fast sweeping algorithms for a class of hamilton jacobi equations we derive a godunov type numerical flux for the class of strictly convex homogeneous hamiltonians that includes h p q sqrt ap 2 bq 2 2cpq c 2 ab we combine our godunov numerical fluxes with simple gauss seidel type iterations for solving the corresponding hamilton jacobi hj equations the resulting algorithm is fast since it does not require a sorting strategy as found e g in the fast marching method in addition it providesa way to compute solutions to a class of hj equations for which the conventional fast marching method is not applicable our experiments indicate convergence after a few iterations even in rather difficult cases

new numerical algorithms for the nonlinear diffusion model of image denoising and segmentation abstract in this paper for the nonlinear diffusion model of image denoising and segmentation we present two high order explicit algorithms and a semi implicit algorithm using algebraic multi grid method the unconditional stability of the semi implicit approximation is given the numerical results of the above three algorithms are demonstrated with comparison to general explicit algorithm in y chen b c vemuri l wang image denoising and segmentation via nonlinear diffusion comput math appl 39 2000 131 149 for real noised images the numerical results show that the implicit algorithm is quicker and sharper than all explicit algorithms

a viscosity solutions approach to shape from shading the problem of recovering a lambertian surface from a single two dimensional image may be written as a first order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions a new approach based on hamilton jacobi bellman equations and viscosity solutions theories enables one to study non uniqueness phenomenon and thus to characterize the surface among the various solutions a consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle and numerical results are presented

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

active contours without edges we propose a new model for active contours to detect objects in a given image based on techniques of curve evolution mumford shah 1989 functional for segmentation and level sets our model can detect objects whose boundaries are not necessarily defined by the gradient we minimize an energy which can be seen as a particular case of the minimal partition problem in the level set formulation the problem becomes a mean curvature flow like evolving the active contour which will stop on the desired boundary however the stopping term does not depend on the gradient of the image as in the classical active contour models but is instead related to a particular segmentation of the image we give a numerical algorithm using finite differences finally we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable also the initial curve can be anywhere in the image and interior contours are automatically detected

heat flows and related minimization problem in image restoration abstract a new anisotropic diffusion model is proposed for image restoration and segmentation which is closely related to the minimization problems for the unconstrained total variation e u x u 2 u i 2 existence uniqueness and stability of the viscosity solutions of the equation are proved the experimental results are given and compared with the existing models in the framework of image restoration the improvement on preserving sharp edges by using the new model is visible

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

crystal growth and dendritic solidification we present a numerical method which computes the motion of complex solid liquid boundaries in crystal growth the model we solve includes physical effects such as crystalline anisotropy surface tension molecular kinetics and undercooling the method is based on two ideas first the equations of motion are recast as a single history dependent boundary integral equation on the solid liquid boundary a fast algorithm is used to solve the integral equation efficiently second the boundary is moved by solving a hamiltonjacobi type equation on a fixed domain formulated by osher and sethian for a function in which the boundary is a particular level set this equation is solved by finite difference schemes borrowed from the technology of hyperbolic conservation laws the two ideas are combined by constructing a smooth extension of the normal velocity off the moving boundary in a way suggested by the physics of the problem our numerical experiments show the evolution of complex crystalline shapes development of large spikes and corners dendrite formation and side branching and pieces of solid merging and breaking off freely

a fast level set method for propagating interfaces a method is introduced to decrease the computational labor of the standard level set method for propagating interfaces the fast approach uses only points close to the curve at every time step we describe this new algorithm and compare its efficiency and accuracy with the standard level set approach

a fast multiphase image segmentation model for gray images abstract in this paper we propose a fast multiphase image segmentation model for gray images our proposed model is based on the piecewise constant multiphase vese chan model the globally convex segmentation method and the split bregman method our proposed model has the advantages of the vese chan model and can deal with piecewise constant multiphase images well besides our proposed model is much more efficient than the vese chan model by applying the globally convex segmentation method and the split bregman method we have tested our proposed model with many synthetic and real images experimental results have demonstrated the efficiency and robustness of our proposed model

constraints on deformable models recovering 3d shape and nongrid motion abstract inferring the 3d structures of nonrigidly moving objects from images is a difficult yet basic problem in computational vision our approach makes use of dynamic elastically deformable object models that offer the geometric flexibility to satisfy a diversity of real world visual constraints we specialize these models to include intrinsic forces inducing a preference for axisymmetry image based constraints are applied as extrinsic forces that mold the symmetry seeking model into shapes consistent with image data we describe an extrinsic force that applies constraints derived from profiles of monocularly viewed objects we generalize this constraint force to incorporate profile information from multiple views and use it to exploit binocular image data for time varying images the force becomes dynamic and the model is able to infer not only depth but nonrigid motion as well we demonstrate the recovery of 3d shape and nonrigid motion from natural imagery

a new eulerian method for the computation of propagating short acoustic and electromagnetic pulses a new method is described to compute short acoustic or electromagnetic pulses that propagate according to geometrical optics the pulses are treated as zero thickness sheets that can propagate over long distances through inhomogeneous media with multiple reflections the method has many of the advantages of lagrangian ray tracing but is completely eulerian typically using a uniform cartesian grid accordingly it can treat arbitrary configurations of pulses that can reflect from surfaces and pass through each other without requiring special computational marker arrays for each pulse also information describing the pulses which are treated as continuous surfaces can be available throughout the computational grid rather than only at isolated individual markers the method uses a new type of representation which we call dynamic surface extension the basic idea is to propagate or broadcast defining fields from each pulse surface through a computational grid these fields carry information about a nearby pulse surface that is used at each node to compute the location of the pulse surfaces and other attributes such as amplitude thus the emphasis is on the dynamics of these propagating defining fields which obey only local eulerian equations at each node the dynamic surface extension representation can be thought of as dual to level set representation the defining fields involve single valued variables which are constant at each time along lines that are normal to the evolving surface whereas level set techniques involve a function which has constant values on the evolving surface and neighboring surfaces in this way the new method overcomes the inability of level set or eikonal methods to treat intersecting pulses that obey a wave equation and can pass through each other while still using only single valued variables propagating thin pulse surfaces in 1 d 2 d and 3 d that can reflect from boundaries and pass through each other are computed using the new method the method was first presented as a new general representation of surfaces filaments and particles by j steinhoff and m fan 1998 eulerian computation of evolving surfaces curves and discontinuous fields utsi preprint

a geometric model for active contours in image processing we propose a new model for active contours based on a geometric partial differential equation our model is intrinsec stable satisfies the maximum principle and permits a rigorous mathematical analysis it enables us to extract smooth shapes we cannot retrieve angles and it can be adapted to find several contours simultaneously moreover as a consequence of the stability we can design robust algorithms which can be engineed with no parameters in applications numerical experiments are presented

image denoising and segmentation via nonlinear diffusion abstract image denoising and segmentation are fundamental problems in the field of image processing and computer vision with numerous applications in this paper we present a nonlinear pde based model for image denoising and segmentation which unifies the popular model of alvarez lions and morel alm for image denoising and the caselles kimmel and sapiro model of geodesic snakes our model includes nonlinear diffusive as well as reactive terms and leads to quality denoising and segmentation results as depicted in the experiments presented here we present a proof for the existence uniqueness and stability of the viscosity solution of this pde based model the proof is in spirit similar to the proof of the alm model however there are several differences which arise due to the presence of the reactive terms that require careful treatment consideration a fast implementation of our model is realized by embedding the model in a scale space and then achieving the solution via a dynamic system governed by a coupled system of first order differential equations the dynamic system finds the solution at a coarse scale and tracks it continuously to a desired fine scale we demonstrate the smoothing and segmentation results on several real images

computing interface motion in compressible gas dynamics a hamilton jacobi level set formulation of the equations of motion for propagating interfaces has been introduced recently by osher and sethian this formulation allows fronts to self intersect develop singularities and change topology the numerical algorithms based on this approach handle topological merging and breaking naturally work in any number of space dimensions and do not require that the moving front be written as a function instead the moving front is embedded as a particular level set in a fixed domain partial differential equation numerical techniques borrowed from hyperbolic conservation laws are then used to accurately capture the complicated surface motion that satisfies the global entropy condition for propagating fronts given by sethian in this paper we analyze the coupling of this level set formulation to a system of conservation laws for compressible gas dynamics we study both conservative and non conservative differencing of the level set function and compare the two approaches to illustrate the capability of the method we study the compressible rayleigh taylor and kelvin helmholtz instabilities for air air and air helium boundaries we perform numerical convergence studies of the method over a range of parameters and analyze the accuracy of this approach applied to these problems

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

a numerical method for solving incompressible flow problems with a surface of discontinuity a numerical method for solving problems in which a moving surface of discontinuity separates regions of incompressible flow is presented the method developed is notable in that it does not introduce any artificial smoothing of the change in fluid properties across the surface of discontinuity this results in an increase in accuracy relative to methods which introduce smoothing effects the method was also shown to be fairly versatile problems describing a free surface an immiscible fluid interface and a premixed flame discontinuity were solved there is a limitation however in that the method appears to be most suitable for application to inviscid problems the reason for this limitation and possible approaches toward resolving it are discussed

spectral optimization for the stekloff laplacian the stability issue abstract we consider the problem of minimizing the first non trivial stekloff eigenvalue of the laplacian among sets with given measure we prove that the brock weinstock inequality asserting that optimal shapes for this spectral optimization problem are balls can be improved by means of a sharp quantitative stability estimate this result is based on the analysis of a certain class of weighted isoperimetric inequalities already proved in betta et al 1999 2 we provide some new sharp quantitative versions of these achieved by means of a suitable calibration technique

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

on hopf s formulas for solutions of hamilton jacobi equations soit h r n r on considere l equation de hamilton jacobi u t h du 0 dans r n 1 r n 0 on etudie les proprietes des formules de hopf pour les solutions

level set methods for the simulation of epitaxial phenomena we introduce a model for epitaxial phenomena based on the motion of island boundaries which is described by the level set method our model treats the growing film as a continuum in the lateral direction but retains atomistic discreteness in the growth direction an example of such an island dynamics model using the level set method is presented and compared with the corresponding rate equation description extensions of our methodology to more general settings are then discussed s1063 651x 98 50212 7

smoothing low snr molecular images via anisotropic median diffusion we propose a new anisotropic diffusion filter for denoising low signal to noise molecular images this filter which incorporates a median filter into the diffusion steps is called an anisotropic median diffusion filter this hybrid filter achieved much better noise suppression with minimum edge blurring compared with the original anisotropic diffusion filter when it was tested on an image created based on a molecular image model the universal quality index proposed in this paper to measure the effectiveness of denoising suggests that the anisotropic median diffusion filter can retain adherence to the original image intensities and contrasts better than other filters in addition the performance of the filter is less sensitive to the selection of the image gradient threshold during diffusion thus making automatic image denoising easier than with the original anisotropic diffusion filter the anisotropic median diffusion filter also achieved good denoising results on a piecewise smooth natural image and real raman molecular images

a simple level set method for solving stefan problems a simple level set method for solving stefan problems is presented this method can be applied to problems involving dendritic solidification our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance our method is accurate with respect to some exact solutions of the stefan problem results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

image quantization using reaction diffusion equations in this paper we present an image quantization model based on a reaction diffusion partial differential equation the quantized image is given by the asymptotic state of this equation existence and uniqueness of the solution are proved in the framework of viscosity solutions we introduce an l infty stable algorithm in order to compute numerically the solution of the equation and some experimental results are shown a new energy functional based on the classical lloyd method is used to compute the quantizer codewords

multicomponent flow calculations by a consistent primitive algorithm the dynamics of inviscid multicomponent fluids may be modelled by the euler equations augmented by one or more additional species equation s attempts to compute solutions for extended euler models in conservation form show strong oscillations and other computational inaccuracies near material interfaces these are due to erroneous pressure fluctuations generated by the conservative wave model this problem does not occur in single component computations and arises only in the presence of several species a nonconservative primitive euler formulation is proposed which results in complete elimination of the oscillations the numerical algorithm uses small viscous perturbations to remove leading order conservation errors and is conservative to the order of numerical approximation numerical experiments show clean monotonic solution profiles with acceptably small conservation error for shocks of weak to moderate strengths

motion of multiple junctions a level set approach abstract a coupled level set method for the motion of multiple junctions is proposed the new method extends the hamilton jacobi level set formulation of osher and sethian it retains the feature of tracking fronts by following level sets and allows the specification of arbitrary velocities on each front the diffusion equation is shown to generate curvature dependent motion and this is used to develop an algorithm to move multiple junctions with curvature dependent speed systems of reaction diffusion equations are shown to possess inherent properties which prohibit efficient numerical solutions when applied to curvature dependent motion

signal matching through scale space given a collection of similar signals that have been deformed with respect to each other the general signal matching problem is to recover the deformation we formulate the problem as the minimization of an energy measure that combines a smoothness term and a similarity term the minimization reduces to a dynamic system governed by a set of coupled first order differential equations the dynamic system finds an optimal solution at a coarse scale and then tracks it continuously to a fine scale among the major themes in recent work on visual signal matching have been the notions of matching as constrained optimization of variational surface reconstruction and of coarse to fine matching our solution captures these in a precise succinct and unified form results are presented for one dimensional signals a motion sequence and a stereo pair

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

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

cauchy noise removal by nonlinear diffusion equations abstract this paper focuses on the problem of image restoration under cauchy noise the variational method which constructs the data fidelity term involving the cauchy distribution by map estimator has been proven to be a successful approach in this paper a nonlinear diffusion equation is proposed to deal with it the main ingredients of the proposed equation are a gray level based diffusivity that estimates the amplitude of the noise and a classical gradient based diffusivity that controls the anisotropic diffusion according to the image s local structure the proposed equation has the nondivergence form and its properties including the existence uniqueness and stability of solutions are established by the notion of viscosity solution experimental results show the superiority of the proposed equation over variational methods in restoring small details of images

a sharp bound for the ratio of the first two dirichlet eigenvalues of a domain in a hemisphere of ⁿ for a domain q contained in a hemisphere of the n dimensional sphere sn we prove the optimal result a2 ai q a2 al 0 for the ratio of its first two dirichlet eigenvalues where q the symmetric rearrangement of q in sn is a geodesic ball in n having the same n volume as q we also show that a2 a1 for geodesic balls of geodesic radius 01 less than or equal to 7r 2 is an increasing function of 01 which runs between the value jn 2 1 in 2 1 1 2 for 1 0 this is the euclidean value and 2 n l n for 01 7r 2 here jv k denotes the kth positive zero of the bessel function j t this result generalizes the payne polya weinberger conjecture which applies to bounded domains in euclidean space and which we had proved earlier our method makes use of symmetric rearrangement of functions and various technical properties of special functions we also prove that among all domains contained in a hemisphere of sn and having a fixed value of a1 the one with the maximal value of a2 is the geodesic ball of the appropriate radius this is a stronger but slightly less accessible isoperimetric result than that for a2 1i various other results for a1 and a2 of geodesic balls in sn are proved in the course of our work

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

periodic solutions for nonlinear evolution equations at resonance abstract we are concerned with periodic problems for nonlinear evolution equations at resonance of the form u t a u t f t u t where a densely defined linear operator a d a x on a banach space x is such that a generates a compact c 0 semigroup and f 0 x x is a nonlinear perturbation imposing appropriate landesman lazer type conditions on the nonlinear term f we prove a formula expressing the fixed point index of the associated translation along trajectories operator in the terms of a time averaging of f restricted to ker a by the formula we show that the translation operator has a nonzero fixed point index and in consequence we conclude that the equation admits a periodic solution

an automatic based nonlinear diffusion equations scheme for skin lesion segmentation the clinical diagnosis of skin lesions is not an easy task for dermatologists a segmentation method that makes the extraction of important characteristics of the skin lesion image can aid dermatologists in the clinical diagnosis of skin cancer in this paper a segmentation method that combines the use of nonlinear diffusion equations and the canny edge detector for the automatic detection of the skin edge lesions is presented experimental results show the efficacy of the proposed method even when the lesion is noisy or when the lesion is covered by hairs

eigenvalue pinching on convex domains in space forms in this paper we show that the convex domains of the hyperbolic space which are almost extremal for the faber krahn or the payne polya weinberger inequalities are close to geodesic balls our proof is also valid in other space forms and allows us to recover known results in euclidean space and on the sphere

fourth order partial differential equations for noise removal a class of fourth order partial differential equations pdes are proposed to optimize the trade off between noise removal and edge preservation the time evolution of these pdes seeks to minimize a cost functional which is an increasing function of the absolute value of the laplacian of the image intensity function since the laplacian of an image at a pixel is zero if the image is planar in its neighborhood these pdes attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image piecewise planar images look more natural than step images which anisotropic diffusion second order pdes uses to approximate an observed image so the proposed pdes are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion although both approaches seem to be comparable in removing speckles in the observed images speckles are more visible in images processed by the proposed pdes because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges speckles can be easily removed by simple algorithms such as the one presented in this paper

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

regularized variational dynamic stochastic resonance method for enhancement of dark and low contrast image abstract dynamic stochastic resonance dsr is a distinctive technique for enhancement of dark and low contrast image noise is necessary for dsr based image enhancement and the level of noise will be enlarged simultaneously with brightness which reduces the perceptual quality of the enhanced image greatly and also increases the difficulty of subsequent denoising because removing high level of noise often leads to serious loss of image details in this paper instead of removing noise after the enhancement process is complete we propose to suppress noise gradually and simultaneously in the process of enhancement we rewrite the traditional partial differential equation pde based dsr model in variational framework firstly and then propose a novel total variation regularized tv dsr method for image enhancement the existence and uniqueness of solution of the tv regularized dsr model is proved theoretically moreover we generalize the tv regularized dsr model in variational framework and in pde framework respectively and therefore we can incorporate more existing denoising methods into our approach numerical comparisons demonstrate that the proposed technique gives significant performance in terms of contrast and brightness enhancement as well as noise suppression and therefore can obtain enhanced image with good perceptual quality

the averaging principle and periodic solutions for nonlinear evolution equations at resonance abstract we study the existence of t periodic solutions t 0 for the first order differential equations being at resonance at infinity where the right hand side is the perturbations of a sectorial operator our aim is to prove an index formula expressing the topological degree of the associated translation along trajectories operator on appropriately large ball in terms of special geometrical assumptions imposed on the nonlinearity we also prove that the geometrical assumptions are generalizations of well known landesman lazer and strong resonance conditions the obtained index formula is used to derive the criteria determining the existence of t periodic solutions for the heat equation being at resonance at infinity

applying multiquadric quasi interpolation for boundary detection in this paper we propose a novel scheme for simulating geometric active contours geometric flow of one kind applying multiquadric mq quasi interpolation we first represent the geometric flow in its parametric form then we obtain the numerical scheme by using the derivatives of the quasi interpolation to approximate the spatial derivative of each dependent variable and a forward difference to approximate the temporal derivative of each dependent variable the resulting scheme is simple efficient and easy to implement also images with complex boundaries can be more easily proposed on the basis of the good properties of the mq quasi interpolation several biomedical and astronomical examples of applications are shown in the paper comparisons with other methods are included to illustrate the validity of the method

scale space tracking and deformable sheet models for computational vision the deformable sheet a physical model that provides a natural framework for addressing many vision problems that can be solved by smoothness constrained optimization is described deformable sheets are characterized by a global energy functional and the smoothness constraint is represented by a linear internal energy term analogous to physical sheets the model sheets are deformed by problem specific external forces and in turn impose smoothness on the applied forces the model unifies the properties of scale and smoothness into a single parameter that makes it possible to perform scale space tracking by properly controlling the smoothness constraint specifically the desired scale space trajectory is found by solving a differential equation in scale the simple analytic dependence on scale also provides a mechanism for adaptive step size control results from application of the deformable sheet model to various problems in computational vision are presented

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

phase transitions and generalized motion by mean curvature we study the limiting behavior of solutions to appropriately rescaled versions of the allen cahn equation a simplified model for dynamic phase transitions we rigorously establish the existence in the limit of a phase antiphase interface evolving according to mean curvature motion this assertion is valid for all positive time the motion interpreted in the generalized sense of evans spruck and chen giga goto after the onset of geometric singularities

a level set formulation of eulerian interface capturing methods for incompressible fluid flows a level set formulation is derived for incompressible immiscible navier stokes equations separated by a free surface the interface is identified as the zero level set of a smooth function eulerian finite difference methods based on this level set formulation are proposed these methods are robust and efficient and are capable of computing interface singularities such as merging and reconnection numerical experiments are presented to demonstrate the effectiveness of the methods

the constrained degree and fixed point index theory for set valued maps abstract in the first part of the paper we give a construction of a topological degree theory for set valued tangent vector fields with convex and nonconvex values defined on nonsmooth closed subsets of a banach space the obtained homotopy invariant is an extension of the classical degree for vector fields on manifolds in the second part we propose a fixed point index for inward maps on arbitrary closed convex sets

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

multigrid anisotropic diffusion a multigrid anisotropic diffusion algorithm for image processing is presented the multigrid implementation provides an efficient hierarchical relaxation method that facilitates the application of anisotropic diffusion to time critical processes through a multigrid v cycle the anisotropic diffusion equations are successively transferred to coarser grids and used in a coarse to fine error correction scheme when a coarse grid with a trivial solution is reached the coarse grid estimates of the residual error can be propagated to the original grid and used to refine the solution the main benefits of the multigrid approach are rapid intraregion smoothing and reduction of artifacts due to the elimination of low frequency error the theory of multigrid anisotropic diffusion is developed then the intergrid transfer functions relaxation techniques diffusion coefficients and boundary conditions are discussed the analysis includes the examination of the storage requirements the computational cost and the solution quality finally experimental results are reported that demonstrate the effectiveness of the multigrid approach

constrained topological degree and positive solutions of fully nonlinear boundary value problems abstract in the first part of the paper we provide a construction of an abstract homotopy invariant detecting zeros of maps of the form a f where a d a e is a densely defined m accretive operator in a banach space e and f u e is a tangent field defined on an open subset u of a neighborhood retract m being invariant with respect to the resolvents of a the construction is performed under the assumption that resolvents of a are completely continuous in the second part we derive index formulae for isolated zeros and apply them to show the existence of nontrivial positive steady state solutions for a class of nonlinear reaction diffusion equations and equations with one dimensional p laplacian with possibly non positive perturbations as well as some controlled neumann like problems

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

quantizer monotonicities and globally optimal scalar quantizer design new monotonicity properties of optimal scalar quantizers are discussed these monotonicities reveal a globally optimal scalar quantizer structure depending on the probability mass functions and on the number of quantizer levels by incorporating the monotone quantizer structure into a dynamic programming process the time complexities of previous algorithms for designing globally optimal scalar quantizers can be significantly reduced for very general classes of distortion measures

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

nonlinear regularized reaction diffusion filters for denoising of images with textures denoising is always a challenging problem in natural imaging and geophysical data processing in this paper we consider the denoising of texture images using a nonlinear reaction diffusion equation and directional wavelet frames in our model a curvelet shrinkage is used for regularization of the diffusion process to preserve important features in the diffusion smoothing and a wave atom shrinkage is used as the reaction in order to preserve and enhance interesting oriented textures we derive a digital reaction diffusion filter that lives on graphs and show convergence of the corresponding iteration process experimental results and comparisons show very good performance of the proposed model for texture preserving denoising

nonlinear image filtering with edge and corner enhancement the authors describe a model of nonlinear image filtering for noise reduction and edge enhancement using anisotropic diffusion the method is designed to enhance not only edges but corners and t junctions as well the method roughly corresponds to a nonlinear diffusion process with backward heat flow across the strong edges such a process is ill posed making the results depend strongly on how the algorithm differs from the diffusion process two ways of modifying the equations using simulations on a variable grid are studied

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

topological degree methods for perturbations of operators generating compact c 0 semigroups abstract the main aim of this paper is to construct a topological degree for maps a f m d a e where a densely defined closed operator a d a e of a banach space e is such that a is the generator of a compact c 0 semigroup and f m e is a locally lipschitz map defined on a neighborhood retract m e if m is a closed convex cone then a degree formula allowing an effective computation of the degree is proved this formula provides an infinite dimensional counterpart of the well known krasnosel skii theorem by the use of the introduced topological degree and an abstract result concerning branching of fixed points the bifurcation of periodic points of the parameterized boundary value problem u au f t u 0 u t m u 0 u t is studied examples of applications to partial differential equations are discussed

stability results for the first eigenvalue of the laplacian on domains in space forms abstract we studied the two known works on stability for isoperimetric inequalities of the first eigenvalue of the laplacian the earliest work is due to a melas who proved the stability of the faber krahn inequality for a convex domain contained in r n with close to the first eigenvalue of the ball b of the same volume the domain must be close to the ball b with respect to the hausdorff distance later y xu studied the stability of the szego weinberger inequality for convex domains in r n and h n where h n denotes hyperbolic space our work consists of extending a melas result to the spaces of constant curvature s 2 and h 2 and y xu s result to domains contained in the polar cap b 4 in s n

capturing the behavior of bubbles and drops using the variational level set approach we reproduce the general behavior of complicated bubble and droplet motions using the variational level set formulation introduced by the authors earlier our approach here ignores inertial effects thus the motion is only correct as an approximation for very viscous problems however the steady states are true equilibrium solutions inertial forces will be added in future work the problems include soap bubbles colliding and merging drops falling or remaining attached to a generally irregular ceiling and liquid penetrating through a funnel in both two and three dimensions each phase is identified with a particular level set function the zero level set of this function is that particular phase boundary the level set functions all evolve in time through a constrained gradient descent procedure so as to minimize an energy functional the functions are coupled through physical constraints and through the requirements that different phases do not overlap and vacuum regions do not develop both boundary conditions and inequality constraints are cast in terms of either local or global equality constraints the gradient projection method leads to a system of perturbed by curvature if surface tension is involved hamilton jacobi equations coupled through a constraint the coupling is enforced using the lagrange multiplier associated with this constraint the numerical implementation requires much of the modern level set technology in particular we achieve a significant speed up by using the fast localization algorithm of h k zhao m kang b merriman d peng and s osher

isophotes selection and reaction diffusion model for object boundaries estimation this paper investigates generic region based segmentation schemes using area minimization constraint and background modeling and develops a computationally efficient framework based on level lines selection coupled with biased anisotropic diffusion a common approach to image segmentation is to construct a cost function whose minima yield the segmented image this is generally achieved by competition of two terms in the cost function one that punishes deviations from the original image and another that acts as a regularization term we propose a variational framework for characterizing global minimizers of a particular segmentation energy that can generates irregular object boundaries in image segmentation our motivation comes from the observation that energy functionals are traditionally complex for which it is usually difficult to precise global minimizers corresponding to best segmentations in this paper we prove that the set of curves that minimizes the basic energy model under concern is a subset of level lines or isophotes i e the boundaries of image level sets the connections of our approach with region growing techniques snakes and geodesic active contours are also discussed moreover it is absolutely necessary to regularize isophotes delimiting object boundaries and to determine piecewise smooth or constant approximations of the image data inside the objects boundaries for vizualization and pattern recognition purposes thus we have constructed a reaction diffusion process based on the perona malik anisotropic diffusion equation in particular a reaction term has been added to force the solution to remain close to the data inside object boundaries and to be constant in non informative regions that is the background region in the overall approach diffusion requires the design of the background and foreground regions obtained by segmentation and segmentation of the adaptively smoothed image is performed after each iteration of the diffusion process from an application point of view the sound initialization free algorithm is shown to perform well in a variety of imaging contexts with variable texture noise and lighting conditions including optical imaging medical imaging and meteorological imaging depending on the context it yields either a reliable segmentation or a good pre segmentation that can be used as initialization for more sophisticated application dependent segmentation models

a continuum method for modeling surface tension abstract a new method for modeling surface tension effects on fluid motion has been developed interfaces between fluids of different properties or colors are represented as transition regions of finite thickness across which the color variable varies continuously at each point in the transition region a force density is defined which is proportional to the curvature of the surface of constant color at that point it is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero the continuum method eliminates the need for interface reconstruction simplifies the calculation of surface tension enables accurate modeling of two and three dimensional fluid flows driven by surface forces and does not impose any modeling restrictions on the number complexity or dynamic evolution of fluid interfaces having surface tension computational results for two dimensional flows are given to illustrate the properties of the method

on the solution sets of differential inclusions and the periodic problem in banach spaces abstract in this paper the topological structure of the solution set of a constrained semilinear differential inclusion in a banach space e is studied it is shown that the set of all mild solutions with values in a closed and in general thin subset d e is an r set provided natural boundary conditions and appropriate geometrical assumptions on d which hold e g when d is convex are satisfied applications to the periodic problem and to the existence of equilibria are given

a boundary condition capturing method for incompressible flame discontinuities abstract in this paper we propose a new numerical method for treating two phase incompressible flow where one phase is being converted into the other e g the vaporization of liquid water we consider this numerical method in the context of treating discontinuously thin flame fronts for incompressible flow this method was designed as an extension of the ghost fluid method 1999 j comput phys 152 457 and relies heavily on the boundary condition capturing technology developed in liu et al 2000 j comput phys 154 15 for the variable coefficient poisson equation and in kang et al in press j comput phys for multiphase incompressible flow our new numerical method admits a sharp interface representation similar to the method proposed in helenbrook et al 1999 j comput phys 148 366 since the interface boundary conditions are handled in a simple and straightforward fashion the code is very robust e g no special treatment is required to treat the merging of flame fronts the method is presented in three spatial dimensions with numerical examples in one two and three spatial dimensions

a well balanced flow equation for noise removal and edge detection an anisotropic nonlinear diffusion equation for image restoration is presented the model has two terms the diffusion term and the forcing term the balance between these terms is made in a selective way in which boundary points and interior points of the objects that make up the image are treated differently the optimal smoothing time concept which allows for finding the ideal stop time for the evolution of the partial differential equation is also proposed numerical results show the proposed model s high performance

markov chain approximations for deterministic control problems with affine dynamics and quadratic cost in the control we consider the construction of markov chain approximations for an important class of deterministic control problems the emphasis is on the construction of schemes that can be easily implemented and which possess a number of highly desirable qualitative properties the class of problems covered is that for which the control is affine in the dynamics and with quadratic running cost this class covers a number of interesting areas of application including problems that arise in large deviations risk sensitive and robust control robust filtering and certain problems in computer vision examples are given as well as a proof of convergence

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

hybrid multifluid algorithms extensions of many successful single component schemes to compute multicomponent gas dynamics suffer from oscillations and other computational inaccuracies near material interfaces that are caused by the failure of the schemes to maintain pressure equilibrium between the fluid components a new algorithm based on the compressible euler equations for multicomponent fluids augmented by the pressure evolution equation is presented the extended set of equations offers two alternative ways to update the pressure field i using the equation of state or ii using the pressure evolution equation in a numerical implementation these two procedures generally yield different answers the former is a standard conservative update but may produce oscillations near material interfaces the latter is nonconservative but becomes exact near interfaces and automatically maintains pressure equilibrium a hybrid scheme which selects from the two pressure update procedures is presented the scheme perfectly conserves total mass and momentum and conserves total energy everywhere except at a finite very small number of grid cells computed solutions exhibit oscillation free interfaces and have em negligible relative conservation errors in total energy even for very strong shocks the proposed hybrid approach and switching strategies are independent of the numerical implementation and may provide a simple framework within which to extend one s favourite scheme to solve multifluid dynamics

existence for reaction diffusion systems a compactness method approach the authors study the existence of weak solutions for the following system ut u f u v vt v g u v in 0 t u v 0 on 0 t u 0 x u0 x v 0 x v0 x in the region rn with smooth boundary the functions r r are assumed to be continuous and nondecreasing with 0 0 0 u0 v0 l f g r2 2r with f an upper semicontinuous mapping u s c the following local existence results are shown 1 for the diffusive case i e when both and are strictly increasing with u s c g 2 for the semi diffusive case only one function is strictly increasing with g being either with separated variables i e having the form of the product or of the sum of two functions g u and h v or globally lipschitz with respect to its second variable i e g u v g u v l v v for each u b r v v r and some l l b additional conditions of linear form on the growth of f and g are indicated to guarantee global existence results

motion of curves in three spatial dimensions using a level set approach the level set method was originally designed for problems dealing with codimension one objects where it has been extremely succesful especially when topological changes in the interface i e merging and breaking occur attempts have been made to modify it to handle objects of higher codimension such as vortex filaments while preserving the merging and breaking property we present numerical simulations of a level set based method for moving curves in r3 the model problem for higher codimension that allows for topological changes a vector valued level set function is used with the zero level set representing the curve our results show that this method can handle many types of curves moving under all types of geometrically based flows while automatically enforcing merging and breaking

regularization of ill posed problems via the level set approach we introduce a new formulation for the motion of curves in r2 easily extendable to the motion of surfaces in r3 when the original motion generally corresponds to an ill posed problem such as the cauchy riemann equations this is in part a generalization of our earlier work in 6 where we applied similar ideas to compute flows with highly concentrated vorticity such as vortex sheets or dipoles for incompressible euler equations our new formulation involves extending the level set method of 12 to problems in which the normal velocity is not intrinsic we obtain a coupled system of two equations one of which is a level surface equation this yields a fixed grid eulerian method which regularizes the ill posed problem in a topological fashion we also present an analysis of curvature regularizations and some other theoretical justification finally we present numerical results showing the stability properties of our approach and the novel nature ofthe regularization including the development o