Results 1  10
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501
A Theoretical Framework for Convex Regularizers in PDEBased Computation of Image Motion
, 2000
"... Many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy. Typically, these energy functionals consist of two terms: a data term, which requires e.g. that a brightness consta ..."
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Cited by 76 (20 self)
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Many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy. Typically, these energy functionals consist of two terms: a data term, which requires e.g. that a brightness constancy assumption holds, and a regularizer that encourages global or piecewise smoothness of the flow field. In this paper we present a systematic classification of rotation invariant convex regularizers by exploring their connection to diffusion filters for multichannel images. This taxonomy provides a unifying framework for datadriven and flowdriven, isotropic and anisotropic, as well as spatial and spatiotemporal regularizers. While some of these techniques are classic methods from the literature, others are derived here for the first time. We prove that all these methods are wellposed: they posses a unique solution that depends in a continuous way on the initial data. An interesting structural relation between isotropic and anisotropic flowdriven regularizers is identified, and a design criterion is proposed for constructing anisotropic flowdriven regularizers in a simple and direct way from isotropic ones. Its use is illustrated by several examples.
Solving monotone inclusions via compositions of nonexpansive averaged operators
 Optimization
, 2004
"... A unified fixed point theoretic framework is proposed to investigate the asymptotic behavior of algorithms for finding solutions to monotone inclusion problems. The basic iterative scheme under consideration involves nonstationary compositions of perturbed averaged nonexpansive operators. The analys ..."
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Cited by 64 (21 self)
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A unified fixed point theoretic framework is proposed to investigate the asymptotic behavior of algorithms for finding solutions to monotone inclusion problems. The basic iterative scheme under consideration involves nonstationary compositions of perturbed averaged nonexpansive operators. The analysis covers proximal methods for common zero problems as well as various splitting methods for finding a zero of the sum of monotone operators.
A douglasRachford splitting approach to nonsmooth convex variational signal recovery
 IEEE Journal of Selected Topics in Signal Processing
, 2007
"... Abstract — Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is propo ..."
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Cited by 49 (15 self)
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Abstract — Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is proposed to solve it. The convergence of the method, which is based on the DouglasRachford algorithm for monotone operatorsplitting, is obtained under general conditions. Applications to nonGaussian image denoising in a tight frame are also demonstrated. Index Terms — Convex optimization, denoising, DouglasRachford, frame, nondifferentiable optimization, Poisson noise,
A Study in the BV Space of a DenoisingDeblurring Variational Problem
, 2001
"... In this paper we study, in the framework of functions of bounded variation, a general variational problem arising in image recovery, introduced in [3]. We prove the existence and the uniqueness of a solution using lower semicontinuity results for convex functionals of measures. We also give a new an ..."
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Cited by 47 (9 self)
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In this paper we study, in the framework of functions of bounded variation, a general variational problem arising in image recovery, introduced in [3]. We prove the existence and the uniqueness of a solution using lower semicontinuity results for convex functionals of measures. We also give a new and fine characterization of the subdifferential of the functional, together with optimality conditions on the solution, using duality techniques of Temam for the theory of timedependent minimal surfaces. We study the associated evolution equation in the context of nonlinear semigroup theory and we give an approximation result in continuous variables, using #convergence. Finally, we discretize the problems by finite differences schemes and we present several numerical results for signal and image reconstruction.
Rates of Convergence of Posterior Distributions
, 1998
"... We compute the rate at which the posterior distribution concentrates around the true parameter value. The spaces we work in are quite general and include infinite dimensional cases. The rates are driven by two quantities: the size of the space, as measure by metric entropy or bracketing entropy, and ..."
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Cited by 47 (0 self)
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We compute the rate at which the posterior distribution concentrates around the true parameter value. The spaces we work in are quite general and include infinite dimensional cases. The rates are driven by two quantities: the size of the space, as measure by metric entropy or bracketing entropy, and the degree to which the prior concentrates in a small ball around the true parameter. We apply the results to several examples. In some cases, natural priors give suboptimal rates of convergence and better rates can be obtained by using sievebased priors such as those introduced by Zhao (1993, 1998). AMS 1990 classification: Primary, 62A15, Secondary: 62E20, 62G15. KEYWORDS: Bayesian inference, asymptotic inference, nonparametric models, sieves. 1 Introduction. Nonparametric Bayesian methods have become quite popular lately, largely because of advances in computing; see Dey, Mueller and Sinha (1998) for a recent account. Because of their growing popularity, it is important to understand ...
Multiple Boundary Peak Solutions For Some Singularly Perturbed Neumann Problems
"... We consider the problem ae " 2 \Deltau \Gamma u + f(u) = 0 in\Omega u ? 0 in\Omega ; @u @ = 0 on @\Omega ; where\Omega is a bounded smooth domain in R N , " ? 0 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike ..."
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Cited by 46 (34 self)
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We consider the problem ae " 2 \Deltau \Gamma u + f(u) = 0 in\Omega u ? 0 in\Omega ; @u @ = 0 on @\Omega ; where\Omega is a bounded smooth domain in R N , " ? 0 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions such that the spike concentrates, as " approaches zero, at a critical point of the mean curvature function H(P ); P 2 @ It is also known that this equation has multiple boundary spike solutions at multiple nondegenerate critical points of H(P ) or multiple local maximum points of H(P ). In this paper, we prove that for any fixed positive integer K there exist boundary K \Gamma peak solutions at a local minimum point of H(P ). This implies that for any smooth and bounded domain there always exist boundary K \Gamma peak solutions. We first use the LiapunovSchmidt method to reduce the problem to finite dimensions. Then we use a maximizing procedure to obtain multiple boundary spikes. 1.
ON METHODS OF SIEVES AND PENALIZATION
, 1997
"... We develop a general theory which provides a unified treatment for the asymptotic normality and efficiency of the maximum likelihood estimates (MLE’s) in parametric, semiparametric and nonparametric models. We find that the asymptotic behavior of substitution estimates for estimating smooth function ..."
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Cited by 44 (1 self)
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We develop a general theory which provides a unified treatment for the asymptotic normality and efficiency of the maximum likelihood estimates (MLE’s) in parametric, semiparametric and nonparametric models. We find that the asymptotic behavior of substitution estimates for estimating smooth functionals are essentially governed by two indices: the degree of smoothness of the functional and the local size of the underlying parameter space. We show that when the local size of the parameter space is not very large, the substitution standard (nonsieve), substitution sieve and substitution penalized MLE’s are asymptotically efficient in the Fisher sense, under certain stochastic equicontinuity conditions of the loglikelihood. Moreover, when the convergence rate of the estimate is slow, the degree of smoothness of the functional needs to compensate for the slowness of the rate in order to achieve efficiency. When the size of the parameter space is very large, the standard and penalized maximum likelihood procedures may be inefficient, whereas the method of sieves may be able to overcome this difficulty. This phenomenon is particularly manifested when the functional of interest is very smooth, especially in the semiparametric case.
Global attractors for damped semilinear wave equations. Partial differential equations and applications, Discrete Contin
 Dyn. Syst
"... Abstract. The existence of a global attractor in the natural energy space is proved for the semilinear wave equation utt + βut − ∆u + f(u) = 0 on a bounded domain Ω ⊂ Rn with Dirichlet boundary conditions. The nonlinear term f is supposed to satisfy an exponential growth condition for n =2,andforn≥ ..."
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Cited by 36 (0 self)
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Abstract. The existence of a global attractor in the natural energy space is proved for the semilinear wave equation utt + βut − ∆u + f(u) = 0 on a bounded domain Ω ⊂ Rn with Dirichlet boundary conditions. The nonlinear term f is supposed to satisfy an exponential growth condition for n =2,andforn≥3thegrowth condition f(u)  ≤c0(u  γ +1), where1≤γ≤ n. No Lipschitz condition on f n−2 is assumed, leading to presumed nonuniqueness of solutions with given initial data. The asymptotic compactness of the corresponding generalized semiflow is proved using an auxiliary functional. The system is shown to possess Kneser’s property, which implies the connectedness of the attractor. In the case n ≥ 3andγ> n the existence of a global attractor is proved under n−2 the (unproved) assumption that every weak solution satisfies the energy equation. Dedicated to M.I. Vishik on the occasion of his 80 th birthday
Metamorphoses Through Lie Group Action
 FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
, 2005
"... We formally analyze a computational problem which has important applications in image understanding and shape analysis. The problem can be summarized as follows. Starting from a group action on a Riemannian manifold M, we introduce a modification of the metric by partly expressing displacements on M ..."
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Cited by 31 (8 self)
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We formally analyze a computational problem which has important applications in image understanding and shape analysis. The problem can be summarized as follows. Starting from a group action on a Riemannian manifold M, we introduce a modification of the metric by partly expressing displacements on M as an effect of the action of some group element. The study of this new structure relates to evolutions on M under the combined effect of the action and of residual displacements, called metamorphoses. This can and has been applied to image processing problems, providing in particular diffeomorphic matching algorithms for pattern recognition.
Implicit Partial Differential Equations,” Birkhäuser
, 1999
"... Abstract. We study a Dirichlet problem associated to some nonlinear partial di¤erential equations under additional constraints that are relevant in non linear elasticity. We also give several examples related to the complex eikonal equation, optimal design, potential wells or nematic elastomers. ..."
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Cited by 26 (5 self)
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Abstract. We study a Dirichlet problem associated to some nonlinear partial di¤erential equations under additional constraints that are relevant in non linear elasticity. We also give several examples related to the complex eikonal equation, optimal design, potential wells or nematic elastomers.