial differential equation describing the electric heating of a conducting body. We prove error bounds of optimal order under minimal regularity assu ..."
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Cited by 9 (1 self)
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. We study a spatially semidiscrete and a completely discrete finite element model for a nonlinear system consisting of an elliptic and a parabolic partial differential equation describing the electric heating of a conducting body. We prove error bounds of optimal order under minimal regularity assumptions when the number of spatial variables d 3. We establish the existence of solutions with the required regularity over arbitrarily long intervals of time when d 2. 1. Introduction In this note we consider the numerical approximation by the finite element method of the following nonlinear ellipticparabolic system (1.1) u t \Gamma \Deltau = oe(u)jrOEj 2 ; \Gamma r \Delta (oe(u)rOE) = 0; x 2\Omega ; t 2 [0; T ]; where u = u(x; t), OE = OE(x; t), u t = @u=@t, r denotes the gradient with respect to the xvariables and \Delta = r \Delta r is the Laplacian. These differential equations are studied for t in a finite interval [0; T ] and for x in a bounded convex polygonal domain\Ome...