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

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

a level set approach to 3d reconstruction from range data this paper presents a method that uses the level sets of volumes to reconstruct the shapes of 3d objects from range data the strategy is to formulate 3d reconstruction as a statistical problem find that surface which is mostly likely given the data and some prior knowledge about the application domain the resulting optimization problem is solved by an incremental process of deformation we represent a deformable surface as the level set of a discretely sampled scalar function of three dimensions i e a volume such level set models have been shown to mimic conventional deformable surface models by encoding surface movements as changes in the greyscale values of the volume the result is a voxel based modeling technology that offers several advantages over conventional parametric models including flexible topology no need for reparameterization concise descriptions of differential structure and a natural scale space for hierarchical representations this paper builds on previous work in both 3d reconstruction and level set modeling it presents a fundamental result in surface estimation from range data an analytical characterization of the surface that maximizes the posterior probability it also presents a novel computational technique for level set modeling called the sparse field algorithm which combines the advantages of a level set approach with the computational efficiency and accuracy of a parametric representation the sparse field algorithm is more efficient than other approaches and because it assigns the level set to a specific set of grid points it positions the level set model more accurately than the grid itself these properties computational efficiency and subcell accuracy are essential when trying to reconstruct the shapes of 3d objects results are shown for the reconstruction objects from sets of noisy and overlapping range maps

isolating segments fixed point index and symbolic dynamics ii homoclinic solutions we present a topological method for detecting homoclinic solutions in nonautonomous ode s as an application we show that there exist infinitely many geometrically distinct homoclinic solutions to the trivial 0 solution in the planar systemz z 1 ei t z 2 for 0 0 495

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

a method for registration of 3 d shapes the authors describe a general purpose representation independent method for the accurate and computationally efficient registration of 3 d shapes including free form curves and surfaces the method handles the full six degrees of freedom and is based on the iterative closest point icp algorithm which requires only a procedure to find the closest point on a geometric entity to a given point the icp algorithm always converges monotonically to the nearest local minimum of a mean square distance metric and the rate of convergence is rapid during the first few iterations therefore given an adequate set of initial rotations and translations for a particular class of objects with a certain level of shape complexity one can globally minimize the mean square distance metric over all six degrees of freedom by testing each initial registration one important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model prior to shape inspection experimental results show the capabilities of the registration algorithm on point sets curves and surfaces

qualocation for a singularly perturbed boundary value problem a singularly perturbed one dimensional two point boundary value problem of reaction convection diffusion type is considered we generate a c 0 collocation like method by combining galerkin with an adapted quadrature rule using lobatto quadrature and splines of degree r we prove on a shishkin mesh for the qualocation method the same error estimate as for the galerkin technique the result is also important for the practical realization of finite element methods on shishkin meshes using quadrature formulas we report the results of numerical experiments that support the theoretical findings

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 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

isolating segments fixed point index and symbolic dynamics iii applications abstract we prove the existence of complicated dynamics in some class of periodic systems on the plane the proof is based on the continuation of the considered system to the model one

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

chaotic dynamics in a vertically driven planar pendulum as a first application we prove the presence of chaotic dynamics for a vertically driven planar pendulum the theory of topological horseshoes and linked twist maps developed so far along with phase plane analysis are used to show the presence of chaotic dynamics we propose in the general setting of topological spaces a definition of two dimensional oriented cell and consider maps which possess a property of stretching along the paths with respect to oriented cells for these maps we prove some theorems on the existence of fixed points periodic points and sequences of iterates which are chaotic in a suitable manner our results motivated by the study of the poincare map associated to some nonlinear equations extend and improve some recent work

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

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

remark on complicated dynamics of some planar system abstract in this paper we study the complicated dynamics generated by the planar periodic system z z 1 e i t z 2 for 0 0 495 we prove the existence of infinitely many geometrically distinct periodic solutions with some special properties

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

reduced bezier element quadrature rules for quadratic and cubic splines in isogeometric analysis abstract we explore the use of various element based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis the rules studied encompass tensor product gauss and gauss lobatto rules and certain so called monomial rules that do no possess a tensor product structure the objective of the study is to determine quadrature strategies which enjoy the same accuracy and stability behavior as full gauss quadrature but with significantly fewer quadrature points several cases emerge that satisfy this objective and also demonstrate superior efficiency compared with standard c 0 continuous finite elements of the same order

a geometric method for detecting chaotic dynamics a new method of detection of chaos in dynamical systems generated by time periodic nonautonomous differential equations is presented it is based on the existence of some sets called periodic isolating segments in the extended phase space satisfying some topological conditions by chaos we mean the existence of a compact invariant set such that the poincare map is semiconjugated to the shift on two symbols and the counterimage by the semiconjugacy of any periodic point in the shift contains a periodic point of the poincare map as an application we prove that the planar equationz 1 ei t z 2 zgenerates chaotic dynamics provided 0 1 288

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

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

isolating segments fixed point index and symbolic dynamics abstract an extension of the recently introduced srzednicki wojcik method for detecting chaotic dynamics in periodically forced ordinary differential equations is presented as an application of the method we construct a topological model for the planar equation z 1 e i t z 2 z z c 1 and we show by a continuation argument that the symbolic dynamics on three symbols for the topological model continues to eq 1 for 0

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

the method of holomorphic foliations in planar periodic systems the case of riccati equations abstract we develop a new method of investigation of periodic solutions of planar holomorphic polynomial differential equations with trigonometric coefficients this method relies on analysis of geometry of leaves of certain holomorphic foliations in cp1 cp1 we concentrate our attention on the case of riccati equations we determine the dynamics of the non autonomous equations using some local invariants of holomorphic vector fields defining the foliations in particular new examples of riccati equations without periodic solutions are presented

planar nonautonomous polynomial equations ii coinciding sectors abstract we give a few sufficient conditions for the existence of periodic solutions of the equation z j 0 n a j t z j where a j s are complex valued we prove the existence of one up to two periodic solutions and heteroclinic ones

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

global dense multiscale reconstruction for a billion points we present a variational approach for surface reconstruction from a set of oriented points with scale information we focus particularly on scenarios with nonuniform point densities due to images taken from different distances in contrast to previous methods we integrate the scale information in the objective and globally optimize the signed distance function of the surface on a balanced octree grid we use a finite element discretization on the dual structure of the octree minimizing the number of variables the tetrahedral mesh is generated efficiently with a lookup table which allows to map octree cells to the nodes of the finite elements we optimize memory efficiency by data aggregation such that robust data terms can be used even on very large scenes the surface normals are explicitly optimized and used for surface extraction to improve the reconstruction at edges and corners

topological entropy for local processes abstract the aim of this article is to formalize definition of chaos in terms of topological entropy for dynamics of processes described by nonautonomous differential equations we state a formal definition of topological entropy in this setting and provide tools for estimation of its value its upper or lower bounds in terms of poincare sections

chaotic dynamics in periodically forced asymmetric ordinary differential equations abstract using a topological approach we prove the existence of infinitely many periodic solutions and the presence of chaotic dynamics for the periodically forced second order ode u b u a u p t the choice of the equation is motivated by the studies about the dancer fucik spectrum and the lazer mckenna suspension bridge model

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

a c 0 collocation like method for two point boundary value problems a method of a collocation type based onc0 piecewise polynomial spaces is presented for a two point boundary value problem of the second order the method has an optimal order of convergence under smoothness requirements on the exact solution which are weaker than forc1 collocation methods if the differential operator is symmetric a modification of this method leads to a symmetric system of linear equations it is shown that if the collocation solution is a piecewise polynomial of degree not greater thanr the method is stable and convergent with orderh r inh1 norm a similar symmetric modification forc0 colloction finite element method 7 is also obtained superconvergence at the nodes is established

positive periodic solutions of second order nonlinear equations with indefinite weight multiplicity results and complex dynamics we prove the existence of a pair of positive t periodic solutions as well as the existence of positive subharmonic solutions of any order and the presence of chaotic like dynamics for the scalar second order ode u a t g u 0 where g x is a positive function on r superlinear at zero and sublinear at infinity and a t is a t periodic and sign indefinite weight of the form a t a t with 0 and large

periodic points of the planar area preserving poincare map inside isolating segment abstract the aim of this note is to study the structure of the set of periodic points of the area preserving poincare map p associated to t periodic local planar process we will use the approach based on the method of periodic isolating segments isolating chains and the fixed point index theory the number of the applications to the processes generated by periodic in time non autonomous planar ode s are given

a collocated c0 finite element method reduced quadrature perspective cost comparison with standard finite elements and explicit structural dynamics summary we demonstrate the potential of collocation methods for efficient higher order analysis on standard nodal finite element meshes we focus on a collocation method that is variationally consistent and geometrically flexible converges optimally embraces concepts of reduced quadrature and leads to symmetric stiffness and diagonal consistent mass matrices at the same time it minimizes the evaluation cost per quadrature point thus reducing formation and assembly effort significantly with respect to standard galerkin finite element methods we provide a detailed review of all components of the technology in the context of elastodynamics that is weighted residual formulation nodal basis functions on gauss lobatto quadrature points and symmetrization by averaging with the ultra weak formulation we quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort such as higher order elastostatic analysis furthermore we illustrate the potential of collocation for efficient higher order explicit dynamics throughout this work we advocate a straightforward implementation based on simple modifications of standard finite element codes we also point out the close connection to spectral element methods where many of the key ideas are already established copyright 2014 john wiley sons ltd

scene understanding by reasoning stability and safety this paper presents a new perspective for 3d scene understanding by reasoning object stability and safety using intuitive mechanics our approach utilizes a simple observation that by human design objects in static scenes should be stable in the gravity field and be safe with respect to various physical disturbances such as human activities this assumption is applicable to all scene categories and poses useful constraints for the plausible interpretations parses in scene understanding given a 3d point cloud captured for a static scene by depth cameras our method consists of three steps i recovering solid 3d volumetric primitives from voxels ii reasoning stability by grouping the unstable primitives to physically stable objects by optimizing the stability and the scene prior and iii reasoning safety by evaluating the physical risks for objects under physical disturbances such as human activity wind or earthquakes we adopt a novel intuitive physics model and represent the energy landscape of each primitive and object in the scene by a disconnectivity graph dg we construct a contact graph with nodes being 3d volumetric primitives and edges representing the supporting relations then we adopt a swendson wang cuts algorithm to partition the contact graph into groups each of which is a stable object in order to detect unsafe objects in a static scene our method further infers hidden and situated causes disturbances in the scene and then introduces intuitive physical mechanics to predict possible effects e g falls as consequences of the disturbances in experiments we demonstrate that the algorithm achieves a substantially better performance for i object segmentation ii 3d volumetric recovery and iii scene understanding with respect to other state of the art methods we also compare the safety prediction from the intuitive mechanics model with human judgement

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

on periodic solutions of planar polynomial differential equations with periodic coefficients abstract we consider the planar equation z ak l t zk z l where ak l is a t periodic complex valued continuous function equal to 0 for almost all k l n we present sufficient conditions imposed on ak l which guarantee the existence of its t periodic solutions and in the case a0 0 0 the conditions for the existence of nonzero ones we use a method which computes the fixed point index of the poincare andronov operator in isolated sets of fixed points generated by so called periodic blocks the method is based on the lefschetz fixed point theorem and the topological principle of wazewski

iterative point matching for registration of free form curves and surfaces a heuristic method has been developed for registering two sets of 3 d curves obtained by using an edge based stereo system or two dense 3 d maps obtained by using a correlation based stereo system geometric matching in general is a difficult unsolved problem in computer vision fortunately in many practical applications some a priori knowledge exists which considerably simplifies the problem in visual navigation for example the motion between successive positions is usually approximately known from this initial estimate our algorithm computes observer motion with very good precision which is required for environment modeling e g building a digital elevation map objects are represented by a set of 3 d points which are considered as the samples of a surface no constraint is imposed on the form of the objects the proposed algorithm is based on iteratively matching points in one set to the closest points in the other a statistical method based on the distance distribution is used to deal with outliers occlusion appearance and disappearance which allows us to do subset subset matching a least squares technique is used to estimate 3 d motion from the point correspondences which reduces the average distance between points in the two sets both synthetic and real data have been used to test the algorithm and the results show that it is efficient and robust and yields an accurate motion estimate

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

adaptively merging large scale range data with reflectance properties in this paper we tackle the problem of geometric and photometric modeling of large intricately shaped objects typical target objects we consider are cultural heritage objects when constructing models of such objects we are faced with several important issues that have not been addressed in the past issues that mainly arise due to the large amount of data that has to be handled we propose two novel approaches to efficiently handle such large amounts of data a highly adaptive algorithm for merging range images and an adaptive nearest neighbor search to be used with the algorithm we construct an integrated mesh model of the target object in adaptive resolution taking into account the geometric and or photometric attributes associated with the range images we use surface curvature for the geometric attributes and laser reflectance values for the photometric attributes this adaptive merging framework leads to a significant reduction in the necessary amount of computational resources furthermore the resulting adaptive mesh models can be of great use for applications such as texture mapping as we will briefly demonstrate additionally we propose an additional test for the k d tree nearest neighbor search algorithm our approach successfully omits back tracking which is controlled adaptively depending on the distance to the nearest neighbor since the main consumption of computational cost lies in the nearest neighbor search the proposed algorithm leads to a significant speed up of the whole merging process in this paper we present the theories and algorithms of our approaches with pseudo code and apply them to several real objects including large scale cultural assets

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

remark on krasnosielski s guiding functions and srzednicki s periodic segments abstract we show that the existence of the complete set of guiding functions guarantees the existence of periodic isolating segment that carries the same information concerning periodic solutions of the non autonomous periodic equations x f x t

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

hyperbolicity and averaging for the srzednicki wojcik equation for the srzednicki wojcik equation the planar nonautonomous ode parameterized by r z z 1 z 2exp i t z t c using averaging we show how the region of hyperbolicity grows with based on this we give bounds on the sizes of bounded orbits

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

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

recovering the position and orientation of free form objects from image contours using 3d distance maps the accurate matching of 3d anatomical surfaces with sensory data such as 2d x ray projections is a basic problem in computer and robot assisted surgery in model based vision this problem can be formulated as the estimation of the spatial pose position and orientation of a 3d smooth object from 2d video images the authors present a new method for determining the rigid body transformation that describes this match the authors method performs a least squares minimization of the energy necessary to bring the set of the camera contour projection lines tangent to the surface to correctly deal with projection lines that penetrate the surface the authors consider the minimum signed distance to the surface along each line i e distances inside the object are negative to quickly and accurately compute distances to the surface the authors introduce a precomputed distance map represented using an octree spline whose resolution increases near the surface this octree structure allows the authors to quickly find the minimum distance along each line using best first search experimental results for 3d surface to 2d projection matching are presented for both simulated and real data the combination of the authors problem formulation in 3d their computation of line to surface distances with the octree spline distance map and their simple minimization technique based on the levenberg marquardt algorithm results in a method that solves the 3d 2d matching problem for arbitrary smooth shapes accurately and quickly

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

isolating chains and chaotic dynamics in planar nonautonomous odes abstract we present a topological method for detecting complicated dynamics in nonautonomous periodic ode s the method is based on the notion of isolating chain as an application we show the chaotic behaviour of processes generated by the equations in the complex plane z z k e i t z where k 3 is an even number and is close to 0

a general surface approach to the integration of a set of range views this paper presents a new and general solution to the problem of range view integration the integration problem consists in computing a connected surface model from a set of registered range images acquired from different viewpoints the proposed method does not impose constraints on the topology of the observed surfaces the position of the viewpoints or the number of views that can be merged the integrated surface model is piecewise estimated by a set of triangulations modeling each canonical subset of the venn diagram of the set of range views the connection of these local models by constrained delaunay triangulations yields g non redundant surface triangulation describing all surface elements sampled by the set of range views experimental results show that the integration technique can be used to build connected surface models of free form objects no integrated models built from objects of such complexity have yet been reported in the literature it is assumed that accurate range views are available and that frame transformations between all pairs of views can be reliably computed

planar nonautonomous polynomial equations the riccati equation abstract we give a few sufficient conditions for the existence of two periodic solutions of the riccati ordinary differential equation in the plane we give also examples of the equation without periodic solutions

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

superlinear indefinite equations on the real line and chaotic dynamics abstract in this paper we are concerned with a differential equation of the form x c x q t g x 0 t a b where a b q has infinitely many zeros in a b and g is superlinear we prove the existence of solutions with prescribed nodal properties in the intervals of negativity and positivity of q when c 0 and q is periodic we show that the equation under consideration exhibits chaotic like dynamics

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

on periodic solutions inside isolating chains abstract we develop a geometric approach to problems concerning the existence of t periodic solutions of a non autonomous time t periodic ordinary differential equation we consider isolating segments subsets of the extended phase space of the equation which in some ways resemble isolating blocks from the theory of isolated invariant sets the union of several contiguous isolating segments is called an isolating chain isolating segments determine some homomorphisms in reduced singular homologies the main theorem asserts that the lefschetz number of the composition of the homomorphisms determined by segments such that their union is a periodic isolating chain is equal to the fixed point index of the poincare map of the equation in the set of initial values of t periodic solutions contained inside the chain we give some applications of the theorem to planar polynomial equations in particular we prove that the equation z z 5 sin 2 t z has four nonzero periodic solutions provided 0 336

quaternionic valued ordinary differential equations the riccati equation abstract we give some sufficient conditions for the existence of at least two periodic solutions of the quaternionic riccati equation in some cases we are able to give a full description of dynamics and detect solutions heteroclinic to the periodic ones we also provide an example of the riccati equation without periodic solutions which appears in the euler vorticity dynamics

hole filling of a 3d model by flipping signs of a signed distance field in adaptive resolution when we use range finders to observe the shape of an object many occluded areas may occur these become holes and gaps in the model and make it undesirable for various applications we propose a novel method to fill holes and gaps to complete this incomplete model as an intermediate representation we use a signed distance field sdf which stores euclidean signed distances from a voxel to the nearest point of the mesh model by using an sdf we can obtain interpolating surfaces for holes and gaps the proposed method generates an interpolating surface that becomes smoothly continuous with real surfaces by minimizing the area of the interpolating surface since the isosurface of an sdf can be identified as being a real or interpolating surface from the magnitude of signed distances our method computes the area of an interpolating surface in the neighborhood of a voxel both before and after flipping the sign of the signed distance of the voxel if the area is reduced by flipping the sign then our method changes the sign for the voxel therefore we minimize the area of the interpolating surface by iterating this computation until convergence unlike methods based on partial differential equations pdes our method does not require any boundary condition and the initial state that we use is automatically obtained by computing the distance to the closest point of the real surface moreover because our method can be applied to an sdf of adaptive resolution our method efficiently interpolates large holes and gaps of high curvature we tested the proposed method with both synthesized and real objects and evaluated the interpolating surfaces

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

structural indexing efficient 3 d object recognition the authors present an approach for the recognition of multiple 3 d object models from three 3 d scene data the approach uses two different types of primitives for matching small surface patches where differential properties can be reliably computed and lines corresponding to depth or orientation discontinuities these are represented by splashes and 3 d curves respectively it is shown how both of these primitives can be encoded by a set of super segments consisting of connected linear segments these super segments are entered into a table and provide the essential mechanism for fast retrieval and matching the issues of robustness and stability of the features are addressed in detail the acquisition of the 3 d models is performed automatically by computing splashes in highly structured areas of the objects and by using boundary and surface edges for the generation of 3 d curves the authors present results with the current system 3 d object recognition based on super segments and discuss further extensions

accurate integration of multi view range images using k means clustering 3d modelling finds a wide range of applications in industry however due to the presence of surface scanning noise accumulative registration errors and improper data fusion reconstructed object surfaces using range images captured from multiple viewpoints are often distorted with thick patches false connections blurred features and artefacts moreover the existing integration methods are often expensive in the sense of both computational time and data storage these shortcomings limit the wide applications of 3d modelling using the latest laser scanning systems in this paper the k means clustering approach from the pattern recognition and machine learning literatures is employed to minimize the integration error and to optimize the fused point locations to initialize the clustering approach an automatic method is developed shifting points in the overlapping areas between neighbouring views towards each other so that the initialized cluster centroids are in between the two overlapping surfaces this results in more efficient and effective integration of data while the overlapping areas were initially detected using a single distance threshold they are then refined using the k means clustering method for more accurate integration results a weighting scheme reflecting the imaging principle is developed to integrate the corresponding points in the overlapping areas the fused point set is finally triangulated using an improved delaunay method guaranteeing a watertight surface a comparative study based on real images shows that the proposed algorithm is efficient in the sense of either running time or memory usage and reduces significantly the integration error while desirably retaining geometric details of 3d object surfaces of interest

example of a suspension bridge ode model exhibiting chaotic dynamics a topological approach abstract using an elementary phase plane analysis combined with some recent results on topological horseshoes and fixed points for planar maps we prove the existence of infinitely many periodic solutions as well as the presence of chaotic dynamics for a simple second order nonlinear ordinary differential equation arising in the study of lazer mckenna suspension bridges model

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

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