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USER’S GUIDE TO VISCOSITY SOLUTIONS OF SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS
, 1992
"... The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking argume ..."
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Cited by 1410 (14 self)
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The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a selfcontained exposition of the basic theory of viscosity solutions.
Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers
, 2010
"... ..."
CoherenceEnhancing Diffusion Filtering
, 1999
"... The completion of interrupted lines or the enhancement of flowlike structures is a challenging task in computer vision, human vision, and image processing. We address this problem by presenting a multiscale method in which a nonlinear diffusion filter is steered by the socalled interest operato ..."
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Cited by 137 (3 self)
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The completion of interrupted lines or the enhancement of flowlike structures is a challenging task in computer vision, human vision, and image processing. We address this problem by presenting a multiscale method in which a nonlinear diffusion filter is steered by the socalled interest operator (secondmoment matrix, structure tensor). An mdimensional formulation of this method is analysed with respect to its wellposedness and scalespace properties. An efficient scheme is presented which uses a stabilization by a semiimplicit additive operator splitting (AOS), and the scalespace behaviour of this method is illustrated by applying it to both 2D and 3D images.
Reliable Estimation of Dense Optical Flow Fields with Large Displacements
, 2001
"... In this paper we show that a classic optical ow technique by Nagel and Enkelmann (1986) can be regarded as an early anisotropic diusion method with a diusion tensor. We introduce three improvements into the model formulation that (i) avoid inconsistencies caused by centering the brightness term and ..."
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Cited by 122 (14 self)
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In this paper we show that a classic optical ow technique by Nagel and Enkelmann (1986) can be regarded as an early anisotropic diusion method with a diusion tensor. We introduce three improvements into the model formulation that (i) avoid inconsistencies caused by centering the brightness term and the smoothness term in dierent images, (ii) use a linear scalespace focusing strategy from coarse to ne scales for avoiding convergence to physically irrelevant local minima, and (iii) create an energy functional that is invariant under linear brightness changes. Applying a gradient descent method to the resulting energy functional leads to a system of diusion{reaction equations. We prove that this system has a unique solution under realistic assumptions on the initial data, and we present an ecient linear implicit numerical scheme in detail. Our method creates ow elds with 100 % density over the entire image domain, it is robust under a large range of parameter variations, and it c...
Variable exponent, linear growth functionals in image processing
 SIAM Journal on Applied Mathematics
, 2004
"... Abstract. We study a functional with variable exponent, 1 ≤ p(x) ≤ 2, which provides a model for image denoising, enhancement, and restoration. The diffusion resulting from the proposed model is a combination of Total Variation based regularization and Gaussian smoothing. The existence, uniqueness, ..."
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Cited by 106 (1 self)
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Abstract. We study a functional with variable exponent, 1 ≤ p(x) ≤ 2, which provides a model for image denoising, enhancement, and restoration. The diffusion resulting from the proposed model is a combination of Total Variation based regularization and Gaussian smoothing. The existence, uniqueness, and longtime behavior of the proposed model are established. Experimental results illustrate the effectiveness of the model in image restoration.
A Review of Nonlinear Diffusion Filtering
, 1997
"... . This paper gives an overview of scalespace and image enhancement techniques which are based on parabolic partial differential equations in divergence form. In the nonlinear setting this filter class allows to integrate apriori knowledge into the evolution. We sketch basic ideas behind the differ ..."
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Cited by 100 (10 self)
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. This paper gives an overview of scalespace and image enhancement techniques which are based on parabolic partial differential equations in divergence form. In the nonlinear setting this filter class allows to integrate apriori knowledge into the evolution. We sketch basic ideas behind the different filter models, discuss their theoretical foundations and scalespace properties, discrete aspects, suitable algorithms, generalizations, and applications. 1 Introduction During the last decade nonlinear diffusion filters have become a powerful and wellfounded tool in multiscale image analysis. These models allow to include apriori knowledge into the scalespace evolution, and they lead to an image simplification which simultaneously preserves or even enhances semantically important information such as edges, lines, or flowlike structures. Many papers have appeared proposing different models, investigating their theoretical foundations, and describing interesting applications. For a n...
Dense Disparity Map Estimation Respecting Image Discontinuities: A PDE and ScaleSpace Based Approach
 JOURNAL OF VISUAL COMMUNICATION AND IMAGE REPRESENTATION
, 2000
"... We present an energy based approach to estimate a dense disparity map between two images while preserving its discontinuities resulting from image boundaries. We first derive a simplied expression for the disparity that allows us to easily estimate it from a stereo pair of images using an energy min ..."
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Cited by 85 (11 self)
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We present an energy based approach to estimate a dense disparity map between two images while preserving its discontinuities resulting from image boundaries. We first derive a simplied expression for the disparity that allows us to easily estimate it from a stereo pair of images using an energy minimization approach. We assume that the epipolar geometry is known, and we include this information in the energy model. Discontinuities are preserved by means of a regularization term based on the NagelEnkelmann operator. We investigate the associated EulerLagrange equation of the energy functional, and we approach the solution of the underlying partial differential equation (PDE) using a gradient descent method. In order to reduce the risk to be trapped within some irrelevant local minima during the iterations, we use a focusing strategy based on a linear scalespace. We prove the existence and uniqueness of the underlying parabolic partial differential equation. Experimental results on bot...
A WEAKTOSTRONGCONVERGENCE PRINCIPLE FOR FEJÉRMONOTONE METHODS IN HILBERT SPACES
, 2001
"... We consider a wide class of iterative methods arising in numerical mathematics and optimization that are known to converge only weakly. Exploiting an idea originally proposed by Haugazeau, we present a simple modification of these methods that makes them strongly convergent without additional assump ..."
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Cited by 83 (13 self)
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We consider a wide class of iterative methods arising in numerical mathematics and optimization that are known to converge only weakly. Exploiting an idea originally proposed by Haugazeau, we present a simple modification of these methods that makes them strongly convergent without additional assumptions. Several applications are discussed.
The ÃLojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, preprint
, 2004
"... Abstract. Given a realanalytic function f: Rn → R and a critical point a ∈ Rn, the Lojasiewicz inequality asserts that there exists θ ∈ [ 1 2, 1) such that the function f − f(a)θ ‖∇f‖−1 remains bounded around a. In this paper, we extend the above result to a wide class of nonsmooth functions (th ..."
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Cited by 76 (18 self)
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Abstract. Given a realanalytic function f: Rn → R and a critical point a ∈ Rn, the Lojasiewicz inequality asserts that there exists θ ∈ [ 1 2, 1) such that the function f − f(a)θ ‖∇f‖−1 remains bounded around a. In this paper, we extend the above result to a wide class of nonsmooth functions (that possibly admit the value +∞), by establishing an analogous inequality in which the derivative ∇f(x) can be replaced by any element x ∗ of the subdifferential ∂f(x) of f. Like its smooth version, this result provides new insights into the convergence aspects of subgradienttype dynamical systems. Provided that the function f is sufficiently regular (for instance, convex or lowerC2), the bounded trajectories of the corresponding subgradient dynamical system can be shown to be of finite length. Explicit estimates of the rate of convergence are also derived. Key words. Lojasiewicz inequality, subanalytic function, nonsmooth analysis, subdifferential, dynamical system, descent method