vertical square functions and other operators associated with an elliptic operator abstract we study the vertical and conical square functions defined via elliptic operators in divergence form in general vertical and conical square functions are equivalent operators just in l 2 but when this square functions are defined through the heat or poisson semigroups that arise from an elliptic operator we are able to find open intervals containing 2 where the equivalence holds the intervals in question depend ultimately on the range where the semigroup is uniformly bounded or has off diagonal estimates as a consequence we obtain new boundedness results for some square functions besides we consider a non tangential maximal function associated with the poisson semigroup and extend the known range where that operator is bounded our methods are based on the use of extrapolation for muckenhoupt weights and change of angle estimates all our results are obtained in the general setting of a degenerate elliptic operator where the degeneracy is given by an a 2 weight in weighted lebesgue spaces of course they are valid in the unweighted and or non degenerate situations which can be seen as special cases and they provide new results even in those particular settings we also consider the square root of a degenerate elliptic operator in divergence form l w and improve the lower bound of the interval where this operator is known to be bounded on l p v d w finally we give unweighted boundedness results for the degenerate operators under consideration

finite element methods for active contour models and balloons for 2 d and 3 d images the use of energy minimizing curves known as snakes to extract features of interest in images has been introduced by kass witkin and terzopoulos 1987 a balloon model was introduced by cohen 1991 as a way to generalize and solve some of the problems encountered with the original method a 3 d generalization of the balloon model as a 3 d deformable surface which evolves in 3 d images is presented it is deformed under the action of internal and external forces attracting the surface toward detected edgels by means of an attraction potential we also show properties of energy minimizing surfaces concerning their relationship with 3 d edge points to solve the minimization problem for a surface two simplified approaches are shown first defining a 3 d surface as a series of 2 d planar curves then after comparing finite element method and finite difference method in the 2 d problem we solve the 3 d model using the finite element method yielding greater stability and faster convergence this model is applied for segmenting magnetic resonance images

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

gaussian bounds for degenerate parabolic equations abstract let a be a real symmetric degenerate elliptic matrix whose degeneracy is controlled by a weight w in the a 2 or qc class we show that there is a heat kernel w t x y associated to the parabolic equation w u t div a u and w t satisfies classic gaussian bounds w t x y c 1 t n 2 exp c 2 x y 2 t we then use this bound to derive a number of other properties of the kernel

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 nonconservative flow field for robust variational image segmentation we introduce a robust image segmentation method based on a variational formulation using edge flow vectors we demonstrate the nonconservative nature of this flow field a feature that helps in a better segmentation of objects with concavities a multiscale version of this method is developed and is shown to improve the localization of the object boundaries we compare and contrast the proposed method with well known state of the art methods detailed experimental results are provided on both synthetic and natural images that demonstrate that the proposed approach is quite competitive

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

an eulerian method for capturing caustics a robust numerical method for the localization of caustics is proposed for general hamiltonians it is based on the direct resolution of a system of partial differential equations obtained through a local change of the time variable in the hamilton jacobi equation and complemented by a set of transport equations numerical results 1 to 3 d are presented

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

gaussian quadrature involving einstein and fermi functions with an application to summation of series polynomials 7rk rk d x k 0 1 2 are constructed which are orthogonal with respect to the weight distributions da t t et 1 dt and da t 1 et 1 dt r 1 2 on 0 oo moment related methods being inadequate a discretized stieltjes procedure is used to generate the coefficients ak pk in the recursion formula 7rk 1 t t ak k t fk7rk1 t k 0 1 2 so t 1 17j l t 0 the discretization is effected by the gauss laguerre and a composite fejer quadrature rule respectively numerical values of ak ipk as well as associated error constants are provided for 0 k 39 these allow the construction of gaussian quadrature formulae including error terms with up to 40 points examples of n point formulae n 5 5 40 are provided in the supplements section at the end of this issue such quadrature formulae may prove useful in solid state physics calculations and can also be applied to sum slowly convergent series

hamilton jacobi equations on a manifold and applications to grid generation or refinement in this paper we consider hamilton jacobi equations on a manifold typically on the graph of some previously computed function z x y and we show how the corresponding level set method allows us to generate and or to refine a mesh in regions where this function z has large derivatives such as it is the method needs to be strongly improved and accelerated but the principle is awfully natural and in principle the method is fully automatic similar ideas are also useful in image processing in particular for the active contours method

the kato problem for operators with weighted ellipticity we consider second order operators lw w 1 divawr with ellipticity controlled by a muckemphout a2 weight w we prove that the kato square root estimate l 1 2 w f l2 w kr fkl2 w holds in the weighted space l 2 w

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

a multiphase level set framework for image segmentation using the mumford and shah model we propose a new multiphase level set framework for image segmentation using the mumford and shah model for piecewise constant and piecewise smooth optimal approximations the proposed method is also a generalization of an active contour model without edges based 2 phase segmentation developed by the authors earlier in t chan and l vese 1999 in scale space 99 m nilsen et al eds lncs vol 1682 pp 141 151 and t chan and l vese 2001 ieee ip 10 2 266 277 the multiphase level set formulation is new and of interest on its own by construction it automatically avoids the problems of vacuum and overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies including triple junctionss in the piecewise smooth case only two level set functions formally suffice to represent any partition based on the four color theorem finally we validate the proposed models by numerical results for signal and image denoising and segmentation implemented using the osher and sethian level set method

convergence analysis for a class of high order semi lagrangian advection schemes the convergence properties of a class of high order semi lagrangian schemes for pure advection equations are studied here in the framework of the theory of viscosity solutions we review the general convergence results for discrete time approximation schemes belonging to that class and we prove some a priori estimates in l infty and l2 for the rate of convergence of fully discrete schemes we prove then that a careful coupling of time and space discretizations can allow large time steps in the numerical integration still preserving the accuracy of the solutions several examples of schemes and numerical tests are presented

an active contour model and its algorithms with local and global gaussian distribution fitting energies in this paper we propose an active contour model and its corresponding algorithms with detailed implementation for image segmentation in the proposed model the local and global region fitting energies are described by the combination of the local and global gaussian distributions with different means and variances respectively in this combination we increase a weighting coefficient by which we can adjust the ratio between the local and global region fitting energies then we present an algorithm for implementing the proposed model directly considering that in practice the selection of the weighting coefficient is troublesome we present a modified algorithm in order to overcome this problem and increase the flexibility by adaptively updating the weighting coefficient and the time step with the contour evolution this algorithm is less sensitive to the initialization of the contour and can speed up the convergence rate besides it is robust to the noise and can be used to extract the desired objects experiment results demonstrate that the proposed model and its algorithms are effective with application to both the synthetic and real world images

a reformulation of the conservative level set reinitialization equation for accurate and robust simulation of complex multiphase flows abstract this paper presents an accurate and robust reinitialization equation for the conservative level set that does not significantly deform stationary surfaces the compression and diffusion term of the reinitialization equation are reformulated to use a distance level set directly mapped from the conservative level set the normals are calculated using a distance level set reconstructed from the interface using a fast marching method increasing robustness and allowing the use of high order non tvd transport schemes using this new reinitialization equation we present results for canonical test cases such as zalesak s disk and spurious currents which show significant improvement a simulation of a liquid gas jet with r e 5000 is also presented to demonstrate the volume conservation properties of the method in more complex flows

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

the solution of the kato problem for degenerate elliptic operators with gaussian bounds we prove the kato conjecture for degenerate elliptic operators on r n more precisely we consider the divergence form operator lw w 1 diva where w is a muckenhoupt a2 weight and a is a complex valued n n matrix such that w 1 a is bounded and uniformly elliptic we show that if the heat kernel of the associated semigroup e tlw satisfies gaussian bounds then the weighted kato square root estimate kl 1 2 w f k l2 w k f k l2 w holds

a new incompressibility discretization for a hybrid particle mac grid representation with surface tension we take a particle based approach to incompressible free surface flow motivated by the fact that an explicit representation of the interface geometry and internal deformations gives precise feedback to an implicit solver for surface tension methods that enforce incompressibility directly on the particles are typically numerically inefficient compared to those that utilize a background grid however background grid discretizations suffer from inaccuracy near the free surface where they do not properly capture the interface geometry therefore our incompressibility discretization utilizes a particle based projection near the interface and a background mac grid based projection for efficiency in the vast interior of the liquid domain as well as a novel method for coupling these two disparate projections together we show that the overall coupled elliptic solver is second order accurate and remains second order accurate when used in conjunction with an appropriate temporal discretization for parabolic problems a similar second order accurate discretization is derived when the mac grid unknowns are located on faces as opposed to cell centers so that navier stokes viscosity can be solved for implicitly as well finally we present a fully implicit approach to surface tension that is robust enough to achieve a steady state solution in a single time step beyond stable implicit surface tension for our novel hybrid discretization we demonstrate preliminary results for both standard front tracking and the particle level set method

a geometrical optics based numerical method for high frequency electromagnetic fields computations near fold caustics part i this paper presents an eulerian numerical method for the computation of a bi valued solution of hamilton jacobi type equation in a particular geometric setting more precisely we consider high frequency electromagnetic fields in the vicinity of fold caustics

divergence of gradient convolution deformable segmentation with arbitrary initializations in this paper we propose a unified approach to deformable model based segmentation the fundamental force field of the proposed method is based on computing the divergence of a gradient convolution field gcf which makes the full use of directional information of the image gradient vectors and their interactions across image domain however instead of directly using such a vector field for deformable segmentation as in the conventional approaches we derive a more salient representation for contour evolution and very importantly we demonstrate that this representation of image force field not only leads to global minimum through convex relaxation but also can achieve the same result using the conventional gradient descent with an intrinsic regularization thus the proposed method can handle arbitrary initializations the proposed external force field for deformable segmentation has both edge based properties in that the gcf is computed from image gradients and the region based attributes since its divergence can be treated as a region indication function moreover nonlinear diffusion can be conveniently applied to gcf to improve its performance in dealing with noise interference we also show the extension of gcf from 2d to 3d in comparison to the state of the art deformable segmentation techniques the proposed method shows greater flexibility in model initialization and optimization realization as well as better performance toward noise interference and appearance variation

approximations for viscosity solutions of hamilton jacobi equations with locally varying time and space grids a new monotone finite difference scheme is introduced that approximates viscosity solutions of first order nonlinear hamilton jacobi equations the main feature of the scheme is that it allows for locally varying time and space grids making it ideal for use with adaptive algorithms explicit a priori error estimates are given to establish convergence numerical examples including adaptive mesh refinement examples demonstrate the effectiveness of the proposed scheme

on certain slowly convergent series occurring in plate contact problems a simple computational procedure is developed for accurately summing series of the form z 0 2k 1 pz2k 1 where z is complex with jzi 1 and p 2 or 3 as well as series of the type

a simple explicit operator splitting method for effective hamiltonians understanding effective hamiltonians quantitatively is essential for the homogenization of hamilton jacobi equations we propose in this article a simple efficient operator splitting method for computing effective hamiltonians when the hamiltonian is either convex or nonconvex in the gradient variable to speed up our lie scheme based operator splitting method we further propose a cascadic initialization strategy so that the steady state of the underlying time dependent hamilton jacobi equation can be reached more rapidly extensive numerical examples demonstrate the efficiency and accuracy of the new algorithm

convergence analysis for a generalized richardson extrapolation process with an application to the d 1 transformation on convergent and divergent logarithmic sequences in an earlier work by the author the generalized richardson extrapolation process grep was introduced and some of its convergence and stability properties were discussed in a more recent work by the author a special case of grep which we now call grep 1 was considered and its properties were reviewed with emphasis on oscillatory sequences in the first part of the present work we give a detailed convergence and stability analysis of grep 1 as it applies to a large class of logarithmic sequences both convergent and divergent in particular we prove several theorems concerning the columns and the diagonals of the corresponding extrapolation table these theorems are very realistic in the sense that they explain the remarkable efficiency of grep 1 in a very precise manner in the second part we apply this analysis to the levin sidi d 1 transformation as the latter is used with a new strategy to accelerate the convergence of infinite series that converge logarithmically or to sum the divergent extensions of such series this is made possible by the observation that when the proper analogy is drawn the d 1 transformation is in fact a grep 1 we append numerical examples that demonstrate the theory

region based strategies for active contour models the variational method has been introduced by kass et al 1987 in the field of object contour modeling as an alternative to the more traditional edge detection edge thinning edge sorting sequence since the method is based on a pre processing of the image to yield an edge map it shares the limitations of the edge detectors it uses in this paper we propose a modified variational scheme for contour modeling which uses no edge detection step but local computations instead only around contour neighborhoods as well as an anticipating strategy that enhances the modeling activity of deformable contour curves many of the concepts used were originally introduced to study the local structure of discontinuity in a theoretical and formal statement by leclerc zucker 1987 but never in a practical situation such as this one the first part of the paper introduces a region based energy criterion for active contours and gives an examination of its implications as compared to the gradient edge map energy of snakes then a simplified optimization scheme is presented accounting for internal and external energy in separate steps this leads to a complete treatment which is described in the last sections of the paper 4 and 5 the optimization technique used here is mostly heuristic and is thus presented without a formal proof but is believed to fill a gap between snakes and other useful image representations such as split and merge regions or mixed line labels image fields

finite difference methods for the infinity laplace and p laplace equations abstract we build convergent discretizations and semi implicit solvers for the infinity laplacian and the game theoretical p laplacian the discretizations simplify and generalize earlier ones we prove convergence of the solution of the wide stencil finite difference schemes to the unique viscosity solution of the underlying equation we build a semi implicit solver which solves the laplace equation as each step it is fast in the sense that the number of iterations is independent of the problem size this is an improvement over previous explicit solvers which are slow due to the cfl condition

a fast marching like algorithm for geometrical shock dynamics we develop a new algorithm for the computation of the geometrical shock dynamics gsd model the method relies on the fast marching paradigm and enables the discrete evaluation of the first arrival time of a shock wave and its local velocity on a cartesian grid the proposed algorithm is based on a first order upwind finite difference scheme and reduces to a local nonlinear system of two equations solved by an iterative procedure reference solutions are built for a smooth radial configuration and for the 2d riemann problem the link between the gsd model and p systems is given numerical experiments demonstrate the efficiency of the scheme and its ability to handle singularities

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

spectral viscosity approximations to hamilton jacobi solutions the spectral viscosity approximate solution of convex hamilton jacobi equations with periodic boundary conditions is studied it is proved in this paper that the approximation and its gradient remain uniformly bounded formally spectral accurate and converge to the unique viscosity solution the l1 convergence rate of the order 1 varepsilon forall varepsilon 0 is obtained

hybrid control synthesis for eventuality specifications using level set methods this paper is concerned with the extraction of controllers for hybrid systems with respect to eventuality specifications given a hybrid system modelled by a hybrid automaton and a target set of states the objective is to compute the maximal set of initial states together with the hybrid control policy such that all the trajectories of the controlled system reach the target in finite time due to the existence of set valued disturbance inputs the problem is studied in a game theoretic framework having shown that a least restrictive solution does not exist we propose a dynamic programming algorithm that computes the maximal initial set and a controller with the desired property to implement the algorithm reachable sets of pursuit evasion differential games need to be computed for that reason level set methods are employed where the boundary of the reachable set is characterized as the zero level set of a hamilton jacobi equation the procedure for the numerical extraction of the controller is presented in detail and examples illustrate the methodology finally to demonstrate the practical character of our results a control design problem in the benchmark system of the batch evaporator is considered as an eventuality synthesis problem and solved using the proposed methodology

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

a new summation method for power series with rational coefficients we show that an asymptotic summation method recently proposed by the authors can be conveniently applied to slowly convergent power series whose coefficients are rational functions of the summation index several numerical examples are presented

a complexity analysis of the gauss bessel quadrature as applied to the evaluation of multi centre integrals over stfs in a previous work bouferguene 2005 j phys a math gen 38 3923 we have shown that in the framework of the gaussian integral transform multi centre integrals over slater type functions can be evaluated to an acceptable accuracy using a tailored gauss quadrature in which the weight function has the form w z z exp z z to be considered a solution worth implementing within a software for routine use in ab initio molecular simulations the method must also prove to be at least as efficient as those methods previously published in the literature two major results are provided in this paper firstly an improvement of the procedure used to generate the roots and weights of the gauss bessel quadrature is proposed secondly a computational cost analysis of the present method and the safouhi 2001 j phys a math gen 34 2801 based approach are compared hence proving the equivalence of the two from a complexity point of view

efficient energies and algorithms for parametric snakes parametric active contour models are one of the preferred approaches for image segmentation because of their computational efficiency and simplicity however they have a few drawbacks which limit their performance in this paper we identify some of these problems and propose efficient solutions to get around them the widely used gradient magnitude based energy is parameter dependent its use will negatively affect the parametrization of the curve and consequently its stiffness hence we introduce a new edge based energy that is independent of the parameterization it is also more robust since it takes into account the gradient direction as well we express this energy term as a surface integral thus unifying it naturally with the region based schemes the unified framework enables the user to tune the image energy to the application at hand we show that parametric snakes can guarantee low curvature curves but only if they are described in the curvilinear abscissa since normal curve evolution do not ensure constant arc length we propose a new internal energy term that will force this configuration the curve evolution can sometimes give rise to closed loops in the contour which will adversely interfere with the optimization algorithm we propose a curve evolution scheme that prevents this condition

nonrigid image registration with crystal dislocation energy the goal of nonrigid image registration is to find a suitable transformation such that the transformed moving image becomes similar to the reference image the image registration problem can also be treated as an optimization problem which tries to minimize an objective energy function that measures the differences between two involved images in this paper we consider image matching as the process of aligning object boundaries in two different images the registration energy function can be defined based on the total energy associated with the object boundaries the optimal transformation is obtained by finding the equilibrium state when the total energy is minimized which indicates the object boundaries find their correspondences and stop deforming we make an analogy between the above processes with the dislocation system in physics the object boundaries are viewed as dislocations line defects in crystal then the well developed dislocation energy is used to derive the energy assigned to object boundaries in images the newly derived registration energy function takes the global gradient information of the entire image into consideration and produces an orientation dependent and long range interaction between two images to drive the registration process this property of interaction endows the new registration framework with both fast convergence rate and high registration accuracy moreover the new energy function can be adapted to realize symmetric diffeomorphic transformation so as to ensure one to one matching between subjects in this paper the superiority of the new method is theoretically proven experimentally tested and compared with the state of the art syn method experimental results with 3 d magnetic resonance brain images demonstrate that the proposed method outperforms the compared methods in terms of both registration accuracy and computation time

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

a simple approach to the summation of certain slowly convergent series summation of series of the form k k l r k is considered where 0 v 1 and r is a rational function by an application of the euler maclaurin summation formula the problem is reduced to the evaluation of gauss hypergeometric function examples are given

snakes shapes and gradient vector flow snakes or active contours are used extensively in computer vision and image processing applications particularly to locate object boundaries problems associated with initialization and poor convergence to boundary concavities however have limited their utility this paper presents a new external force for active contours largely solving both problems this external force which we call gradient vector flow gvf is computed as a diffusion of the gradient vectors of a gray level or binary edge map derived from the image it differs fundamentally from traditional snake external forces in that it cannot be written as the negative gradient of a potential function and the corresponding snake is formulated directly from a force balance condition rather than a variational formulation using several two dimensional 2 d examples and one three dimensional 3 d example we show that gvf has a large capture range and is able to move snakes into boundary concavities

region competition unifying snakes region growing and bayes mdl for multiband image segmentation we present a novel statistical and variational approach to image segmentation based on a new algorithm named region competition this algorithm is derived by minimizing a generalized bayes minimum description length mdl criterion using the variational principle the algorithm is guaranteed to converge to a local minimum and combines aspects of snakes balloons and region growing the classic snakes balloons and region growing algorithms can be directly derived from our approach we provide theoretical analysis of region competition including accuracy of boundary location criteria for initial conditions and the relationship to edge detection using filters it is straightforward to generalize the algorithm to multiband segmentation and we demonstrate it on gray level images color images and texture images the novel color model allows us to eliminate intensity gradients and shadows thereby obtaining segmentation based on the albedos of objects it also helps detect highlight regions

a class of slowly convergent series and their summation by gaussian quadrature series are considered whose general term is a rational function multiplied by a fractional power the summation of such series is reduced via laplace transformation techniques to a problem of quadrature which is then solved by gaussian quadrature relative to einstein and fermi weight functions a number of examples are worked out in detail

discrete time high order schemes for viscosity solutions of hamilton jacobi bellman equations a general method for constructing high order approximation schemes for hamilton jacobi bellman equations is given the method is based on a discrete version of the dynamic programming principle we prove a general convergence result for this class of approximation schemes also obtaining under more restrictive assumptions an estimate in l infty of the order of convergence and of the local truncation error the schemes can be applied in particular to the stationary linear first order equation in bbb r n we present several examples of schemes belonging to this class and with fast convergence to the solution

summation of power series with rational coefficients via divided differences abstract we obtain new identities involving divided differences and as an application we provide a closed form representation of power series with rational coefficients in terms of divided differences involving special functions

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

an eulerian approach for computing the finite time lyapunov exponent we propose efficient eulerian methods for approximating the finite time lyapunov exponent ftle the idea is to compute the related flow map using the level set method and the liouville equation there are several advantages of the proposed approach unlike the usual lagrangian type computations the resulting method requires the velocity field defined only at discrete locations no interpolation of the velocity field is needed also the method automatically stops a particle trajectory in the case when the ray hits the boundary of the computational domain the computational complexity of the algorithm is o dx d 1 with d the dimension of the physical space since there are the same number of mesh points in the x t space the computational complexity of the proposed eulerian approach is optimal in the sense that each grid point is visited for only o 1 time we also extend the algorithm to compute the ftle on a co dimension one manifold the resulting algorithm does not require computation on any local coordinate system and is simple to implement even for an evolving manifold

geometrically induced force interaction for three dimensional deformable models in this paper we propose a novel 3 d deformable model that is based upon a geometrically induced external force field which can be conveniently generalized to arbitrary dimensions this external force field is based upon hypothesized interactions between the relative geometries of the deformable model and the object boundary characterized by image gradient the evolution of the deformable model is solved using the level set method so that topological changes are handled automatically the relative geometrical configurations between the deformable model and the object boundaries contribute to a dynamic vector force field that changes accordingly as the deformable model evolves the geometrically induced dynamic interaction force has been shown to greatly improve the deformable model performance in acquiring complex geometries and highly concave boundaries and it gives the deformable model a high invariancy in initialization configurations the voxel interactions across the whole image domain provide a global view of the object boundary representation giving the external force a long attraction range the bidirectionality of the external force field allows the new deformable model to deal with arbitrary cross boundary initializations and facilitates the handling of weak edges and broken boundaries in addition we show that by enhancing the geometrical interaction field with a nonlocal edge preserving algorithm the new deformable model can effectively overcome image noise we provide a comparative study on the segmentation of various geometries with different topologies from both synthetic and real images and show that the proposed method achieves significant improvements against existing image gradient techniques

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

uniform approximation on unit disk of series arising in plate contact problems slowly convergent series on the unit disk are reconsidered typical cases arise in plate contact problems uniformly converging sequences of approximants which are expressible by recurrence relations are established accurate error estimates which prove to be realistic are also derived several numerical examples confirm the theoretical predictions and show clearly the effectiveness of the method

weighted sobolev spaces and pseudodifferential operators with smooth symbols absmacr let u i be the fefferman stein sharp function of u and for 1 0 such that the pointwise estimate au x cm u x holds for all x e ri and all schwartz functions u this estimate implies the boundedness of 0 order pseudodifferential operators on weighted lp spaces whenever the weight function belongs to muckenhoupt s class ap having established this we construct weighted sobolev spaces of fractional order in r and on a compact manifold prove a version of sobolev s theorem and exhibit coercive weighted estimates for elliptic pseudodifferential operators

space time discontinuous galerkin finite element method for two fluid flows a novel numerical method for two fluid flow computations is presented which combines the space time discontinuous galerkin finite element discretization with the level set method and cut cell based interface tracking the space time discontinuous galerkin stdg finite element method offers high accuracy an inherent ability to handle discontinuities and a very local stencil making it relatively easy to combine with local hp refinement the front tracking is incorporated via cut cell mesh refinement to ensure a sharp interface between the fluids to compute the interface dynamics the level set method lsm is used because of its ability to deal with merging and breakup also the lsm is easy to extend to higher dimensions small cells arising from the cut cell refinement are merged to improve the stability and performance the interface conditions are incorporated in the numerical flux at the interface and the stdg discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative the numerical method is applied to one and two dimensional two fluid test problems using the euler equations

finite element approximations of general fully nonlinear second order elliptic partial differential equations based on the vanishing moment method the vanishing moment method was introduced by the authors in feng and neilan 2009 as a reliable methodology for computing viscosity solutions of fully nonlinear second order partial differential equations pdes it is based on the simple idea of approximating a hard to handle fully nonlinear second order pde by a family parameterized by a small parameter e of easy to handle quasilinear fourth order pdes the primary objective of this article is to present a comprehensive finite element analysis for the vanishing moment approximation of general fully nonlinear second order elliptic pdes which fulfill some structure conditions abstract methodological and convergence analysis frameworks of conforming finite element methods are first developed for fully nonlinear second order pdes in a general setting the abstract framework is then applied to three prototypical nonlinear equations namely the monge ampere equation the equation of prescribed gauss curvature and the infinity laplacian equation numerical experiments are presented for each problem to validate the theoretical error estimate results and to gauge the efficiency of the proposed numerical methods and the vanishing moment methodology

the spiral of theodorus numerical analysis and special functions theodorus of cyrene ca 460 399 b c teacher of plato und theaetetus is known for his proof of the irrationality of n n 2 3 5 17 he may have known also of a discrete spiral today named after him whose construction is based on the square roots of the numbers n 1 2 3 the subject of this lecture is the problem of interpolating this discrete angular spiral by a smooth if possible analytic spiral an interesting solution was proposed in 1993 by p j davis which is based on an infinite product the computation of this product gives rise to problems of numerical analysis in particular the summation of slowly convergent series and the identification of the product raises questions regarding special functions the former are solved by a method of integration in particular gaussian integration the latter by means of dawson s integral und the bose einstein distribution number theoretic questions also loom behind this work

motion of vortex lines in the ginzburg landau model equations of motion for vortex lines in the ginzburg landau theory are derived we construct asymptotic approximations for the complex order parameter which are valid in the core region and in the far field using the method of matched asymptotic expansions we show that to leading order the line moves in the binormal direction with a curvature dependent velocity we also consider the contribution of remote parts of the line interaction between several vortex lines and interaction with external fields

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

an active contour model for image segmentation based on elastic interaction the task of image segmentation is to partition an image into non overlapping regions based on intensity or textural information the active contour methods provide an effective way for segmentation in which the boundaries of the objects are detected by evolving curves in this paper we propose a new edge based active contour method which uses a long range and orientation dependent interaction between image boundaries and the moving curves while maintaining the edge fidelity as a result this method has a large capture range and is able to detect sharp features of the images the velocity field for the moving curves generated by this elastic interaction is calculated using the fast fourier transform fft method level set representation is used for the moving curves so that the topological changes during the evolution are handled automatically this new method is derived based on the elastic interaction between line defects in solids dislocations although it is derived originally for two dimensional segmentation we also extend it to three dimensions the features of the new method are examined by experiments on both synthetic images and medical images of blood vessels comparisons are made with the existing active contour methods

area and length minimizing flows for shape segmentation a number of active contour models have been proposed that unify the curve evolution framework with classical energy minimization techniques for segmentation such as snakes the essential idea is to evolve a curve in two dimensions or a surface in three dimensions under constraints from image forces so that it clings to features of interest in an intensity image the evolution equation has been derived from first principles as the gradient flow that minimizes a modified length functional tailored to features such as edges however because the flow may be slow to converge in practice a constant hyperbolic term is added to keep the curve surface moving in the desired direction we derive a modification of this term based on the gradient flow derived from a weighted area functional with image dependent weighting factor when combined with the earlier modified length gradient flow we obtain a partial differential equation pde that offers a number of advantages as illustrated by several examples of shape segmentation on medical images in many cases the weighted area flow may be used on its own with significant computational savings

factored singularities and high order lax friedrichs sweeping schemes for point source traveltimes and amplitudes in the high frequency regime the geometrical optics approximation for the helmholtz equation with a point source results in an eikonal equation for traveltime and a transport equation for amplitude because the point source traveltime field has an upwind singularity at the source point all formally high order finite difference eikonal solvers exhibit first order convergence and relatively large errors in this paper we propose to first factor out the singularities of traveltimes takeoff angles and amplitudes and then we design high order lax friedrichs sweeping schemes for point source traveltimes takeoff angles and amplitudes numerical examples are presented to demonstrate the performance of our new method

unconditionally stable methods for hamilton jacobi equations we present new numerical methods for constructing approximate solutions to the cauchy problem for hamilton jacobi equations of the form ut h dxu 0 the methods are based on dimensional splitting and front tracking for solving the associated non strictly hyperbolic system of conservation laws pt dx h p 0 where p dx u in particular our methods depend heavily on a front tracking method for one dimensional scalar conservation laws with discontinuous coefficients the proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as large time step godunov type or front tracking methods we present several numerical examples illustrating the main features of the proposed methods we also compare our methods with several methods from the literature

weighted local variance based edge detection and its application to vascular segmentation in magnetic resonance angiography accurate detection of vessel boundaries is particularly important for a precise extraction of vasculatures in magnetic resonance angiography mra in this paper we propose the use of weighted local variance wlv based edge detection scheme for vessel boundary detection in mra the proposed method is robust against changes of intensity contrast of edges and capable of giving high detection responses on low contrast edges these robustness and capabilities are essential for detecting the boundaries of vessels in low contrast regions of images which can contain intensity inhomogeneity such as bias field interferences induced from other tissues or fluctuation of the speed related vessel intensity the performance of the wlv based edge detection scheme is studied and shown to be able to return strong and consistent detection responses on low contrast edges in the experiments the proposed edge detection scheme can be embedded naturally in the active contour models for vascular segmentation the wlv based vascular segmentation method is tested using mra image volumes it is experimentally shown that the wlv based edge detection approach can achieve high quality segmentation of vasculatures in mra images

a uniformly second order fast sweeping method for eikonal equations a uniformly second order method with a local solver based on the piecewise linear discontinuous galerkin formulation is introduced to solve the eikonal equation with dirichlet boundary conditions the method utilizes an interesting phenomenon referred as the superconvergence phenomenon that the numerical solution of monotone upwind schemes for the eikonal equation is first order accurate on both its value and gradient when the solution is smooth this phenomenon greatly simplifies the local solver based on the discontinuous galerkin formulation by reducing its local degrees of freedom from two 1 d or three 2 d or four 3 d to one with the information of the gradient frozen when considering the eikonal equation with point source conditions we further utilize a factorization approach to resolve the source singularities of the eikonal by decomposing it into two parts either multiplicatively or additively one part is known and captures the source singularities the other part serves as a correction term that is differentiable at the sources and satisfies the factored eikonal equations we extend the second order method to solve the factored eikonal equations to compute the correction term with second order accuracy then recover the eikonal with second order accuracy numerical examples are presented to demonstrate the performance of the method

a variant of the level set method and applications to image segmentation in this paper we propose a variant of the level set formulation for identifying curves separating regions into different phases in classical level set approaches the sign of n level set functions are utilized to identify up to 2 phases the novelty in our approach is to introduce a piecewise constant level set function and use each constant value to represent a unique phase if 2 phases should be identified the level set function must approach 2 predetermined constants we just need one level set function to represent 2 unique phases and this gains in storage capacity further the reinitializing procedure requested in classical level set methods is superfluous using our approach the minimization functional for our approach is locally convex and differentiable and thus avoids some of the problems with the nondifferentiability of the delta and heaviside functions numerical examples are given and we also compare our method with related approaches

new high resolution semi discrete central schemes for hamilton jacobi equations we introduce a new high resolution central scheme for multidimensional hamilton jacobi equations the scheme retains the simplicity of the non oscillatory central schemes developed by c t lin and e tadmor in press siam j sci comput yet it enjoys a smaller amount of numerical viscosity independent of 1 t by letting t 0 we obtain a new second order central scheme in the particularly simple semi discrete form along the lines of the new semi discrete central schemes recently introduced by the authors in the context of hyperbolic conservation laws fully discrete versions are obtained with appropriate runge kutta solvers the smaller amount of dissipation enables efficient integration of convection diffusion equations where the accumulated error is independent of a small time step dictated by the cfl limitation the scheme is non oscillatory thanks to the use of nonlinear limiters here we advocate the use of such limiters on second discrete derivatives which is shown to yield an improved high resolution when compared to the usual limitation of first derivatives numerical experiments demonstrate the remarkable resolution obtained by the proposed new central scheme

asymptotic summation of power series we give an asymptotic expansion in powers of n 1 of the remainder sum j n infty f jz j when the sequence f n has a similar expansion contrary to previous results explicit formulas for the computation of the coefficients are presented in the case of numerical series z 1 rigorous error estimates for the asymptotic approximations are also provided we apply our results to the evaluation of s z j 0 nu a b p sum j j 0 infty z j j b nu 1 j a p which generalizes various summation problems appeared in the recent literature on convergence acceleration of numerical and power series

computation of nonclassical solutions to hamilton jacobi problems this paper is devoted to the construction of numerical methods for the approximation of nonclassical solutions to multidimensional hamilton jacobi equations for both scalar and vectorial problems recent theoretical results have yielded existence of solutions in many cases for which the usual viscosity approach was ill suited or not applicable the selection criterion used here is based on a viscoelasticity capillarity approach common in solid mechanics numerical methods adapted to this framework are built consistency of the model equation with the given selection criterion is essential it is achieved here through the use of high order finite difference schemes by considering applications to potential well problems the convergence of the methods are investigated

acceleration of convergence of a family of logarithmically convergent sequences the asymptotic behavior of several sequence transformations is in vestigated as n oo when applied to a certain family of logarithmically convergent sequences the transformations considered are the iterations of the transformations e an of shanks and wn of lubkin the algorithm of brezinski the levin u and u transforms and generalizations of the p algorithm and the neville table computational results are given for both real and complex sequences

gradient vector flow fast geometric active contours in this paper we propose an edge driven bidirectional geometric flow for boundary extraction to this end we combine the geodesic active contour flow and the gradient vector flow external force for snakes the resulting motion equation is considered within a level set formulation can deal with topological changes and important shape deformations an efficient numerical schema is used for the flow implementation that exhibits robust behavior and has fast convergence rate promising results on real and synthetic images demonstrate the potentials of the flow