a lower bound on the undetected error probability and strictly optimal codes error detection is a simple technique used in various communication and memory systems to enhance reliability we study the probability that a q ary linear or nonlinear block code of length n and size m fails to detect an error a lower bound on this undetected error probability is derived in terms of q n and m the new bound improves upon other bounds mentioned in the literature even those that hold only for linear codes block codes whose undetected error probability equals the new lower bound are investigated we call these codes strictly optimal codes and give a combinatorial characterization of them we also present necessary and sufficient conditions for their existence in particular we find all values of n and m for which strictly optimal binary codes exist and determine the structure of all of them for example we construct strictly optimal binary coded decimal codes of length four and five and we show that these are the only possible lengths of such codes

motion of a set by the curvature of its boundary abstract the study of a crystal shrinking or growing in a melt gives rise to equations relating the normal velocity of the motion to the curvature of the crystal boundary often these equations are anisotropic indicating the preferred directions of the crystal structure in the isotropic case this equation is called the curve shortening or the mean curvature flow equation and has been studied by differential geometric tools in general there are no classical solutions to these equations in this paper we develop a weak theory for the generalized mean curvature equation using the newly developed theory of viscosity solutions our approach is closely related to that of osher and sethian chen giga and goto and evans and spruck who view the boundary of the crystal as the level set of a solution to a nonlinear parabolic equation although we use their results in an essential way we give an intrinsic definition our main results are the existence of a solution large time asymptotics of this solution and its connection to the level set solution of osher and sethian chen giga and goto and evans and spruck in general there is no uniqueness even for classical solutions but we prove a uniqueness result under restrictive assumptions we also construct a class of explicit solutions which are dilations of wulff crystals

invariant geometric evolutions of surfaces and volumetric smoothing the study of geometric flows for smoothing multiscale representation and analysis of two and three dimensional objects has received much attention in the past few years in this paper we first survey the geometric smoothing of curves and surfaces via geometric heat type flows which are invariant under the groups of euclidean and affine motions second using the general theory of differential invariants we determine the general formula for a geometric hypersurface evolution which is invariant under a prescribed symmetry group as an application we present the simplest affine invariant flow for convex surfaces in three dimensional space which like the affine invariant curve shortening flow will be of fundamental importance in the processing of three dimensional images

singularities of lagrangian mean curvature flow zero maslov class case we study singularities of lagrangian mean curvature flow in n when the initial condition is a zero maslov class lagrangian we start by showing that in this setting singularities are unavoidable more precisely we construct lagrangians with arbitrarily small lagrangian angle and lagrangians which are hamiltonian isotopic to a plane that nevertheless develop finite time singularities under mean curvature flow

curvature evolution of nonconvex lens shaped domains we study the curvature flow of planar nonconvex lens shaped domains con sidered as special symmetric networks with two triple junctions we show that the evolv ing domain becomes convex in finite time then it shrinks homothetically to a point as proved in 22 our theorem is the analog of the result of grayson 13 for curvature flow of closed planar embedded curves

poor error correction codes are poor error detection codes corresp we show that binary group codes that do not satisfy the asymptotic varshamov gilbert bound have an undesirable characteristic when used as error detection codes for transmission over the binary symmetric channel

sturm attractors for quasilinear parabolic equations abstract the goal of this paper is to construct explicitly the global attractors of quasilinear parabolic equations as it was done for the semilinear case by brunovsk and fiedler 1986 8 and generalized by fiedler and rocha 1996 13 in particular we construct heteroclinic connections between hyperbolic equilibria stating necessary and sufficient conditions for heteroclinics to occur such conditions can be computed through a permutation of the equilibria lastly an example is computed yielding the well known chafee infante attractor

optical flow a curve evolution approach a novel approach for the computation of optical flow based on an l sup 1 type minimization is presented it is shown that the approach has inherent advantages since it does not smooth the flow velocity across the edges and hence preserves edge information a numerical approach based on computation of evolving curves is proposed for computing the optical flow field computations are carried out on a number of real image sequences in order to illustrate the theory as well as the numerical approach

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 affine heat equation for non convex curves 1998 american mathematical society first published in journal of the american mathematical society vol 11 no 3 july 1998 published by the american mathematical society

necessary conditions for good error detection the problem of determining the error detection capabilities of linear codes where the channel is a binary symmetric channel is addressed necessary conditions are given on the number of codewords in an n k linear code and its dual for the probability of an undetected error to be upper bounded by 2 sup n k

a stable manifold theorem for the curve shortening equation on presente une famille de solutions homothetiques de l equation pour une courbe planaire x kn et on demontre l existence de varietes non lineaires stables et instables autour de telles solutions

parabolic equations for curves on surfaces 2 intersections blow up and generalized solutions we describe a theory for parabolic equations for immersed curves on surfaces which generalizes the curve shortening or flow by mean curvature problem as well as several models in the theory of phase transitions in two dimensions a class of equations is described for which the initial value problem is well posed for rough initial data for which one can give a description of the way a smooth solution becomes singular and for which one can define generalized solutions i e solutions which are smooth except at a discrete set of times the methods which are used in this paper are more geometrical than those of part i by comparing arbitrary solutions with certain special solutions and by considering the way they intersect we derive estimates for the curvature and the tangent which allow one to study the initial value problem and the way solutions become singular

front propagation and phase field theory the connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction diffusion equations is rigorously established two different scalings are studied in the first the limiting geometric equation is a first order equation in the second it is a generalization of the mean curvature equation intrinsic definitions for the geometric equations are obtained and uniqueness under a geometric condition on the initial surface is proved in particular in the case of the mean curvature equation this condition is satisfied by surfaces that are strictly starshaped that have positive mean curvature or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions

on the monotonic property of the probability of undetected error for a shortened code the monotonic property of the probability of undetected error is considered when a shortened code of a linear code over gf q is used for error detection on a q ary symmetric channel some conditions are presented under which the probability of undetected error is or is not monotonic with respect to the code length or the symbol error probability it is shown that the probability of undetected error for a maximum distance separable code is monotonic with respect to the codelength it is also shown that the probability of undetected error of a shortened code of any binary cyclic hamming code generated by a trinomial is not monotonic with respect to the bit error rate if the code length is short

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

shortening three dimensional curves via two dimensional flows in this paper a curve evolution approach for the computation of geodesic curves on 3d surfaces is presented the algorithm is based on deforming via the curve shortening flow an arbitrary initial curve ending at two given surface points the 3d curve shortening flow is first transformed into an equivalent 2d one this 2d flow is implemented using an efficient numerical algorithm for curve evolution with fixed end points

discrete active models and applications abstract optimization processes based on active models play central roles in many areas of computational vision as well as computational geometry unfortunately current models usually require highly complex and sophisticated mathematical machinery and at the same time they suffer from a number of limitations which impose restrictions on their applicability in this paper a simple class of discrete active models called migration processes mps is presented the processes are based on iterated averaging over neighborhoods defined by constant geodesic distance it is demonstrated that the mp model a system of self organizing active particles has a number of advantages over previous models both parametric active models snakes and implicit contour evolution models due to the generality of the mp model the process can be applied to derive natural solutions to a variety of optimization problems including defining minimal surface patches given their boundary curves finding shortest paths joining sets of points and decomposing objects into primitive parts

symmetry reductions and exact solutions of the affine heat equation lie symmetry group method is applied to study the affine heat equation for surface its symmetry groups and corresponding optimal systems are determined and group invariant solutions associated to the symmetries are obtained and classified

anisotropic flows for convex plane curves where v and k are respectively the normal velocity and curvature of the interface equation 1 is sometimes called the curve shortening problem because it is the negative l2 gradient flow of the length of the interface by drawing pictures one can be easily convinced that a simple closed curve stays simple and smooth along 1 and shrinks to a point in finite time with the limiting shape of a circle however a mathematical treatment of 1 turned out to be rather delicate in fact a rigorous study did not exist until the early 1980s when differential geometers considered 1 as a tool in the search for simple closed geodesics on surfaces it was also regarded as a model case for more general curvature flows which are believed to be important in the topological classification of low dimensional manifolds as a first attempt gage 10 proved that the isoperimetric ratio decreases along convex curves then in gage and hamilton 12 it was shown that a convex curve stays convex and shrinks to a point in finite time moreover if one normalizes the flow by dilating it so that the enclosing area is constant the normalized flow converges smoothly to a circle finally grayson 14 completed this line of investigation by showing that a simple curve evolves into a convex one before shrinks to a point for an alternative approach to this result one may consult hamilton 18 recently mullins s theory was generalized by gurtin 15 16 and by angenent and gurtin 4 5 see also the monograph of gurtin 17 to include anisotropy and the possibility of a difference in bulk energies between phases anisotropy is indispensable in dealing with crystalline materials for perfect conductors the temperatures in both phases are constant the equation becomes

dynamic active contours for visual tracking visual tracking using active contours is usually set in a static framework the active contour tracks the object of interest in a given frame of an image sequence a subsequent prediction step ensures good initial placement for the next frame this approach is unnatural the curve evolution gets decoupled from the actual dynamics of the objects to be tracked true dynamical approaches exist all being marker particle based and thus prone to the shortcomings of such particle based implementations in particular topological changes are not handled naturally in this framework the now classical level set approach is tailored for evolutions of manifolds of codimension one however dynamic curve evolution is at least a codimension two problem we propose an efficient level set based approach for dynamic curve evolution which addresses the artificial separation of segmentation and prediction while retaining all the desirable properties of the level set formulation it is based on a new energy minimization functional which for the first time puts dynamics into the geodesic active contour framework

a global exponential fuzzy observer design for time delay takagi sugeno uncertain discrete fuzzy bilinear systems with disturbance a robust h fuzzy observer is proposed for a class of time delay takagi sugeno t s uncertain discrete fuzzy bilinear systems with disturbance utilizing the lipschitz conditions and some variable transformations a linear matrix inequality lmi approach is developed and a sufficient condition is obtained to design the robust h fuzzy observer the robust h fuzzy observer for time delay t s uncertain discrete fuzzy bilinear systems with disturbance can be obtained directly through the lmis with no preassigned matrices the convergent rate of the error state can be estimated from the initial conditions and the delay time of t s uncertain discrete fuzzy bilinear systems the designed robust h fuzzy observer can guarantee the dynamics of the observer error to be globally exponentially stable and has a prescribed h performance a numerical example is given to demonstrate the feasibility and effectiveness of the proposed observer design method

on the probability of undetected error for the maximum distance separable codes in this paper we investigate the performance of maximum distance separable codes with symbols from gf q when they are used for pure error detection or for simultaneous error correction and detection over a q input and q output discret memoryless channel with symbol error probability e first we show that the probability of undetected error for an mds code used for pure error detection is upper bounded by q r and decreases monotonically as edecreases from q 1 q to 0 where r is the number of parity check symbols of the code then we show that the probability of undetected error for an mds code used for correcting t or fewer symbol errors is upper bounded by q r sum min i 0 max t min i max n q 1 i and decreases monotonically as e decreases from q 1 q to 0 these results show that the mds codes are effective for both pure error detection and simultaneous error correction and detection

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

evolution of plane curves driven by a nonlinear function of curvature and anisotropy in this paper we study evolution of plane curves satisfying a geometric equation v beta k nu where v is the normal velocity and k and nu are the curvature and tangential angle of a plane curve gamma we follow the direct approach and we analyze the so called intrinsic heat equation governing the motion of plane curves obeying such a geometric equation the intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional alpha we show how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities from an analytical point of view we present some new results on short time existence of a regular family of evolving curves in the degenerate case when beta k nu gamma nu k m 0 m le 2 and the governing system of equations includes a nontrivial tangential velocity functional

minimal surfaces a geometric three dimensional segmentation approach a novel geometric approach for three dimensional object segmentation is presented the scheme is based on geometric deformable surfaces moving towards the objects to be detected we show that this model is related to the computation of surfaces of minimal area local minimal surfaces the space where these surfaces are computed is induced from the three dimensional image in which the objects are to be detected the general approach also shows the relation between classical deformable surfaces obtained via energy minimization and geometric ones derived from curvature flows in the surface evolution framework the scheme is stable robust and automatically handles changes in the surface topology during the deformation results related to existence uniqueness stability and correctness of the solution to this geometric deformable model are presented as well based on an efficient numerical algorithm for surface evolution we present a number of examples of object detection in real and synthetic images

what is the best causal scale space for three dimensional images we study the unique affine invariant morphological scale space in three dimensions we discuss its properties and show that it improves the dynamic shape model we explain the algorithms and display the first numerical experiments

finite time singularities for lagrangian mean curvature flow given any embedded lagrangian on a four dimensional compact calabi yau we find another lagrangian in the same hamiltonian isotopy class that develops a finite time singularity under mean curvature flow this contradicts a weaker version of the thomas yau conjecture regarding long time existence and convergence of lagrangian mean curvature flow

finsler active contours in this paper we propose an image segmentation technique based on augmenting the conformal or geodesic active contour framework with directional information in the isotropic case the euclidean metric is locally multiplied by a scalar conformal factor based on image information such that the weighted length of curves lying on points of interest typically edges is small the conformal factor that is chosen depends only upon position and is in this sense isotropic although directional information has been studied previously for other segmentation frameworks here we show that if one desires to add directionality in the conformal active contour framework then one gets a well defined minimization problem in the case that the factor defines a finsler metric optimal curves may be obtained using the calculus of variations or dynamic programming based schemes finally we demonstrate the technique by extracting roads from aerial imagery blood vessels from medical angiograms and neural tracts from diffusion weighted magnetic resonance imagery

area and length preserving geometric invariant scale spaces in this paper area preserving multi scale representations of planar curves are described this allows smoothing without shrinkage at the same time preserving all the scale space properties the representations are obtained deforming the curve via geometric heat flows while simultaneously magnifying the plane by a homethety which keeps the enclosed area constant when the euclidean geometric heat flow is used the resulting representation is euclidean invariant and similarly it is affine invariant when the affine one is used the flows are geometrically intrinsic to the curve and exactly satisfy all the basic requirements of scale space representations in the case of the euclidean heat flow it is completely local as well the same approach is used to define length preserving geometric flows a similarity scale invariant geometric heat flow is studied as well in this work

multiplicity of the second schrodinger eigenvalue on closed surfaces we improve roughly by a factor 2 the known bound on the multiplicity of the second eigenvalue of schrodinger operators i e laplace plus potential on closed surfaces this gives four new topological types of surfaces for which colin de verdiere s conjecture relating the maximal multiplicity to the chromatic number of the surface is verified the proof goes by defining a space cal p of nodal splittings of the surface equipped with a double covering to which a borsuk ulam type theorem is applied