modified curvature motion for image smoothing and enhancement we formulate a general modified mean curvature based equation for image smoothing and enhancement the key idea is to consider the image as a graph in some r sup 72 and apply a mean curvature type motion to the graph we consider some special cases relevant to grey scale and color images

level set approach to mean curvature flow in arbitrary codimension we develop a level set theory for the mean curvature evolution of surfaces with arbitrary co dimension thus generalizing the previous work 8 15 on hypersurfaces the main idea is to surround the evolving surface of co dimension k in r by a family of hypersurfaces the level sets of a function evolving with normal velocity equal to the sum of the d k smallest principal curvatures the existence and the uniqueness of a weak level set solution is easily established using mainly the results of 8 and the theory of viscosity solutions for second order nonlinear parabolic equations the level set solutions coincide with the classical solutions whenever the latter exist the proof of this connection uses a careful analysis of the squared distance from the surfaces it is also shown that varifold solutions constructed by brakke 7 are included in the level set solutions the idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of de giorgi an introduction to the theory of barriers and his connection to the level set solutions is also provided

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

a quasistatic frictional problem with normal compliance a friction and contact problem will be considered for an elastic solid body which under the influence of volume and surface forces is in contact with a rigid foundation

boundary optimal control for nonlinear antiplane problems abstract we consider a nonlinear antiplane problem which models the deformation of an elastic cylindrical body in frictional contact with a rigid foundation the contact is modelled with tresca s law of dry friction in which the friction bound is slip dependent the aim of this article is to study an optimal control problem which consists of leading the stress tensor as close as possible to a given target by acting with a control on the boundary of the body the existence of at least one optimal control is proved next we introduce a regularized problem depending on a small parameter and we study the convergence of the optimal controls when tends to zero an optimality condition is delivered for the regularized problem

a general framework for low level vision we introduce a new geometrical framework based on which natural flows for image scale space and enhancement are presented we consider intensity images as surfaces in the x i space the image is thereby a two dimensional 2 d surface in three dimensional 3 d space for gray level images and 2 d surfaces in five dimensions for color images the new formulation unifies many classical schemes and algorithms via a simple scaling of the intensity contrast and results in new and efficient schemes extensions to multidimensional signals become natural and lead to powerful denoising and scale space algorithms

dali deformation and light invariant descriptor recent advances in 3d shape analysis and recognition have shown that heat diffusion theory can be effectively used to describe local features of deforming and scaling surfaces in this paper we show how this description can be used to characterize 2d image patches and introduce dali a novel feature point descriptor with high resilience to non rigid image transformations and illumination changes in order to build the descriptor 2d image patches are initially treated as 3d surfaces patches are then described in terms of a heat kernel signature which captures both local and global information and shows a high degree of invariance to non linear image warps in addition by further applying a logarithmic sampling and a fourier transform invariance to photometric changes is achieved finally the descriptor is compacted by mapping it onto a low dimensional subspace computed using principal component analysis allowing for an efficient matching a thorough experimental validation demonstrates that dali is significantly more discriminative and robust to illuminations changes and image transformations than state of the art descriptors even those specifically designed to describe non rigid deformations

curve evolution implementation of the mumford shah functional for image segmentation denoising interpolation and magnification we first address the problem of simultaneous image segmentation and smoothing by approaching the mumford shah 1989 paradigm from a curve evolution perspective in particular we let a set of deformable contours define the boundaries between regions in an image where we model the data via piecewise smooth functions and employ a gradient flow to evolve these contours each gradient step involves solving an optimal estimation problem for the data within each region connecting curve evolution and the mumford shah functional with the theory of boundary value stochastic processes the resulting active contour model offers a tractable implementation of the original mumford shah model i e without resorting to elliptic approximations which have traditionally been favored for greater ease in implementation to simultaneously segment and smoothly reconstruct the data within a given image in a coupled manner various implementations of this algorithm are introduced to increase its speed of convergence we also outline a hierarchical implementation of this algorithm to handle important image features such as triple points and other multiple junctions next by generalizing the data fidelity term of the original mumford shah functional to incorporate a spatially varying penalty we extend our method to problems in which data quality varies across the image and to images in which sets of pixel measurements are missing this more general model leads us to a novel pde based approach for simultaneous image magnification segmentation and smoothing thereby extending the traditional applications of the mumford shah functional which only considers simultaneous segmentation and smoothing

user s guide to viscosity solutions of second order partial differential equations 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 self contained exposition of the basic theory of viscosity solutions

behavioral analysis of anisotropic diffusion in image processing in this paper we analyze the behavior of the anisotropic diffusion model of perona and malik 1990 the main idea is to express the anisotropic diffusion equation as coming from a certain optimization problem so its behavior can be analyzed based on the shape of the corresponding energy surface we show that anisotropic diffusion is the steepest descent method for solving an energy minimization problem it is demonstrated that an anisotropic diffusion is well posed when there exists a unique global minimum for the energy functional and that the ill posedness of a certain anisotropic diffusion is caused by the fact that its energy functional has an infinite number of global minima that are dense in the image space we give a sufficient condition for an anisotropic diffusion to be well posed and a sufficient and necessary condition for it to be ill posed due to the dense global minima the mechanism of smoothing and edge enhancement of anisotropic diffusion is illustrated through a particular orthogonal decomposition of the diffusion operator into two parts one that diffuses tangentially to the edges and therefore acts as an anisotropic smoothing operator and the other that flows normally to the edges and thus acts as an enhancement operator

on the reconstruction of height functions and terrain maps from dense range data this paper describes a method for combining multiple dense range images to create surface reconstructions of height functions height functions are a special class of three dimensional 3 d surfaces where one 3 d coordinate is a function of the other two they are relevant for application domains such as terrain modeling or two and half dimensional surface reconstruction dense range maps are produced by either a range measuring device combined with a scanning mechanism or a triangulation scheme such as active or passive stereo the proposed method follows from a statistical formulation that characterizes the optimal surface estimate as the one that maximizes the posterior probability conditional on the input data and prior information about the application domain because the domain of the reconstruction is a two dimensional 2 d scalar function the optimal surface can be expressed as an image and the variational form of that optimization produces a 2 d partial differential equation pde the pde consists of two parts a first order data term and a second order smoothing term thus optimal surface reconstruction is formulated as the solution to a second order nonlinear pde on an image which is related to the family of pde based image processing algorithms in the literature this paper presents the theory for reconstruction and some particular aspects of the numerical implementation it also analyzes results on both synthetic and real data sets which show a 75 95 reduction of the rms sensor error

optimal control in some variational inequalities we study a nonconvex nondifferentiable problem of optimal control where the state of the system is defined by an elliptic variational inequality with obstacle and where the cost function is quadr

fundamental relationship between bilateral filtering adaptive smoothing and the nonlinear diffusion equation in this paper the relationship between bilateral filtering and anisotropic diffusion is examined the bilateral filtering approach represents a large class of nonlinear digital image filters we first explore the connection between anisotropic diffusion and adaptive smoothing and then the connection between adaptive smoothing and bilateral filtering previously adaptive smoothing was considered to be an inconsistent approximation to the nonlinear diffusion equation we extend adaptive smoothing to make it consistent thus enabling a unified viewpoint that relates nonlinear digital image filters and the nonlinear diffusion equation

boundary optimal control for quasistatic bilateral frictional contact problems abstract we consider a mathematical model describing the quasistatic process of bilateral contact with friction between an elastic body and a rigid foundation our goal is to study a related optimal control problem which allows us to obtain a desired field of displacements on the contact boundary by acting with a control on another part of the boundary of the body using penalization and regularization techniques we derive the necessary conditions of optimality

anisotropic diffusion of multivalued images with applications to color filtering a general framework for anisotropic diffusion of multivalued images is presented we propose an evolution equation where at each point in time the directions and magnitudes of the maximal and minimal rate of change in the vector image are first evaluated these are given by eigenvectors and eigenvalues of the first fundamental form in the given image metric then the image diffuses via a system of coupled differential equations in the direction of minimal change the diffusion strength is controlled by a function that measures the degree of dissimilarity between the eigenvalues we apply the proposed framework to the filtering of color images represented in cie l a b space

diffusion of general data on non flat manifolds viaharmonic maps theory the direction diffusion case 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 l 2 norm and edge preserving diffusion obtained from an l norm in general and an l 1 norm in particular in contrast with previous approaches the framework is valid for directions in any dimensions supports non smooth 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 non flat 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

color image enhancement via chromaticity diffusion a novel approach for color image denoising is proposed in this paper the algorithm is based on separating the color data into chromaticity and brightness and then processing each one of these components with partial differential equations or diffusion flows in the proposed algorithm each color pixel is considered as an n dimensional vector the vectors direction a unit vector gives the chromaticity while the magnitude represents the pixel brightness the chromaticity is processed with a system of coupled diffusion equations adapted from the theory of harmonic maps in liquid crystals this theory deals with the regularization of vectorial data while satisfying the intrinsic unit norm constraint of directional data such as chromaticity both isotropic and anisotropic diffusion flows are presented for this n dimensional chromaticity diffusion flow the brightness is processed by a scalar median filter or any of the popular and well established anisotropic diffusion flows for scalar image enhancement we present the underlying theory a number of examples and briefly compare with the current literature

locally monotonic diffusion anisotropic diffusion affords an efficient adaptive signal smoothing technique that can be used for signal enhancement signal segmentation and signal scale space creation this paper introduces a novel partial differential equation pde based diffusion method for generating locally monotonic signals unlike previous diffusion techniques that diverge or converge to trivial signals locally monotonic lomo diffusion converges rapidly to well defined lomo signals of the desired degree the property of local monotonicity allows both slow and rapid signal transitions ramp and step edges while excluding outliers due to noise in contrast with other diffusion methods lomo diffusion does not require an additional regularization step to process a noisy signal and uses no ad hoc thresholds or parameters in the paper we develop the lomo diffusion technique and provide several salient properties including stability and a characterization of the root signals the convergence of the algorithm is well behaved nonoscillatory and is independent of the signal length in contrast with the median filter a special case of lomo diffusion is identical to the optimal solution achieved via regression experimental results validate the claim that lomo diffusion can produce denoised lomo signals with low error using less computation than the median order statistic approach

noise removal with gauss curvature driven diffusion in this paper we propose the use of the gauss curvature in a gauss curvature driven diffusion equation for noise removal the proposed scheme uses the gauss curvature as the conductance term and controls the amount of diffusion the main advantage of the scheme is that it preserves important structures such as straight edges curvy edges ramps corners small scaled features etc

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

asymptotic behavior of flows by powers of the gaussian curvature we consider a 1 parameter family of strictly convex hypersurfaces in mathbb r n 1 moving with speed k alpha where denotes the outward pointing unit normal vector and alpha geqslant 1 n 2 for alpha gt 1 n 2 we show that the flow converges to a round sphere after rescaling in the affine invariant case alpha 1 n 2 our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling

resultant operators and the bezout equation for analytic matrix functions i this paper is a continuation of 4 it contains our results on resultant operators for a family of matrix functions that are analytic in a finitely connected domain the proofs are based on the results about the bezout equation obtained in 4 it is assumed that the reader is familiar with the concepts and results from 4 let f denote a cauchy contour in the complex plane consisting of several non intersecting simple smooth closed contours which form the positively oriented boundary of a finitely connected bounded domain a throughout this chapter we assume for convenience that 0 e a the general case can be obtained by a shift in the complex plane let a a be regular n x n matrix functions which are analytic in a and continuous in a u r and assume that c a n r 3 the paper consists of two sections in section 1 we show that the role of the resultant operator of the matrix functions a a is played by the operator r l t l 1 p co defined by

common eigenvalues divisors and multiples of matrix polynomials a review abstract the main objective of this paper is to review some results obtained in the last decade using methods of spectral theory and concerning common multiples common divisors and common eigenvalues for matrix polynomials this includes discussion of resultant and bezoutian matrices for matrix polynomials and descriptions of their kernels in terms of spectral data some extensions to operator polynomials whose coefficients are bounded linear operators on a hilbert space are also included

optimal control for variational inequalities consider the problem of maximizing a functional which depends on a control function k and on the solution of an elliptic variational inequality with k appearing in the data the variational problem for k is nondiflerentiable and nonconvex we obtain necessary conditions on a maximizes k 0 and then use them to determine the structure of k 0 in some cases

evolution of nonparametric surfaces with speed depending on curvature ii the mean curvature case we consider an evolution which starts as a flow of smooth surfaces in nonparametric form propagating in space with normal speed equal to the mean curvature of the current surface the boundaries of the surfaces are assumed to remain fixed g huisken has shown that if the boundary of the domain over which this flow is considered satisfies the mean curvature condition of h jenkins and j serrin that is the boundary of the domain is convex in the mean then the corresponding initial boundary value problem with dirichlet boundary data and smooth initial data admits a smooth solution for all time in this paper we consider the case of arbitrary domains with smooth boundaries not necessarily satisfying the condition of jenkins serrin in this case even if the flow starts with smooth initial data and homogeneous dirichlet boundary data singularities may develop in finite time at the boundary of the domain and the solution will not satisfy the boundary condition we prove however existence of solutions that are smooth inside the domain for all time and become smooth up to the boundary after elapsing of a sufficiently long period of time from that moment on such solutions assume the boundary values in the classical sense we also give sufficient conditions that guarantee the existence of classical solutions for all time t 0 in addition we establish estimates of the rate at which solutions tend to zero as t

optimal control of an elastic contact problem involving tresca friction law in this paper we establish existence results and necessary optimality conditions for a class of optimal control problems where the state is described by elliptic variational inequalities modelling the frictional contact between an elastic body and a rigid foundation we use the tresca s law of dry friction in which the friction bound is prescribed for the contact condition we assume that slowly varying time dependent volume forces and surface tractions act on the body so that the acceleration in the system will be neglected existence and uniqueness results for initial and boundary value problems involving tresca friction law were obtained in 2 6 10 in the case of an elastic body in 4 in the case of elasto viscoplastic models and in 3 in the case of perfectly plastic solids the aim of this paper is to study optimal control problems for mechanical models describing the quasistatic process of bilateral frictional contact between an elastic body and a rigid foundation the state of this system is governed by a variational inequality

vertex based diffusion for 3 d mesh denoising we present a vertex based diffusion for 3 d mesh denoising by solving a nonlinear discrete partial differential equation the core idea behind our proposed technique is to use geometric insight in helping construct an efficient and fast 3 d mesh smoothing strategy to fully preserve the geometric structure of the data illustrating experimental results demonstrate a much improved performance of the proposed approach in comparison with existing methods currently used in 3 d mesh smoothing

the control variational approach for differential systems in this work a new approach to generalized naghdi shell and curved rod models is discussed the method is based on optimal control theory and has a wide range of applications some abstract variants are also indicated

on the evolution of curves via a function of curvature i the classical case abstract the problem of curve evolution as a function of its local geometry arises naturally in many physical applications a special case of this problem is the curve shortening problem which has been extensively studied here we consider the general problem and prove an existence theorem for the classical solution the main theorem rests on lemmas that bound the evolution of length curvature and how far the curve can travel

bezout equation vandermonde operators and common multiples for operator polynomials soit h un espace de hilbert complexe on considere des polynomes d operateurs de type p j 0 m i p j ou p j sont des operateurs agissant dans h et est un parametre complexe on considere des p pour lequel le spectre p est un ensemble compact