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

three dimensional non planar crack growth by a coupled extended finite element and fast marching method a numerical technique for non planar three dimensional linear elastic crack growth simulations is proposed this technique couples the extended finite element method x fem and the fast marching method fmm in crack modeling using x fem the framework of partition of unity is used to enrich the standard finite element approximation by a discontinuous function and the two dimensional asymptotic crack tip displacement fields the initial crack geometry is represented by two level set functions and subsequently signed distance functions are used to maintain the location of the crack and to compute the enrichment functions that appear in the displacement approximation crack modeling is performed without the need to mesh the crack and crack propagation is simulated without remeshing crack growth is conducted using fmm unlike a level set formulation for interface capturing no iterations nor any time step restrictions are imposed in the fmm planar and non planar quasi static crack growth simulations are presented to demonstrate the robustness and versatility of the proposed technique copyright 2008 john wiley sons ltd

a localized re initialization equation for the conservative level set method the conservative level set methodology for interface transport is modified to allow for localized level set re initialization this approach is suitable to applications in which there is a significant amount of spatial variability in level set transport the steady state solution of the modified re initialization equation matches that of the original conservative level set provided an additional eikonal equation is solved which can be done efficiently through a fast marching method fmm implemented within the context of the accurate conservative level set method acls desjardins et al 2008 6 the fmm solution of this eikonal equation comes at no additional cost a metric for the appropriate amount of local re initialization is proposed based on estimates of local flow deformation and numerical diffusion the method is compared to standard global re initialization for two test cases yielding the expected results that minor differences are observed for zalesak s disk and improvements in both mass conservation and interface topology are seen for a drop deforming in a vortex finally the method is applied to simulation of a viscously damped standing wave and a three dimensional drop impacting on a shallow pool negligible differences are observed for the standing wave as expected for the last case results suggest that spatially varying re initialization provides a reduction in spurious interfacial corrugations improvements in the prediction of radial growth of the splashing lamella and a reduction in conservation errors as well as a reduction in overall computational cost that comes from improved conditioning of the pressure poisson equation due to the removal of spurious corrugations

on the estimation of optical flow relations between different approaches and some new results abstract a common basis is suggested for the optical flow estimation approaches of nagel 1983 haralick and lee 1983 and tretiak and pastor 1984 based on a discussion of these approaches an exact solution for the system of partial differential equations proposed by horn and schunck 1981 is given at gray value corners and extrema the insight gained by this solution results in a modification of the oriented smoothness approach of nagel 1983 which thereby becomes considerably simpler in addition the optical flow estimation approach of hildreth 1983 1984 can be shown to represent a kind of special case of this modified oriented smoothness approach in a more direct manner than discussed in nagel 1984

optimal covering tours with turn costs we give the first algorithmic study of a class of covering tour problems related to the geometric traveling salesman problem find a polygonal tour for a cutter so that it sweeps out a specified region pocket in order to minimize a cost that depends mainly on the number of turns these problems arise naturally in manufacturing applications of computational geometry to automatic tool path generation and automatic inspection systems as well as arc routing postman problems with turn penalties we prove the np completeness of minimum turn milling and give efficient approximation algorithms for several natural versions of the problem including a polynomial time approximation scheme based on a novel adaptation of the m guillotine method

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

fast marching methods for stationary hamilton jacobi equations with axis aligned anisotropy the fast marching method fmm has proved to be a very efficient algorithm for solving the isotropic eikonal equation because it is a minor modification of dijkstra s algorithm for finding the shortest path through a discrete graph fmm is also easy to implement in this paper we describe a new class of hamilton jacobi hj pdes with axis aligned anisotropy which satisfy a causality condition for standard finite difference schemes on orthogonal grids and can hence be solved using the fmm the only modification required to the algorithm is in the local update equation for a node this class of hj pdes has applications in anelliptic wave propagation and robotic path planning and brief examples are included since our class of hj pdes and grids permit asymmetries we also examine some methods of improving the efficiency of the local update that do not require symmetric grids and pdes finally we include explicit update formulas for variations of the eikonal equation that use the manhattan euclidean and infinity norms on orthogonal grids of arbitrary dimension and with variable node spacing

a finite element level set redistancing method based on gradient recovery we introduce a new redistancing method for level set functions this method applies in a finite element setting and uses a gradient recovery technique based on the recovered gradient a quasi normal field on the zero level of the finite element level set function is defined and from this an approximate signed distance function is determined for this redistancing method rigorous error bounds are derived for example the distance between the original zero level and the zero level after redistancing can be shown to be bounded by ch k 1 if finite elements of degree k are used in the discretization

approximation schemes for viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of applications including the calculus of variations control theory and differential games recently m g crandall and p l lions established the correct notion of generalized solutions for these equations this article discusses the convergence of general approximation schemes to this solution and gives under certain hypotheses explicit error estimates these results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes with error estimates

semi lagrangian schemes for hamilton jacobi equations discrete representation formulae and godunov methods we study a class of semi lagrangian schemes which can be interpreted as a discrete version of the hopf lax oleinik representation formula for the exact viscosity solution of first order evolutive hamilton jacobi equations that interpretation shows that the scheme is potentially accurate to any prescribed order we discuss how the method can be implemented for convex and coercive hamiltonians with a particular structure and how this method can be coupled with a discrete legendre trasform we also show that in one dimension the first order semi lagrangian scheme coincides with the integration of the godunov scheme for the corresponding conservation laws several test illustrate the main features of semi lagrangian schemes for evolutive hamilton jacobi equations

the use of a legendre pseudospectral viscosity technique to solve a class of nonlinear dynamic hamilton jacobi equations in this research we study the problem of finding the approximate solution of a class of hamilton jacobi equations namely the eikonal equation we employ the legendre pseudospectral viscosity method to solve this problem this method basically consists of adding a spectral viscosity to the equation this spectral viscosity which is sufficiently small to retain the formal spectral accuracy is large enough to stabilize the numerical scheme several test problems are considered and the numerical results are given to show the efficiency of the proposed method

a hybrid extended finite element level set method for modeling phase transformations a hybrid numerical method for modelling the evolution of sharp phase interfaces on fixed grids is presented we focus attention on two dimensional solidification problems where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front the enrichment strategies of the extended finite element method x fem are employed to represent the jump in the temperature gradient that governs the velocity of the phase boundary a new approach with the x fem is suggested for this class of problems whereby the partition of unity is constructed with c1 polynomials and enriched with a c0 function this approach leads to jumps in temperature gradient occurring only at the phase boundary and is shown to significantly improve estimates for the front velocity temporal derivatives of the temperature field in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment in conjunction with a finer finite difference grid the level set method is used to represent the evolution of the phase interface an iterative procedure is adopted to satisfy the constraints on the temperature field on the phase boundary the robustness and utility of the method is demonstrated with several benchmark problems of phase transformation copyright 2002 john wiley sons ltd

l 1 approximation of stationary hamilton jacobi equations we describe a nonlinear finite element technique to approximate the solutions of stationary hamilton jacobi equations in two space dimensions using continuous finite elements of arbitrary degree the method consists of minimizing a functional containing the l 1 norm of the hamiltonian plus a discrete entropy it is shown that the approximate sequence converges to the unique viscosity solution under appropriate hypotheses on the hamiltonian and the mesh family

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

a fast marching approach to multidimensional extrapolation a computationally efficient approach to extrapolating a data field with second order accuracy is presented this is achieved through the sequential solution of non homogeneous linear static hamilton jacobi equations which can be performed rapidly using the fast marching methodology in particular the method relies on a fast marching calculation of the distance from the manifold c that separates the subdomain w i n over which the quanity is known from the subdomain w o u t over which the quantity is to be extrapolated a parallel algorithm is included and discussed in the appendices results are compared to the multidimensional partial differential equation pde extrapolation approach of aslam aslam 2004 31 it is shown that the rate of convergence of the extrapolation within a narrow band near c is controlled by both the number of successive extrapolations performed and the order of accuracy of the spatial discretization for m successive extrapolating steps and a spatial discretization scheme of order n the rate of convergence in a narrow band is shown to be min n 1 m 1 results show that for a wide range of error levels the fast marching extrapolation strategy leads to dramatic improvements in computational cost when compared to the pde approach

fast computation of weighted distance functions and geodesics on implicit hyper surfaces 730 an algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hyper surfaces is introduced in this paper the basic idea is to approximate the intrinsic weighted distance by the euclidean weighted distance computed in a band surrounding the implicit hyper surface in the embedding space thereby performing all the computations in a cartesian grid with classical and efficient numerics based on work on geodesics on riemannian manifolds with boundaries we bound the error between the two distance functions we show that this error is of the same order as the theoretical numerical error in computationally optimal hamilton jacobi based algorithms for computing distance functions in cartesian grids therefore we can use these algorithms modified to deal with spaces with boundaries and obtain also for the case of intrinsic distance functions on implicit hyper surfaces a computationally efficient technique the approach can be extended to solve a more general class of hamilton jacobi equations defined on the implicit surface following the same idea of approximating their solutions by the solutions in the embedding euclidean space the framework here introduced thereby allows for the computations to be performed on a cartesian grid with computationally optimal algorithms in spite of the fact that the distance and hamilton jacobi equations are intrinsic to the implicit hyper surface c 2001 academic press

on the minimum link length rectilinear spanning path problem complexity and algorithms the parameterized minimum link length rectilinear spanning path problem in the mbi d dimensional euclidean space mbi bbr d mbi d rsp for a given set mbi s of mbi n points in mbi bbr d and a positive integer mbi k is to find a piecewise linear path mbi p with at most mbi k line segments that covers i e contains all points in mbi s where all line segments in mbi p are axis parallel we first prove that the problem 2 rsp is np complete improving the previously known result that the problem 10 rsp is np complete we then consider a constrained mbi d rsp problem in which each line segment mbi s in the spanning path must cover all the points in the given set mbi s that share the same line with mbi s we present a new parameterized algorithm with running time mbi o ast 2d k for the constrained mbi d rsp problem which significantly improves the previous best result and is the first parameterized algorithm of running time mbi o ast 2 o k for the constrained mbi d rsp problem for a fixed mbi d we show that these results can be extended to the minimum link length rectilinear traveling salesman problem

fast full search in motion estimation by hierarchical use of minkowski s inequality humi abstract in this paper we extend the idea of successive elimination algorithm sea to obtain a fast full search fs algorithm accelerating the block matching procedure of motion estimation based on the monotonic relation between the accumulated absolution distortions aad obtained for distinct layers of a pyramid structure the proposed method successfully rejects many impossible candidates considered in the fs the derivation of the monotonicity relation repeatedly uses in a four dimensional vector space the l 1 version of minkowski s inequality an inequality which is quite well known in the field of mathematics simulation results show that the processing speed is faster than that of several well known fast full search methods including the sea that uses just once the minkowski s inequality in a vector space of 256 dimension when the block size is 16 16 the processing speed of the proposed method is also competitive with that of the three step search tss which is often used for block matching in interframe video coding although the visual quality performance of tss is usually a little poorer than that of the fs

fast sweeping methods for static hamilton jacobi equations we propose a new sweeping algorithm which discretizes the legendre transform of the numerical hamiltonian using an explicit formula this formula yields the numerical solution at a grid point using only its immediate neighboring grid values and is easy to implement numerically the minimization that is related to the legendre transform in our sweeping scheme can either be solved analytically or numerically we illustrate the efficiency and accuracy approach with several numerical examples in two and three dimensions

another look at velocity extensions in the level set method the level set method s osher and j a sethian j comput phys 79 1988 pp 12 49 has become a widely used numerical method for moving interfaces e g see the many examples in j sethian level set methods evolving interfaces in geometry fluid mechanics computer vision and material science cambridge university press london 1996 s osher and r fedkiw level set methods and dynamic implicit surfaces springer verlag heidelberg 2002 for many applications the velocity of the interface is known only on the interface while the level set method requires information about the interface speed at least in a neighborhood of grid points near the interface to address this issue velocity extensions are used to map the velocity information on the interface into the rest of the computational domain d adalsteinsson and j sethian j comput phys 118 1995 pp 269 277 this allows the level set method to proceed the velocity extension method presented in d adalsteinsson and j sethian j comput phys 118 1995 pp 269 277 uses the fast marching method j sethian proc natl acad sci 93 1996 pp 1591 1595 j sethian siam rev 41 1999 pp 199 235 and is only a first order approximation for the velocity field near the interface furthermore it can lead to unexpected behavior in some cases this is primarily due to the strictly local solution for the characteristics of the flow near the interface in this paper we look more closely at the characteristics near the interface and then present a modified velocity extension method also based on the fast marching method which handles the characteristcs more accurately in turn this new method will lead to some interesting new possibilities for the fast marching method

the lax friedrichs sweeping method for optimal control problems in continuous and hybrid dynamics abstract we implement the lax friedrichs sweeping lfs method to approximate the solution to the hamilton jacobi hj equation arising in an infinite horizon optimal control problem the fast sweeping method approximates the viscosity solution of a boundary value problem for hj equations the static hj equation is discretized by using the lax friedrichs numerical hamiltonian and the solution is approximated iteratively by using a gauss seidel like new updating process assuming that the hamiltonian is strictly convex in the control variable we derive a pair of hj equations in continuous time through the dynamic programming formulation the lfs method is applied to solve the coupled pdes by successive iteration of the optimal cost and control the lfs scheme also applies to optimal control problems in hybrid systems in which it admits the viscosity solution for the hj equation we demonstrate the efficiency of the method through some numerical examples in both continuous and hybrid dynamics

an efficient computation constrained block based motion estimation algorithm for low bit rate video coding we present an efficient computation constrained block based motion vector estimation algorithm for low bit rate video coding that yields good tradeoffs between motion estimation distortion and number of computations a reliable predictor determines the search origin localizing the search process an efficient search pattern exploits structural constraints within the motion field a flexible cost measure used to terminate the search allows simultaneous control of the motion estimation distortion and the computational cost experimental results demonstrate the viability of the proposed algorithm in low bit rate video coding applications the resulting low bit rate video encoder yields essentially the same levels of rate distortion performance and subjective quality achieved by the ubc h 263 video coding reference software however the proposed motion estimation algorithm provides substantially higher encoding speed as well as graceful computational degradation capabilities

estimation of general 2d affine motion using fourier descriptors abstract in the present paper a general affine motion estimation algorithm using the fourier descriptors is proposed it is able to estimate in addition to the translation scaling and rotation the stretching parameters or equivalently the scale the shift and the four coefficients of the general affine matrix a modified claire test sequence will serve as a useful test to evaluate the proposed motion algorithm this sequence was selected by international expert groups and was provided by the centre national d etudes des telecommunications cnet france

fast image and video colorization using chrominance blending colorization the task of coloring a grayscale image or video involves assigning from the single dimension of intensity or luminance a quantity that varies in three dimensions such as red green and blue channels mapping between intensity and color is therefore not unique and colorization is ambiguous in nature and requires some amount of human interaction or external information a computationally simple yet effective approach of colorization is presented in this paper the method is fast and it can be conveniently used on the fly permitting the user to interactively get the desired results promptly after providing a reduced set of chrominance scribbles based on the concepts of luminance weighted chrominance blending and fast intrinsic distance computations high quality colorization results for still images and video are obtained at a fraction of the complexity and computational cost of previously reported techniques possible extensions of the algorithm introduced here included the capability of changing the colors of an existing color image or video as well as changing the underlying luminance and many other special effects demonstrated here

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

a novel two phase hilbert scan based search algorithm for block motion estimation using ctf data structure motion estimation is one of the kernel issues in mpeg series in this correspondence a novel two phase hilbert scan based search algorithm for block motion estimation is presented first in the intra phase a segmentation of the hilbert curve is applied to the current block then a novel coarse to fine data structure is developed to eliminate the impossible reference blocks in the search window of the reference frame in the inter phase a new prediction scheme for estimating the initial motion vector of the current block is presented experimental results reveal that when compared to the gapd algorithm our proposed algorithm has better execution time and estimation accuracy performance under the same estimation accuracy our proposed algorithm has better execution time performance when compared to the fs algorithm in addition when comparing with the tss algorithm our proposed algorithm has better estimation accuracy performance but has worse execution time performance

improved parameterized algorithms for minimum link length rectilinear spanning path problem the parameterized minimum link length rectilinear spanning path problem in the d dimensional euclidean space r d d rsp for a given set s of n points in r d and a positive integer k is to find a rectilinear spanning path p with at most k line segments that cover all points in s where all line segments in p are axis parallel in this paper we study a constrained d rsp problem constrained d rsp problem in which each line segment l in the spanning path must cover all the points in s that share the same line with l by applying the branch and search and dynamic programming techniques a parameterized algorithm with running time o 1 1 4 d 1 2 k is given for the constrained d rsp problem which significantly improves the current best result o 0 74 d k k

ordered upwind methods for static hamilton jacobi equations abstract we introduce a family of fast ordered upwind methods for approximating solutions to a wide class of static hamilton jacobi equations with dirichlet boundary conditions standard techniques often rely on iteration to converge to the solution of a discretized version of the partial differential equation our fast methods avoid iteration through a careful use of information about the characteristic directions of the underlying partial differential equation these techniques are of complexity o m log m where m is the total number of points in the domain we consider anisotropic test problems in optimal control seismology and paths on surfaces

a fast sweeping method for eikonal equations in this paper a fast sweeping method for computing the numerical solution of eikonal equations on a rectangular grid is presented the method is an iterative method which uses upwind difference for discretization and uses gauss seidel iterations with alternating sweeping ordering to solve the discretized system the crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding eikonal equation in a certain direction simultaneously the method has an optimal complexity of o n for n grid points and is extremely simple to implement in any number of dimensions monotonicity and stability properties of the fast sweeping algorithm are proven convergence and error estimates of the algorithm for computing the distance function is studied in detail it is shown that 2 n gauss seidel iterations is enough for the distance function in n dimensions an estimation of the number of iterations for general eikonal equations is also studied numerical examples are used to verify the analysis

weighted distance maps computation on parametric three dimensional manifolds we propose an efficient computational solver for eikonal equations on parametric three dimensional manifolds our approach is based on the fast marching method for solving the eikonal equation in o nlogn steps on n grid points by numerically simulating wavefront propagation the obtuse angle splitting problem is reformulated as a set of small integer linear programs that can be solved in o n numerical simulations demonstrate the accuracy of the proposed algorithm

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

faster scaling algorithms for network problems this paper presents algorithms for the assignment problem the transportation problem and the minimum cost flow problem of operations research the algorithms find a minimum cost solution yet run in time close to the best known bounds for the corresponding problems without costs for example the assignment problem equivalently minimum cost matching in a bipartite graph can be solved in o sqrt nm log nn time where n m and n denote the number of vertices number of edges and largest magnitude of a cost costs are assumed to be integral the algorithms work by scaling as in the work of goldberg and tarjan in each scaled problem an approximate optimum solution is found rather than an exact optimum

robust non local tv l 1 optical flow estimation with occlusion detection in this paper we propose a robust non local tv l 1 optical flow method with occlusion detection to address the problem of weak robustness of optical flow estimation with motion occlusion first a tv l 1 form for flow estimation is defined using a combination of the brightness constancy and gradient constancy assumptions in the data term and by varying the weight under the charbonnier function in the smoothing term second to handle the potential risk of the outlier in the flow field a general non local term is added in the tv l 1 optical flow model to engender the typical non local tv l 1 form third an occlusion detection method based on triangulation is presented to detect the occlusion regions of the sequence the proposed non local tv l 1 optical flow model is performed in a linearizing iterative scheme using improved median filtering and a coarse to fine computing strategy the results of the complex experiment indicate that the proposed method can overcome the significant influence of non rigid motion motion occlusion and large displacement motion results of experiments comparing the proposed method and existing state of the art methods by respectively using middlebury and mpi sintel database test sequences show that the proposed method has higher accuracy and better robustness

a region based selective optical flow back projection for genuine motion vector estimation motion vector plays one significant feature in moving object segmentation however the motion vector in this application is required to represent the actual motion displacement rather than regions of visually significant similarity in this paper region based selective optical flow back projection rsofb which back projects optical flows in a region to restore the region s motion vector from gradient based optical flows is proposed to obtain genuine motion displacement the back projection is performed based on minimizing the projection mean square errors of the motion vector on gradient directions as optical flows of various magnitudes and directions provide various degrees of reliability in the genuine motion restoration the optical flows to be used in the rsofb are optimally selected based on their sensitivity to noises and their tendency in causing motion estimation errors in this paper a deterministic solution is also derived for performing the minimization and obtaining the genuine motion magnitude and motion direction

fast sweeping methods for eikonal equations on triangular meshes the original fast sweeping method which is an efficient iterative method for stationary hamilton jacobi equations relies on natural ordering provided by a rectangular mesh we propose novel ordering strategies so that the fast sweeping method can be extended efficiently and easily to any unstructured mesh to that end we introduce multiple reference points and order all the nodes according to their l p metrics to those reference points we show that these orderings satisfy the two most important properties underlying the fast sweeping method 1 these orderings can cover all directions of information propagating efficiently 2 any characteristic can be decomposed into a finite number of pieces and each piece can be covered by one of the orderings we prove the convergence of the new algorithm the computational complexity of the algorithm is nearly optimal in the sense that the total computational cost consists of o m flops for iteration steps and o m rm log m flops for sorting at the predetermined initialization step which can be efficiently optimized by adopting a linear time sorting method where m is the total number of mesh points extensive numerical examples demonstrate that the new algorithm converges in a finite number of iterations independent of mesh size

a second order discontinuous galerkin fast sweeping method for eikonal equations in this paper we construct a second order fast sweeping method with a discontinuous galerkin dg local solver for computing viscosity solutions of a class of static hamilton jacobi equations namely the eikonal equations our piecewise linear dg local solver is built on a dg method developed recently y cheng c w shu a discontinuous galerkin finite element method for directly solving the hamilton jacobi equations journal of computational physics 223 2007 398 415 for the time dependent hamilton jacobi equations the causality property of eikonal equations is incorporated into the design of this solver the resulting local nonlinear system in the gauss seidel iterations is a simple quadratic system and can be solved explicitly the compactness of the dg method and the fast sweeping strategy lead to fast convergence of the new scheme for eikonal equations extensive numerical examples verify efficiency convergence and second order accuracy of the proposed method

a two dimensional multi species model for different listeria monocytogenes biofilm structures and its numerical simulation abstract in this work we propose a two dimensional multi species model to describe the dynamics of biofilms formed by the pathogenic bacteria listeria monocytogenes different listeria monocytogenes strains produce biofilms with different structures namely flat honeycomb and clustered previous works showed that glucose impaired uptake and the appearance of damaged or dead cells are critical mechanisms underlying listeria monocytogenes biofilm dynamics here we explicitly propose an extension of the two dimensional multi species model proposed by alpkist and klapper to account for those mechanisms the result is a continuous two dimensional multi species model with non linear detachment and mass action nutrient consumption moreover we also propose a set of efficient numerical methods to solve the coupled model and we have developed their computer implementation from scratch in c c mainly based on finite differences schemes these numerical techniques include crank nicolson schemes for time discretization gibou s ghost node techniques and level set methods to cope with the free boundary associated to the determination of the time dependent biofilm domain to finish with we compare our simulation results with the dynamics of real biofilms as observed in the laboratory more precisely by using model parameters calibrated to experiments the numerical results clearly illustrate the performance of the proposed model and the numerical methods to reproduce the real dynamics of flat clustered and honeycomb structures shown by different listeria monocytogenes strains

parallel re initialization of level set functions on distributed unstructured tetrahedral grids level set functions are employed to track interfaces in various application areas including simulation of two phase flows and image segmentation often a re initializing algorithm is incorporated to transform a numerically instable level set function to a signed distance function in this note we present a parallel algorithm for re initializing level set functions on unstructured three dimensional tetrahedral grids the main idea behind this new domain decomposition approach is to combine a parallel brute force re initializing algorithm with an efficient way to compute distances between the interface and grid points time complexity and error analysis of the algorithm are investigated detailed numerical experiments demonstrate the accuracy and scalability on up to 128 processes

image flow segmentation and estimation by constraint line clustering image flow is the velocity field in the image plane caused by the motion of the observer objects in the scene or apparent motion and can contain discontinuities due to object occlusion in the scene an algorithm that can estimate the image flow velocity field when there are discontinuities due to occlusions is described the constraint line clustering algorithm uses a statistical test to estimate the image flow velocity field in the presence of step discontinuities in the image irradiance or velocity field particular emphasis is placed on motion estimation and segmentation in situations such as random dot patterns where motion is the only cue to segmentation experimental results on a demanding synthetic test case and a real image are presented a smoothing algorithm for improving the velocity field estimate is also described the smoothing algorithm constructs a smooth estimate of the velocity field by approximating a surface between step discontinuities it is noted that the velocity field estimate can be improved using surface reconstruction between velocity field boundaries

a fast parametric motion estimation algorithm with illumination and lens distortion correction methods for estimating motion in video sequences that are based on the optical flow equation ofe assume that the scene illumination is uniform and that the imaging optics are ideal when these assumptions are appropriate these methods can be very accurate but when they are not the accuracy of the motion field drops off accordingly this paper extends the models upon which the ofe methods are based to include irregular time varying illumination models and models for imperfect optics that introduce vignetting gamma and geometric warping such as are likely to be found with inexpensive pc cameras the resulting optimization framework estimates the motion parameters illumination parameters and camera parameters simultaneously in some cases these models can lead to nonlinear equations which must be solved iteratively in other cases the resulting optimization problem is linear for the former case an efficient hierarchical iterative framework is provided that can be used to implement the motion estimator

simplex free adaptive tree fast sweeping and evolution methods for solving level set equations in arbitrary dimension we introduce simplex free adaptive tree numerical methods for solving static and time dependent hamilton jacobi equations arising in level set problems in arbitrary dimension the data structure upon which our method is built in a generalized n dimensional binary tree but it does not require the complicated splitting of cubes into simplices aka generalized n dimensional triangles or hypertetrahedrons that current tree based methods require it has enough simplicity that minor variants of standard numerical hamiltonians developed for uniform grids can be applied yielding consistent monotone convergent schemes combined with the fast sweeping strategy the resulting tree based methods are highly efficient and accurate thus without changing more than a few lines of code when changing dimension we have obtained results for calculations in up to n 7 dimensions

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

vials an eulerian tool based on total variation and the level set method for studying dynamical systems abstract we propose a new eulerian tool to study complicated dynamical systems based on the average growth in the surface area of a family of level surfaces represented implicitly by a level set function since this proposed quantity determines the temporal variation of the averaged surface area of all level surfaces we name the quantity the variation of the integral over area of level surfaces vials numerically all these infinitely many level surfaces are advected according to the given dynamics by solving one single linear advection equation to develop a computationally efficient approach we apply the coarea formula and rewrite the surface area integral as a simple integral relating the total variation tv of the level set function the proposed method can be easily incorporated with a recent eulerian algorithm for efficient computation of flow maps to speed up our approach we will also prove that the proposed vials is closely related to the computation of the finite time lyapunov exponent ftle in the lagrangian coherent structure lcs extraction this connects our proposed eulerian approach to widely used lagrangian techniques for understanding complicated dynamical systems

the cross search algorithm for motion estimation image coding a fast block matching algorithm for motion estimation is presented it is based on a logarithmic step where in each search step only four locations are tested for a motion displacement of w pels frame this technique requires 5 4 log sub 2 w computations to locate the best match using sequences of cif standard pictures the interframe motion compensated prediction error with this technique is compared to the other fast methods the computational complexity of this algorithm is also compared against those methods

ordered upwind methods for static hamilton jacobi equations theory and algorithms we develop a family of fast methods for approximating the solutions to a wide class of static hamilton jacobi pdes these fast methods include both semi lagrangian and fully eulerian versions numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations our techniques which we refer to as ordered upwind methods oums use partial information about the characteristic directions to decouple these nonlinear systems greatly reducing the computational labor our techniques are considered in the context of control theoretic and front propagation problems we begin by discussing existing oums focusing on those designed for isotropic problems we then introduce a new class of oums which decouple systems for general anisotropic problems we prove convergence of one such scheme to the viscosity solution of the corresponding hamilton jacobi pde next we introduce a set of finite differences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems the performance of the methods is analyzed and computational experiments are performed using test problems from computational geometry and seismology

hamilton paths in grid graphs a grid graph is a node induced finite subgraph of the infinite grid it is rectangular if its set of nodes is the product of two intervals given a rectangular grid graph and two of its nodes we give necessary and sufficient conditions for the graph to have a hamilton path between these two nodes in contrast the hamilton path and circuit problem for general grid graphs is shown to be np complete this provides a new relatively simple proof of the result that the euclidean traveling salesman problem is np complete

efficient algorithms for globally optimal trajectories we present serial and parallel algorithms for solving a system of equations that arises from the discretization of the hamilton jacobi equation associated to a trajectory optimization problem of the following type a vehicle starts at a prespecified point x sub o and follows a unit speed trajectory x t inside a region in spl rscr sup m until an unspecified time t that the region is exited a trajectory minimizing a cost function of the form spl int sub 0 sup t r x t dt q x t is sought the discretized hamilton jacobi equation corresponding to this problem is usually solved using iterative methods nevertheless assuming that the function r is positive we are able to exploit the problem structure and develop one pass algorithms for the discretized problem the first algorithm resembles dijkstra s shortest path algorithm and runs in time o n log n where n is the number of grid points the second algorithm uses a somewhat different discretization and borrows some ideas from a variation of dial s shortest path algorithm 1969 that we develop here it runs in time o n which is the best possible under some fairly mild assumptions finally we show that the latter algorithm can be efficiently parallelized for two dimensional problems and with p processors its running time becomes o n p provided that p o spl radic n log n

finite difference quasi p traveltimes for anisotropic media the first arrival quasi p wave traveltime field in an anisotropic elastic solid solves a first order nonlinear partial differential equation the q p eikonal equation the difficulty in solving this eikonal equation by a finite difference method is that for anisotropic media the ray group velocity direction is not the same as the direction of the traveltime gradient so that the traveltime gradient can no longer serve as an indicator of the group velocity direction in extrapolating the traveltime field however establishing an explicit relation between the ray velocity vector and the phase velocity vector overcomes this difficulty furthermore the solution of the paraxial q p eikonal equation an evolution equation in depth gives the first arrival traveltime along downward propagating rays a second order upwind finite difference scheme solves this paraxial eikonal equation in o n floating point operations where n is the number of grid points numerical experiments using 2 d and 3 d transversely is

a computational framework and an algorithm for the measurement of visual the robust measurement of visual motion from digitized image sequences has been an important but difficult problem in computer vision this paper describes a hierarchical computational framework for the determination of dense displacement fields from a pair of images and an algorithm consist ent with that framework our framework is based on the separation of the image intensity information as well as the process of measuring motion according to scale the large scale intensity information is first used to obtain rough estimates of image motion which are then refined by using intensity information at smaller scales the estimates are in the form of displacement or velocity vectors for pixels and are accompanied by a direction dependent confidence measure a smoothness constraint is employed to propagate the measurements with high confidence to their neigboring areas where the confidences are low at all levels the computations are pixel parallel uniform across the image and based on information from a small neighborhood of a pixel for our algorithm the local displacement vectors are determind by mini mizing the sum of squared differences ssd of intensities the confidence measures are derived from the shape of the ssd surface and the smoothness constraint is cast in the form of energy minimization results of applying our algorithm to pairs of real images are included in addition to our own

uniformly accurate discontinuous galerkin fast sweeping methods for eikonal equations in f li c w shu y t zhang h zhao j comput phys 227 2008 pp 8191 8208 we developed a fast sweeping method based on a hybrid local solver which is a combination of a discontinuous galerkin dg finite element solver and a first order finite difference solver for eikonal equations the method has second order accuracy in the l 1 norm and a very fast convergence speed but only first order accuracy in the l infty norm for the general cases this is an obstacle to the design of higher order dg fast sweeping methods in this paper we overcome this problem by developing uniformly accurate dg fast sweeping methods for solving eikonal equations we design novel causality indicators which guide the information flow directions for the dg local solver the values of these indicators are initially provided by the first order finite difference fast sweeping method and they are updated during iterations along with the solution we observe both a uniform second order accuracy in the l infty norm in smooth regions and the fast convergence speed linear computational complexity in the numerical examples

estimating piecewise smooth optical flow with global matching and graduated optimization this paper presents a new method for estimating piecewise smooth optical flow we propose a global optimization formulation with three frame matching and local variation and develop an efficient technique to minimize the resultant global energy this technique takes advantage of local gradient global gradient and global matching methods and alleviates their limitations experiments on various synthetic and real data show that this method achieves highly competitive accuracy

3 d kalman filter for image motion estimation this paper presents a new three dimensional 3 d markov model for motion vector fields the three dimensions consist of the two space dimensions plus a scale dimension we use a compound signal model to handle motion discontinuity in this 3 d markov random field mrf for motion estimation we use an extended kalman filter as a pel recursive estimator since a single observation can be sensitive to local image characteristics especially when the model is not accurate we employ windowed multiple observations at each pixel to increase accuracy these multiple observations employ different weighting values for each observation since the uncertainty in each observation is different finally we compare this 3 d model with earlier proposed one dimensional 1 d coarse to fine scale and two dimensional 2d spatial compound models in terms of motion estimation performance on a synthetic and a real image sequence

a generalized fast marching method for dislocation dynamics in this paper we consider a generalized fast marching method gfmm as a numerical method to compute dislocation dynamics the dynamics of a dislocation hypersurface in mathbb r n with n 2 for physical applications is given by its normal velocity which is a nonlocal function of the whole shape of the hypersurface itself for this dynamics we show a convergence result of the gfmm as the mesh size goes to zero we also provide some numerical simulations in dimension n 2

fast semi lagrangian schemes for the eikonal equation and applications we introduce and analyze a fast version of the semi lagrangian algorithm for front propagation originally proposed in m falcone the minimum time problem and its applications to front propagation in motion by mean curvature and related topics a visintin and g buttazzo eds de gruyter berlin 1994 pp 70 88 the new algorithm is obtained using the local definition of the approximate solution typical of semi lagrangian schemes and redefining the set of neighboring nodes necessary for fast marching schemes a new proof of convergence is needed since that definition produces a new narrow band centered at the interphase which is larger than the one used in fast marching methods based on finite differences we show that the new algorithm converges to the viscosity solution of the problem and that its complexity is o n log n nb as it is for the fast marching method based on finite difference n and n nb being respectively the total number of nodes and the number of nodes in the narrow band a new sufficient condition for the convergence of the standard finite difference fast marching method is also given we present several tests comparing the two algorithms and other fast methods e g fast sweeping on a series of benchmarks which include the minimum time problem and the shape from shading problem

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

fast sweeping method for the factored eikonal equation we develop a fast sweeping method for the factored eikonal equation by decomposing the solution of a general eikonal equation as the product of two factors the first factor is the solution to a simple eikonal equation such as distance or a previously computed solution to an approximate eikonal equation the second factor is a necessary modification correction appropriate discretization and a fast sweeping strategy are designed for the equation of the correction part the key idea is to enforce the causality of the original eikonal equation during the gauss seidel iterations using extensive numerical examples we demonstrate that 1 the convergence behavior of the fast sweeping method for the factored eikonal equation is the same as for the original eikonal equation i e the number of iterations for the gauss seidel iterations is independent of the mesh size 2 the numerical solution from the factored eikonal equation is more accurate than the numerical solution directly computed from the original eikonal equation especially for point sources

fast legendre fenchel transform and applications to hamilton jacobi equations and conservation laws we are interested in the study of a fast algorithm introduced by brenier computing the discrete legendre fenchel transform of a real function we present convergence results and show how the order of convergence grows with the regularity of the function to be transformed applications to hamilton jacobi equations for front propagation problems and conservation laws are presented

an efficient interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow in sussman smereka and osher j comp phys 94 1994 pp 146 159 a numerical scheme was presented for computing incompressible air water flows using the level set method crucial to the above method was a new iteration method for maintaining the level set function as the signed distance from the zero level set in this paper we implement a constraint along with higher order difference schemes in order to make the iteration method more accurate and efficient accuracy is measured in terms of the new computed signed distance function and the original level set function having the same zero level set we apply our redistancing scheme to incompressible flows with noticeably better resolved results at reduced cost we validate our results with experiment and theory we show that our distance level set scheme with the added constraint competes well with available interface tracking schemes for basic advection of an interface we perform basic accuracy checks and more stringent tests involving complicated interfacial structures as with all level set schemes our method is easy to implement

multiple initial point prediction based search pattern selection for fast motion estimation a novel computationally efficient and robust scheme for multiple initial point prediction has been proposed in this paper a combination of spatial and temporal predictors has been used for initial motion vector prediction determination of magnitude and direction of motion and search pattern selection initially three predictors from the spatio temporal neighboring blocks are selected if all these predictors point to the same quadrant then a simple search pattern based on the direction and magnitude of the predicted motion vector is selected however if the predictors belong to different quadrants then we start the search from multiple initial points to get a clear idea of the location of minimum point we have also defined local minimum elimination criteria to avoid being trapped in local minimum in this case multiple rood search patterns are selected the predictive search center is closer to the global minimum and thus decreases the effect of monotonic error surface assumption and its impact on the motion field its additional advantage is that it moves the search closer to the global minimum hence increases the computation speed further computational speed up has been obtained by considering the zero motion threshold for no motion blocks the image quality measured in terms of psnr also shows good results

motion of curves constrained on surfaces using a level set approach the level set method has been successfully applied to a variety of problems that deal with curves in r2 or surfaces in r3 we present here a combination of these two cases creating a level set representation for curves constrained to lie on surfaces we study primarily geometrically based motions of these curves on stationary surfaces while allowing topological changes in the curves i e merging and breaking to occur applications include finding geodesic curves and shortest paths and curve shortening on surfaces further applications can be arrived at by extending those for curves moving in r2 to surfaces the problem of moving curves on surfaces can also be viewed as a simple constraint problem and may be useful in studying more difficult versions results show that our representation can accurately handle many geometrically based motions of curves on a wide variety of surfaces while automatically enforcing topological changes in the curves when they occur and automatically fixing the curves to lie on the surfaces the method can also be easily extended to higher dimensions

multiscale modeling and estimation of motion fields for video coding we present a systematic approach to forward motion compensated predictive video coding the first step is the definition of a flexible model that compactly represents motion fields the inhomogeneity and spatial coherence properties of motion fields are captured using linear multiscale models one possible design is based on linear finite elements and yields a multiscale extension of the triangle motion compensation tmc method the second step is the choice of a computational technique that identifies the coefficients of the linear model we study a modified optical flow technique and minimize a cost function closely related to horn and schunck s 1981 criterion the cost function balances accuracy and complexity of the motion compensated predictor and is viewed as a measure of goodness of the motion field it determines not only the coefficients of the model but also the quantization method we formulate the estimation and quantization problems jointly as a discrete optimization problem and solve it using a fast multiscale relaxation algorithm a hierarchical extension of the algorithm allows proper handling of large displacements simulations on a variety of video sequences have produced improvements over tmc and over the half pel accuracy full search block matching algorithm in excess of 0 5 db in average the results are visually superior as well in particular the reconstructed video is entirely free of blocking artifacts

a level set projection model of lipid vesicles in general flows a new numerical method to model the dynamic behavior of lipid vesicles under general flows is presented a gradient augmented level set method is used to model the membrane motion to enforce the volume and surface incompressibility constraints a four step projection method is developed to integrate the full navier stokes equations this scheme is implemented on an adaptive non graded cartesian grid convergence results are presented along with sample two dimensional results of vesicles under various flow conditions

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

geometric observers for dynamically evolving curves this paper proposes a deterministic observer design for visual tracking based on nonparametric implicit level set curve descriptions the observer is continuous discrete with continuous time system dynamics and discrete time measurements its state space consists of an estimated curve position augmented by additional states e g velocities associated with every point on the estimated curve multiple simulation models are proposed for state prediction measurements are performed through standard static segmentation algorithms and optical flow computations special emphasis is given to the geometric formulation of the overall dynamical system the discrete time measurements lead to the problem of geometric curve interpolation and the discrete time filtering of quantities propagated along with the estimated curve interpolation and filtering are intimately linked to the correspondence problem between curves correspondences are established by a laplace equation approach the proposed scheme is implemented completely implicitly by eulerian numerical solutions of transport equations and thus naturally allows for topological changes and subpixel accuracy on the computational grid

alternating evolution methods for static hamilton jacobi equations abstract we design analyze and numerically validate alternating evolution schemes for solving static hamilton jacobi equations this class of high resolution numerical schemes is based on the alternating evolution reformulation of the original hamilton jacobi equation stability and convergence properties are proven for first order schemes numerical implementation of up to third order schemes is carried out for a set of widely used examples further application of the proposed method to the kinetic hamilton jacobi equation is given while the hamiltonian is determined by an integral in phase space

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

multistencils fast marching methods a highly accurate solution to the eikonal equation on cartesian domains a wide range of computer vision applications require an accurate solution of a particular hamilton jacobi hj equation known as the eikonal equation in this paper we propose an improved version of the fast marching method fmm that is highly accurate for both 2d and 3d cartesian domains the new method is called multistencils fast marching msfm which computes the solution at each grid point by solving the eikonal equation along several stencils and then picks the solution that satisfies the upwind condition the stencils are centered at each grid point and cover its entire nearest neighbors in 2d space two stencils cover 8 neighbors of the point whereas in 3d space six stencils cover its 26 neighbors for those stencils that are not aligned with the natural coordinate system the eikonal equation is derived using directional derivatives and then solved using higher order finite difference schemes the accuracy of the proposed method over the state of the art fmm based techniques has been demonstrated through comprehensive numerical experiments

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 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

what shape is your conjugate a survey of computational convex analysis and its applications computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields to further communications between practitioners we review the field of computational convex analysis which focuses on the numerical computation of fundamental transforms arising from convex analysis current models use symbolic numeric and hybrid symbolic numeric algorithms our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing computing the distance transform the generalized distance transform and mathematical morphology operators partial differential equations solving hamilton jacobi equations and using differential equations numerical schemes to compute the convex envelope max plus algebra computing the equivalent of the fast fourier transform multifractal analysis etc the fields of applications include among others computer vision robot navigation thermodynamics electrical networks medical imaging and network communication

a static pde approach for multidimensional extrapolation using fast sweeping methods a static partial differential equation pde approach is presented for multidimensional extrapolation under the assumption that a level set function exists which separates the region of known value

efficient algorithms for coding hilbert curve of arbitrary sized image and application to window query previously several efficient hilbert scan based operations were presented but they all suffer from the constraint that the image size must be 2 rx2 r considering an image with arbitrary size i 1xi 2 this paper first presents an efficient snake scan based algorithm for coding the hilbert curve of the given image the proposed coding algorithm takes o k logu time to code the hilbert order of one pixel where k denotes the number of the quadrants and u min i 1 i 2 next a memory saving hilbert curve representation called hcgl is presented for representing the encoded hilbert curve and it can be constructed in o l 2logl time where l max i 1 i 2 based on the hcgl representation of arbitrary sized image an application to window query is presented and the proposed window query algorithm takes o mlogl p time where m denotes the number of generated maximal quadtree blocks and p denotes the number of output codes under the same psnr peak signal to noise ratio experimental results demonstrate that our proposed hcgl representation outperforms some existing related algorithms in terms of execution time and bpp bit per pixel in addition our proposed window query algorithm has been justified in the gis geographical information system application