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Diffusion of General Data on NonFlat Manifolds via Harmonic Maps Theory: The Direction Diffusion Case
 Journal Computer Vision
, 2000
"... Abstract. In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to denoise directional data and to obtain multiscale representati ..."
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Cited by 52 (6 self)
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Abstract. In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to denoise directional data and to obtain multiscale representations of it. The basic idea is to apply and extend results from the theory of harmonic maps, and in particular, harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the intrinsic unit norm constraint of directional data. We show the corresponding variational and partial differential equations formulations for isotropic diffusion, obtained from an L2 norm, and edge preserving diffusion, obtained from an L p norm in general and an L1 norm in particular. In contrast with previous approaches, the framework is valid for directions in any dimensions, supports nonsmooth data, and gives both isotropic and anisotropic formulations. In addition, the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general nonflat manifolds, that is, functions between two general manifolds. We present a number of theoretical results, open questions, and examples for gradient vectors, optical flow, and color images.
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...
The Dirichlet Problem for the Total Variation Flow
, 2001
"... We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use th ..."
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Cited by 20 (7 self)
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We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions.
A variational approach to remove multiplicative noise
, 2006
"... Abstract. This paper focuses on the problem of multiplicative noise removal. We draw our inspiration from the modeling of speckle noise. By using a MAP estimator, we can derive a functional whose minimizer corresponds to the denoised image we want to recover. Although the functional is not convex, w ..."
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Cited by 19 (0 self)
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Abstract. This paper focuses on the problem of multiplicative noise removal. We draw our inspiration from the modeling of speckle noise. By using a MAP estimator, we can derive a functional whose minimizer corresponds to the denoised image we want to recover. Although the functional is not convex, we prove the existence of a minimizer and we show the capability of our model on some numerical examples. We study the associated evolution problem, for which we derive existence and uniqueness results for the solution. We prove the convergence of an implicit scheme to compute the solution. Key words. Calculus of variation, functional analysis, BV, variational approach, multiplicative noise, speckle noise, image restoration. AMS subject classifications. 68U10, 94A08, 49J40, 35A15, 35B45, 35B50. 1. Introduction. Image
A Mathematical Study of the Relaxed Optical Flow Problem in the Space BV(Ω)
 SIAM J. MATH. ANAL
, 1999
"... This paper describes a variational approach for estimating a discontinuous optical flow from a sequence of images. Defined as the apparent motion of the image brightness pattern, the optical flow is very important in the computer vision community where its accurate estimation is strongly needed. Aft ..."
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Cited by 9 (0 self)
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This paper describes a variational approach for estimating a discontinuous optical flow from a sequence of images. Defined as the apparent motion of the image brightness pattern, the optical flow is very important in the computer vision community where its accurate estimation is strongly needed. After a fast overview of existing methods, we present a new variational method that we study in the space of Bounded Variations. We first present an integral representation of the optical flow problem which appears to be not lower semicontinuous. The relaxed functional is then calculated. We conclude by challenging questions about the possible numerical analysis of the abstract results.
A Parabolic Quasilinear Problem for Linear Growth Functionals
"... We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of energy functional we consider is the one given by the nonparametric area integrand f(x; ) = p 1 + kk ..."
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Cited by 7 (2 self)
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We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of energy functional we consider is the one given by the nonparametric area integrand f(x; ) = p 1 + kk 2 , which corresponds with the timedependent minimal surface equation. We also study the asymptotic behavoiur of the solutions.
Existence and Uniqueness of Solution for a Parabolic Quasilinear Problem for Linear Growth Functionals with L¹ Data
, 2001
"... We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of energy functional we consider is the one given by the nonparametric area integrand f(x; ) = p 1 + kk ..."
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Cited by 2 (0 self)
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We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of energy functional we consider is the one given by the nonparametric area integrand f(x; ) = p 1 + kk 2 , which corresponds with the timedependent minimal surface equation. We also study the asimptotic behavoiur of the solutions.
NONTRIVIAL SOLUTIONS OF QUASILINEAR EQUATIONS IN BV
"... Abstract. The existence of a nontrivial critical point is proved for a functional containing an areatype term. Techniques of nonsmooth critical point theory are applied. 1. Introduction. Let Ω be a bounded open subset of R n (n ≥ 3) and g: Ω × R → R a Carathéodory function with g(x,0) = 0. A class ..."
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Abstract. The existence of a nontrivial critical point is proved for a functional containing an areatype term. Techniques of nonsmooth critical point theory are applied. 1. Introduction. Let Ω be a bounded open subset of R n (n ≥ 3) and g: Ω × R → R a Carathéodory function with g(x,0) = 0. A classical result of Ambrosetti and Rabinowitz [1, 12, 13] says that the semilinear problem −∆u = g(x,u) in Ω
STRICT INTERIOR APPROXIMATION OF SETS OF FINITE PERIMETER AND FUNCTIONS OF BOUNDED VARIATION
"... Abstract. It is well known that sets of finite perimeter can be strictly approximated by smooth sets, while, in general, one cannot hope to approximate an open set Ω of finite perimeter in R n strictly from within. In this note we show that, nevertheless, the latter type of approximation is possible ..."
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Abstract. It is well known that sets of finite perimeter can be strictly approximated by smooth sets, while, in general, one cannot hope to approximate an open set Ω of finite perimeter in R n strictly from within. In this note we show that, nevertheless, the latter type of approximation is possible under the mild hypothesis that the (n−1)dimensional Hausdorff measure of the topological boundary ∂Ω equals the perimeter of Ω. We also discuss an optimality property of this hypothesis, and we establish a corresponding result on strict approximation of BVfunctions from a prescribed Dirichlet class.