euler s elastica and curvature based inpainting image inpainting is a special image restoration problem for which image prior models play a crucial role euler s elastica was first introduced to computer vision by mumford algebraic geometry and its applications springer verlag new york 1994 pp 491 506 as a curve prior model by functionalizing the elastica energy masnou and morel proceedings of the 5th ieee international conference image processing 3 1998 pp 259 263 proposed an elastica based variational inpainting model the current paper is intended to contribute to the development of its mathematical foundation and the study of its properties and connections to the earlier works of bertalmio sapiro caselles and ballester siggraph 2000 acm press new york 2000 and chan and shen j visual comm image rep 12 2001 pp 436 449 a computational scheme based on numerical pdes is presented which allows the automatic handling of topologically complex inpainting domains

the piecewise smooth mumford shah functional on an arbitrary graph the mumford shah functional has had a major impact on a variety of image analysis problems including image segmentation and filtering and despite being introduced over two decades ago it is still in widespread use present day optimization of the mumford shah functional is predominated by active contour methods until recently these formulations necessitated optimization of the contour by evolving via gradient descent which is known for its overdependence on initialization and the tendency to produce undesirable local minima in order to reduce these problems we reformulate the corresponding mumford shah functional on an arbitrary graph and apply the techniques of combinatorial optimization to produce a fast low energy solution in contrast to traditional optimization methods use of these combinatorial techniques necessitates consideration of the reconstructed image outside of its usual boundary requiring additionally the inclusion of regularization for generating these values the energy of the solution provided by this graph formulation is compared with the energy of the solution computed via traditional gradient descent based narrow band level set methods this comparison demonstrates that our graph formulation and optimization produces lower energy solutions than the traditional gradient descent based contour evolution methods in significantly less time finally we demonstrate the usefulness of the graph formulation to apply the mumford shah functional to new applications such as point clustering and filtering of nonuniformly sampled images

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

a review of statistical approaches to level set segmentation integrating color texture motion and shape since their introduction as a means of front propagation and their first application to edge based segmentation in the early 90 s level set methods have become increasingly popular as a general framework for image segmentation in this paper we present a survey of a specific class of region based level set segmentation methods and clarify how they can all be derived from a common statistical framework region based segmentation schemes aim at partitioning the image domain by progressively fitting statistical models to the intensity color texture or motion in each of a set of regions in contrast to edge based schemes such as the classical snakes region based methods tend to be less sensitive to noise for typical images the respective cost functionals tend to have less local minima which makes them particularly well suited for local optimization methods such as the level set method we detail a general statistical formulation for level set segmentation subsequently we clarify how the integration of various low level criteria leads to a set of cost functionals we point out relations between the different segmentation schemes in experimental results we demonstrate how the level set function is driven to partition the image plane into domains of coherent color texture dynamic texture or motion moreover the bayesian formulation allows to introduce prior shape knowledge into the level set method we briefly review a number of advances in this domain

shortest paths with higher order regularization this paper describes a new method of finding thin elongated structures in images and volumes we use shortest paths to minimize very general functionals of higher order curve properties such as curvature and torsion our method uses line graphs to find the optimal path on a given discretization often in the order of seconds on a single computer the curves are then refined using local optimization making it possible to recover very smooth curves we are able to place constraints on our curves such as maximum integrated curvature or a maximum curvature at any point of the curve to our knowledge we are the first to perform experiments in three dimensions with curvature and torsion regularization the largest graphs we process have over a hundred billion arcs experiments on medical images and in multi view reconstruction show the significance and practical usefulness of higher order regularization

using dynamic programming for solving variational problems in vision dynamic programming is discussed as an approach to solving variational problems in vision dynamic programming ensures global optimality of the solution is numerically stable and allows for hard constraints to be enforced on the behavior of the solution within a natural and straightforward structure as a specific example of the approach s efficacy applying dynamic programming to the energy minimizing active contours is described the optimization problem is set up as a discrete multistage decision process and is solved by a time delayed discrete dynamic programming algorithm a parallel procedure for decreasing computational costs is discussed

completion energies and scale the detection of smooth curves in images and their completion over gaps are two important problems in perceptual grouping we examine the notion of completion energy of curve elements showing and exploiting its intrinsic dependence on length and width scales we introduce a fast method for computing the most likely completion between two elements by developing novel analytic approximations and a fast numerical procedure for computing the curve of least energy we then use our newly developed energies to find the most likely completions in images through a generalized summation of induction fields this is done through multiscale procedures i e separate processing at different scales with some interscale interactions such procedures allow the summation of all induction fields to be done in a total of only o n log n operations where n is the number of pixels in the image more important such procedures yield a more realistic dependence of the induction field on the length and width scales the field of a long element is very different from the sum of the fields of its composing short segments

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

progressive minimal path method for segmentation of 2d and 3d line structures we propose a novel minimal path method for the segmentation of 2d and 3d line structures minimal path methods perform propagation of a wavefront emanating from a start point at a speed derived from image features followed by path extraction using backtracing usually the computation of the speed and the propagation of the wave are two separate steps and point features are used to compute a static speed we introduce a new continuous minimal path method which steers the wave propagation progressively using dynamic speed based on path features we present three instances of our method using an appearance feature of the path a geometric feature based on the curvature of the path and a joint appearance and geometric feature based on the tangent of the wavefront these features have not been used in previous continuous minimal path methods we compute the features dynamically during the wave propagation and also efficiently using a fast numerical scheme and a low dimensional parameter space our method does not suffer from discretization or metrication errors we performed qualitative and quantitative evaluations using 2d and 3d images from different application areas

an explicit time evolution method for acoustic wave propagation cost effective waveform modeling is the key to practical reverse time migration rtm and full waveform inversion fwi implementations we evaluated an explicit time evolution ete method to efficiently simulate wave propagation in acoustic media with high temporal accuracy we started from the constant density acoustic wave equation and obtained an analytical time marching scheme in the wavenumber domain we then formulated an ete scheme in the time space domain by introducing a cosine function approximation although the ete operator appears to be similar to the second order temporal finite difference fd operator the exact nature of the ete formula ensures high accuracy in time we further introduced a set of optimum stencils and coefficients by minimizing evolution errors in a least squares sense our numerical tests indicated that ete can achieve similar waveform accuracy as fd with four times larger time steps meanwhile the compact ete operator keeps the computation efficient the efficiency and capability to handle complex velocity field make ete an attractive engine in acoustic rtm and fwi

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

trace inference curvature consistency and curve detection an approach is described for curve inference that is based on curvature information the inference procedure is divided into two stages a trace inference stage which is the subject of the present work and a curve synthesis stage it is shown that recovery of the trace of a curve requires estimating local models for the curve at the same time and that tangent and curvature information are sufficient these make it possible to specify powerful constraints between estimated tangents to a curve in terms of a neighborhood relationship called cocircularity and between curvature estimates in terms of a curvature consistency relation because all curve information is quantized special care must be taken to obtain accurate estimates of trace points tangents and curvatures this issue is addressed specifically to the introduction of a smoothness constraint and a maximum curvature constraint the procedure is applied to two types of images artificial images designed to evaluate curvature and noise sensitivity and natural images

isoperimetric graph partitioning for image segmentation spectral graph partitioning provides a powerful approach to image segmentation we introduce an alternate idea that finds partitions with a small isoperimetric constant requiring solution to a linear system rather than an eigenvector problem this approach produces the high quality segmentations of spectral methods but with improved speed and stability

the optimized expansion based low rank method for wavefield extrapolation abstractspectral methods are fast becoming an indispensable tool for wavefield extrapolation especially in anisotropic media because it tends to be dispersion and artifact free as well as highly accurate when solving the wave equation however for inhomogeneous media we face difficulties in dealing with the mixed space wavenumber domain extrapolation operator efficiently to solve this problem we evaluated an optimized expansion method that can approximate this operator with a low rank variable separation representation the rank defines the number of inverse fourier transforms for each time extrapolation step and thus the lower the rank the faster the extrapolation the method uses optimization instead of matrix decomposition to find the optimal wavenumbers and velocities needed to approximate the full operator with its explicit low rank representation as a result we obtain lower rank representations compared with the standard low rank method within reasonable accuracy and thus cheaper extrapola

a new time space domain high order finite difference method for the acoustic wave equation a new unified methodology was proposed in finkelstein and kastner 2007 39 to derive spatial finite difference fd coefficients in the joint time space domain to reduce numerical dispersion the key idea of this method is that the dispersion relation is completely satisfied at several designated frequencies we develop this new time space domain fd method further for 1d 2d and 3d acoustic wave modeling using a plane wave theory and the taylor series expansion new spatial fd coefficients are frequency independent though they lead to a frequency dependent numerical solution we prove that the modeling accuracy is 2nd order when the conventional 2m th order space domain fd and the 2nd order time domain fd stencils are directly used to solve the acoustic wave equation however under the same discretization the new 1d method can reach 2m th order accuracy and is always stable the 2d method can reach 2m th order accuracy along eight directions and has better stability similarly the 3d method can reach 2m th order accuracy along 48 directions and also has better stability than the conventional fd method the advantages of the new method are also demonstrated by the results of dispersion analysis and numerical modeling of acoustic wave equation for homogeneous and inhomogeneous acoustic models in addition we study the influence of the fd stencil length on numerical modeling for 1d inhomogeneous media and derive an optimal fd stencil length required to balance the accuracy and efficiency of modeling a new time space domain high order staggered grid fd method for the 1d acoustic wave equation with variable densities is also developed which has similar advantages demonstrated by dispersion analysis stability analysis and modeling experiments the methodology presented in this paper can be easily extended to solve similar partial difference equations arising in other fields of science and engineering

computing the nucleolus of some combinatorially structured games in this paper we introduce the prenucleolus for a transferable utility game n v where 2 n the prenucleolus is a straightforward generalization of the ordinary prenucleolus where only the coalitions in determine the outcome we impose a combinatorial structure on the collection which enables us to compute the prenucleolus in n 3 time the algorithm can be used for computing the nucleolus of several classes of games among which is the class of minimum cost spanning tree games

cost allocation for a tree network with heterogeneous customers we analyze a cost allocation problem which could naturally arise from a situation wherein a tree networkt n 0 e serving heterogeneous customers has to be constructed the customers located atn require some service from a central supplier located at vertex 0 the customers have heterogeneous preferences for the level or quality of service received from the central supplier we formulate the above cost allocation problem as a cooperative game referred to as anextended tree game the extended tree game is a proper extension of megiddo s 1978 tree game wherein all the customers have identical preferences regarding the level of service received we prove that an extended tree game is convex and we show that its shapley value can be computed in p n time wherep is the number of distinct preference levels we further provide a complete facial description of the core polytope of an extended tree game and demonstrate that even when there are only two classes of customers the number of nonredundant core constraints could be exponential in n nevertheless we are able to construct an p n algorithm to check the core membership of an arbitrary cost allocation which can be used to construct an p n 3 combinatorial algorithm to compute the nucleolus of an extended tree game finally we show that the complements of the facet defining coalitions for the core are all connected in an auxiliary tree graph with node setn

three dimensional elastic modeling by the fourier method earlier work on three dimensional forward modeling is extended to elastic waves using the equations of con servation of momentum and the stress strain relations for an isotropic elastic medium undergoing infinitesimal deformation in addition to arbitrary compressional or p wave velocity and density variation in lateral and vertical directions elastic modeling permits shear or s wave velocity variation as well the elastic wave equation is solved using a generalization of the method for the acoustic case computation of each time step begins by computing six strain components by per forming nine spatial partial differentiation operations on the three displacement components from the pre vious time step the six strains and two lame constants are linearly combined to yield six stress components

fourier finite difference migration many existing migration schemes cannot simultaneously handle the two most important problems of migration imaging of steep dips and imaging in media with arbitrary velocity variations in all directions for example phase shift k migration is accurate for nearly all dips but it is limited to very simple velocity functions on the other hand finite difference schemes based on one way wave equations consider arbitrary velocity functions but they attenuate steeply dipping events we propose a new hybrid migration method named fourier finite difference migration wherein the downward continuation operator is split into two downward continuation operators one operator is a phase shift operator for a chosen constant background velocity and the other operator is an optimized finite difference operator for the varying component of the velocity function if there is no variation of velocity then only a phase shift operator will be applied automatically on the other hand if there is a strong variatio

the elastic ratio introducing curvature into ratio based image segmentation we present the first ratio based image segmentation method that allows imposing curvature regularity of the region boundary our approach is a generalization of the ratio framework pioneered by jermyn and ishikawa so as to allow penalty functions that take into account the local curvature of the curve the key idea is to cast the segmentation problem as one of finding cyclic paths of minimal ratio in a graph where each graph node represents a line segment among ratios whose discrete counterparts can be globally minimized with our approach we focus in particular on the elastic ratio 0l c i c s c s ds vl c 0l c c s qds that depends given an image i on the oriented boundary c of the segmented region candidate minimizing this ratio amounts to finding a curve neither small nor too curvy through which the brightness flux is maximal we prove the existence of minimizers for this criterion among continuous curves with mild regularity assumptions we also prove that the discrete minimizers provided by our graph based algorithm converge as the resolution increases to continuous minimizers in contrast to most existing segmentation methods with computable and meaningful i e nondegenerate global optima the proposed approach is fully unsupervised in the sense that it does not require any kind of user input such as seed nodes numerical experiments demonstrate that curvature regularity allows substantial improvement of the quality of segmentations furthermore our results allow drawing conclusions about global optima of a parameterization independent version of the snakes functional the proposed algorithm allows determining parameter values where the functional has a meaningful solution and simultaneously provides the corresponding global solution

a nonreflecting boundary condition for discrete acoustic and elastic wave equations one of the nagging problems which arises in application of discrete solution methods for wave propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh these undesired events eventually override the actual seismic signals which propagate in the modeled region the solution to avoiding boundary effects used to be to enlarge the numerical mesh thus delaying the side reflections and wraparound longer than the range of times involved in the modeling obviously this solution considerably increases the expense of computation more recently nonreflecting boundary conditions were introduced for the finite difference method clayton and enquist 1977 reynolds 1978 these boundary conditions are based on replacing the wave equation in the boundary region by one way wave equations which do not permit energy to propagate from the boundaries into the numerical mesh this approach has been relatively successful except that its effectiveness degrades for event

modeling of pseudoacoustic p waves in orthorhombic media with a low rank approximation abstractwavefield extrapolation in pseudoacoustic orthorhombic anisotropic media suffers from wave mode coupling and stability limitations in the parameter range we use the dispersion relation for scalar wave propagation in pseudoacoustic orthorhombic media to model acoustic wavefields the wavenumber domain application of the laplacian operator allows us to propagate the p waves exclusively without imposing any conditions on the parameter range of stability it also allows us to avoid dispersion artifacts commonly associated with evaluating the laplacian operator in space domain using practical finite difference stencils to handle the corresponding space wavenumber mixed domain operator we apply the low rank approximation approach considering the number of parameters necessary to describe orthorhombic anisotropy the low rank approach yields space wavenumber decomposition of the extrapolator operator that is dependent on space location regardless of the parameters a feature necessary for orthorhomb

minimal surfaces based object segmentation a geometric approach for 3d object segmentation and representation is presented the segmentation is obtained by deformable surfaces moving towards the objects to be detected in the 3d image the model is based on curvature motion and the computation of surfaces with minimal areas better known as minimal surfaces the space where the surfaces are computed is induced from the 3d image volumetric data in which the objects are to be detected the model links between classical deformable surfaces obtained via energy minimization and intrinsic ones derived from curvature based flows the new approach is stable robust and automatically handles changes in the surface topology during the deformation

polynomial time algorithms for ratio regions and a variant of normalized cut in partitioning clustering and grouping problems a typical goal is to group together similar objects or pixels in the case of image processing at the same time another goal is to have each group distinctly dissimilar from the rest and possibly to have the group size fairly large these goals are often combined as a ratio optimization problem one example of such a problem is a variant of the normalized cut problem another is the ratio regions problem we devise here the first polynomial time algorithms solving optimally the ratio region problem and the variant of normalized cut as well as a few other ratio problems the algorithms are efficient and combinatorial in contrast with nonlinear continuous approaches used in the image segmentation literature which often employ spectral techniques such techniques deliver solutions in real numbers which are not feasible to the discrete partitioning problem furthermore these continuous approaches are computationally expensive compared to the algorithms proposed here the algorithms presented here use as a subroutine a minimum s t cut procedure on a related graph which is of polynomial size the output consists of the optimal solution to the respective ratio problem as well as a sequence of nested solutions with respect to any relative weighting of the objectives of the numerator and denominator

contrast driven elastica for image segmentation minimization of boundary curvature is a classic regularization technique for image segmentation in the presence of noisy image data techniques for minimizing curvature have historically been derived from gradient descent methods which could be trapped by a local minimum and therefore required a good initialization recently combinatorial optimization techniques have overcome this barrier by providing solutions that can achieve a global optimum however curvature regularization methods can fail when the true object has high curvature in these circumstances existing methods depend on a data term to overcome the high curvature of the object unfortunately the data term may be ambiguous in some images which causes these methods also to fail to overcome these problems we propose a contrast driven elastica model including curvature which can accommodate high curvature objects and an ambiguous data model we demonstrate that we can accurately segment extremely challenging synthetic and real images with ambiguous data discrimination poor boundary contrast and sharp corners we provide a quantitative evaluation of our segmentation approach when applied to a standard image segmentation data set

euler spiral for shape completion in this paper we address the curve completion problem e g the geometric continuation of boundaries of objects which are temporarily interrupted by occlusion also known as the gap completion or shape completion problem this problem is a significant element of perceptual grouping of edge elements and has been approached by using cubic splines or biarcs which minimize total curvature squared elastica as motivated by a physical analogy our approach is motivated by railroad design methods of the early 1900 s which connect two rail segments by transition curves and by the work of knuth on mathematical typography we propose that in using an energy minimizing solution completion curves should not penalize curvature as in elastica but curvature variation the minimization of total curvature variation leads to an euler spiral solution a curve whose curvature varies linearly with arclength the construction of this curve from a pair of points and tangents at these points is reduced to a nonlinear system of equations involving fresnel integrals whose solution relies on optimization from a suitable initial condition constrained to satisfy given boundary conditions since the choice of an appropriate initial curve is critical in this optimization we analytically derive an optimal solution in the class of biarc curves which is then used as the initial curve the resulting interpolations yield intuitive interpolation across gaps and occlusions and are extensible in contrast to scale invariant elastica in addition euler spiral segments can be used in other applications of curve completions e g modeling boundary segments between curvature extrema or skeletal branch geometry

mac magnetostatic active contour model we propose an active contour model using an external force field that is based on magnetostatics and hypothesized magnetic interactions between the active contour and object boundaries the major contribution of the method is that the interaction of its forces can greatly improve the active contour in capturing complex geometries and dealing with difficult initializations weak edges and broken boundaries the proposed method is shown to achieve significant improvements when compared against six well known and state of the art shape recovery methods including the geodesic snake the generalized version of gradient vector flow gvf snake the combined geodesic and gvf snake and the charged particle model

a k space operator based least squares staggered grid finite difference method for modeling scalar wave propagation abstracttwo staggered grid finite difference sgfd schemes with fourth and sixth order accuracies in time have been developed recently based on new sgfd stencils the sgfd coefficients of the two schemes are determined by a taylor series expansion te approach which is accurate only near a zero wavenumber we have adopted the same sgfd stencils and determined the sgfd coefficients by minimizing the errors between the wavenumber responses of the sgfd operators and the first order k wavenumber space operator in a least squares ls sense we solved the ls problems by performing a weighted pseudoinverse of nonsquare matrices to obtain the sgfd coefficients and to yield ls based sgfd methods we have developed an efficient ls implementation by using a small number of representative wavenumbers which makes the computational time for estimating the coefficients negligible dispersion analysis and numerical examples demonstrate that our ls based sgfd methods can preserve the original temporal accuracy and

an acoustic wave equation for anisotropic media a wave equation derived using the acoustic medium assumption for p waves in transversely isotropic ti media with a vertical symmetry axis vti media yields a good kinematic approximation to the familiar elastic wave equation for vti media the wavefield solutions obtained using this vti acoustic wave equation are free of shear waves which significantly reduces the computation time compared to the elastic wavefield solutions for exploding reflector type applications from this vti acoustic wave equation the eikonal and transport equations that describe the ray theoretical aspects of wave propagation in a ti medium are derived these equations based on the acoustic assumption shear wave velocity 0 are much simpler than their elastic counterparts yet they yield an accurate description of traveltimes and geometrical amplitudes numerical examples prove the usefulness of this acoustic equation in simulating the kinematic aspects of wave propagation in complex ti models

time evolution of the wave equation using rapid expansion method forward modeling of seismic data and reverse time migration are based on the time evolution of wavefields for the case of spatially varying velocity we have worked on two approaches to evaluate the time evolution of seismic wavefields an exact solution for the constant velocity acoustic wave equation can be used to simulate the pressure response at any time for a spatially varying velocity a one step method can be developed where no intermediate time responses are required using this approach we have solved for the pressure response at intermediate times and have developed a recursive solution the solution has a very high degree of accuracy and can be reduced to various finite difference time derivative methods depending on the approximations used although the two approaches are closely related each has advantages depending on the problem being solved

lowrank seismic wave extrapolation on a staggered grid we evaluated a new spectral method and a new finite difference fd method for seismic wave extrapolation in time using staggered temporal and spatial grids we derived a wave extrapolation operator using a lowrank decomposition for a first order system of wave equations and designed the corresponding fd scheme the proposed methods extend previously proposed lowrank and lowrank fd wave extrapolation methods from the cases of constant density to those of variable density dispersion analysis demonstrated that the proposed methods have high accuracy for a wide wavenumber range and significantly reduce the numerical dispersion the method of manufactured solutions coupled with mesh refinement was used to verify each method and to compare numerical errors tests on 2d synthetic examples demonstrated that the proposed method is highly accurate and stable the proposed methods can be used for seismic modeling or reverse time migration

flux maximizing geometric flows several geometric active contour models have been proposed for segmentation in computer vision and image analysis the essential idea is to evolve a curve in 2d or a surface in 3d under constraints from image forces so that it clings to features of interest in an intensity image recent variations on this theme take into account properties of enclosed regions and allow for multiple curves or surfaces to be simultaneously represented however it is still unclear how to apply these techniques to images of narrow elongated structures such as blood vessels where intensity contrast may be low and reliable region statistics cannot be computed to address this problem we derive the gradient flows which maximize the rate of increase of flux of an appropriate vector field through a curve in 2d or a surface in 3d the key idea is to exploit the direction of the vector field along with its magnitude the calculations lead to a simple and elegant interpretation which is essentially parameter free and has the same form in both dimensions we illustrate its advantages with several level set based segmentations of 2d and 3d angiography images of blood vessels

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

filling in by joint interpolation of vector fields and gray levels a variational approach for filling in regions of missing data in digital images is introduced the approach is based on joint interpolation of the image gray levels and gradient isophotes directions smoothly extending in an automatic fashion the isophote lines into the holes of missing data this interpolation is computed by solving the variational problem via its gradient descent flow which leads to a set of coupled second order partial differential equations one for the gray levels and one for the gradient orientations the process underlying this approach can be considered as an interpretation of the gestaltist s principle of good continuation no limitations are imposed on the topology of the holes and all regions of missing data can be simultaneously processed even if they are surrounded by completely different structures applications of this technique include the restoration of old photographs and removal of superimposed text like dates subtitles or publicity examples of these applications are given we conclude the paper with a number of theoretical results on the proposed variational approach and its corresponding gradient descent flow

disocclusion a variational approach using level lines object recognition robot vision image and film restoration may require the ability to perform disocclusion we call disocclusion the recovery of occluded areas in a digital image by interpolation from their vicinity it is shown in this paper how disocclusion can be performed by means of the level lines structure which offers a reliable complete and contrast invariant representation of images level lines based disocclusion yields a solution that may have strong discontinuities the proposed method is compatible with kanizsa s 1979 amodal completion theory

one step extrapolation method for reverse time migration we have proposed a new method a one step extrapolation algorithm to solve the acoustic wave equation by introducing a square root operator the two way wave equation can be formulated as a first order partial differential equation in time which is similar to the one way wave equation to solve the new wave equation we used a stable explicit extrapolation method in the time direction and handled lateral velocity variations in the space and wavenumber domains unlike the conventional explicit finite difference schemes the new method does not suffer from numerical instability or numerical dispersion problems it can be used to design cost effective and high quality reverse time migration or modeling code

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

on computing the nucleolus of a balanced connected game the question of fairly allocating costs or cost savings among the participants of a joint enterprise is frequently answered by finding the nucleolus of a related cooperative game in connection with various routing sequencing and scheduling situations connected games often arise their common feature is that all the essential coalitions whose breaking up into subcoalitions is disadvantageous for its members are connected with respect to a fixed order of the players any player between two members of a coalition is also a member of the coalition in the literature several special classes of connected games are discussed that are shown to be balanced in this paper an algorithm is presented that computes the nucleolus of a general n person balanced connected game the input is the array of the explicitly given values of the n n 1 2 connected coalitions the algorithm generates a sequence of at most n n 1 2 payoff vectors starting from a special vertex of the nonempty core and ending in the nucleolus by exploiting the special structure of connected games the initial as well as the subsequent payoff vectors are obtained by evaluating some simple recursive formulas it is shown that the algorithm requires at most o n4 time and o n2 space a 5 person illustrative example is also given

fourier finite difference wave propagation we introduce a novel technique for seismic wave extrapolation in time the technique involves cascading a fourier transform operator and a finite difference operator to form a chain operator fourier finite differences ffd we derive the ffd operator from a pseudoanalytical solution of the acoustic wave equation two dimensional synthetic examples demonstrate that the ffd operator can have high accuracy and stability in complex velocity media applying the ffd method to the anisotropic case overcomes some disadvantages of other methods such as the coupling of qp waves and qsv waves the ffd method can be applied to enhance accuracy and stability of seismic imaging by reverse time migration

analysis of higher order finite difference schemes in 3 d reverse time migration the design and usefulness of practical algorithms for 3 d reverse time depth migration is examined while demonstrating data applications of the algorithm we evaluate quantitatively the accuracy of the finite difference operator from second order to eighth order for the scalar wave equation by comparing numerical and analytical solutions the results clearly show the advantage of using higher order finite difference schemes especially from second order to fourth order for space derivatives hence a finite difference method with the accuracy of fourth order in space and second order in time is applied to 3 d full scalar wave equations in reverse time migration considerable savings in cpu and memory are obtained by using larger horizontal grid spacings than the vertical spacings therefore we derive dispersion and stability conditions for such unequal grid spacing the 3 d reverse time migration of data from the hibernia field shows an image improvement over 2 d migrations despite the fact that much of the structure was considered to be approximately 2 d stable nondispersive finite difference methods of higher order can allow for tractable and efficient 3 d reverse time migration solutions

coupled equations for reverse time migration in transversely isotropic media reverse time migration rtm images reflectors by using time extrapolation modeling codes to synthesize source and receiver wavefields in the subsurface asymptotic analysis of wave propagation in transversely isotropic ti media yields a dispersion relation describing coupled p and sv wave modes this dispersion relation can be converted into a fourth order scalar partial differential equation pde increased computational efficiency can be achieved using equivalent coupled second order pdes analysis of the corresponding dispersion relations as matrix eigenvalue systems allows one to characterize all possible coupled linear second order systems equivalent to a given linear fourth order pde and to determine which ones yield optimally efficient finite difference implementations setting the shear velocity along the axis of symmetry to zero yields a simpler approximate ti wave equation that is more efficient to implement this simpler approximation however can become unstable for some plausible combinations of anisotropic parameters the same eigensystem analysis can be applied using finite vertical shear velocity to obtain solutions that avoid these instability problems

globally optimal regions and boundaries as minimum ratio weight cycles we describe a new form of energy functional for the modeling and identification of regions in images the energy is defined on the space of boundaries in the image domain and can incorporate very general combinations of modeling information both from the boundary intensity gradients etc and from the interior of the region texture homogeneity etc we describe two polynomial time digraph algorithms for finding the global minima of this energy one of the algorithms is completely general minimizing the functional for any choice of modeling information it runs in a few seconds on a 256 spl times 256 image the other algorithm applies to a subclass of functionals but has the advantage of being extremely parallelizable neither algorithm requires initialization

radial projection an efficient update rule for relaxation labeling relaxation labeling uses contextual information for finding consistent labelings of graphs although relaxation labeling is parallel and iterative the complexity of updating with standard rules is too costly for practical implementation a description is given of a computationally more efficient updating rule that utilizes radial projection instead of normal projection to avoid the complexities incurred by previous update rules when boundaries to the labeling space are encountered this reduction in complexity is achieved by first restricting support vectors to the positive quadrant and then using radial projection onto the constraint instead of normal projection crucial order information is conserved through smooth convergence towards the optimum and a rate of convergence proportional to the magnitudes of the support functions

recursive integral time extrapolation methods for waves a comparative review abstractwe compared several families of algorithms for recursive integral time extrapolation rite algorithms for waves in isotropic and anisotropic media these methods allow simulating accurate wave extrapolation with little numerical dispersion even when using larger time steps than are usually possible for conventional finite difference methods these various rite algorithms all share the use of mixed space wavenumber domain operators derived from fourier integral solutions of single mode wave equations we evaluated a taxonomy for rite methods based on how they approximated the influence of medium heterogeneity one family of methods uses mixed domain series expansions to provide accurate approximations to heterogeneous extrapolators even for large time steps we compared several methods for deriving coefficients for such series approximations another family of methods uses interpolation between different homogeneous extrapolations to approximate heterogeneous time extrapolation such methods can b

winding number constrained contour detection salient contour detection can benefit from the integration of both contour cues and region cues however this task is difficult due to different nature of region representations and contour representations to solve this problem this paper proposes an energy minimization framework based on winding number constraints in this framework both region cues such as color texture homogeneity and contour cues such as local contrast and continuity are represented in a joint objective function which has both region and contour labels the key problem is how to design constraints that ensure the topological consistency of the two kinds of labels our technique is based on the topological concept of winding number using a fast method for winding number computation a small number of linear constraints are derived to ensure label consistency our method is instantiated by ratio based energy functions by successfully integrating both region and contour cues our method shows advantages over competitive methods our method is extended to incorporate user interaction which leads to further improvements

traveling salesman games in this paper we discuss the problem of how to divide the total cost of a round trip along several institutes among the institutes visited we introduce two types of cooperative games fixed route traveling salesman games and traveling salesman games as a tool to attack this problem under very mild conditions we prove that fixed route traveling salesman games have non empty cores if the fixed route is a solution of the classical traveling salesman problem core elements provide us with fair cost allocations a traveling salesman game may have an empty core even if the cost matrix satisfies the triangle inequality in this paper we introduce a class of matrices defining ts games with non empty cores

globally optimal finite difference schemes based on least squares abstractspatial finite difference fd coefficients are usually determined by the taylor series expansion te or optimization methods the former can provide high accuracy on a smaller wavenumber or frequency zone and the latter can give moderate accuracy on a larger zone present optimization methods applied to calculate fd coefficients are generally gradient like or global optimization like algorithms and thus iterations are involved they are more computationally expensive and sometimes the global solution may not be found i examined second order spatial derivatives and computed the optimized spatial fd coefficients over the given wavenumber range using the least squares ls method the results indicated that the fd accuracy increased with increasing operator length and decreasing wavenumber range therefore for the given error and operator length globally optimal spatial fd coefficients can be easily obtained some optimal fd coefficients were given i developed schemes to obtain optimized ls

third order symplectic integration method with inverse time dispersion transform for long term simulation the symplectic integration method is popular in high accuracy numerical simulations when discretizing temporal derivatives however it still suffers from time dispersion error when the temporal interval is coarse especially for long term simulations and large scale models we employ the inverse time dispersion transform itdt to the third order symplectic integration method to reduce the time dispersion error first we adopt the pseudospectral algorithm for the spatial discretization and the third order symplectic integration method for the temporal discretization then we apply the itdt to eliminate time dispersion error from the synthetic data as a post processing method the itdt can be easily cascaded in traditional numerical simulations we implement the itdt in one typical exiting third order symplectic scheme and compare its performances with the performances of the conventional second order scheme and the rapid expansion method theoretical analyses and numerical experiments show that the itdt can significantly reduce the time dispersion error especially for long travel times the implementation of the itdt requires some additional computations on correcting the time dispersion error but it allows us to use the maximum temporal interval under stability conditions thus its final computational efficiency would be higher than that of the traditional symplectic integration method for long term simulations with the aid of the itdt we can obtain much more accurate simulation results but with a lower computational cost we use the inverse time dispersion transform to the 3rd order symplectic integration it can greatly reduce the time dispersion error especially for long travel times it is superior to the conventional 2nd and high order finite difference methods it allows the maximum temporal interval under stability conditions thus is efficient we can obtain more accurate simulation results but with a lower computational cost

pseudo analytical finite difference elastic wave extrapolation based on the k space method abstractcost effective elastic wave modeling is the key to practical elastic reverse time migration and full waveform inversion implementations we have developed an efficient elastic pseudo analytical finite difference pafd scheme for elastic wave extrapolation the elastic pafd scheme is based on a modified pseudo spectral method k space method in which a pseudo analytical operator is used to ensure the high accuracy of elastic wave extrapolation however the k space method is motivated for a pure wave mode and thus its application in coupled first order elastic wave equations may cause the elastic pseudo analytical operators to suffer from crosstalk between the p and s wavefields the approaches presented attempt to overcome these shortcomings by introducing two improvements to achieve the goal this is done first by performing a predictor corrector strategy in first order elastic wave equations to eliminate those errors during wave extrapolation considering the massive computational cost in

contour completion without region segmentation contour completion plays an important role in visual perception where the goal is to group fragmented low level edge elements into perceptually coherent and salient contours most existing methods for contour completion have focused on pixelwise detection accuracy in contrast fewer methods have addressed the global contour closure effect despite psychological evidences for its importance this paper proposes a purely contour based higher order crf model to achieve contour closure through local connectedness approximation this leads to a simplified problem structure where our higher order inference problem can be transformed into an integer linear program and be solved efficiently compared with the methods based on the same bottom up edge detector our method achieves a superior contour grouping ability measured by rand index a comparable precision recall performance and more visually pleasing results our results suggest that contour closure can be effectively achieved in contour domain in contrast to a popular view that segmentation is essential for this purpose

acoustic and elastic modeling by optimal time space domain staggered grid finite difference schemes abstractstaggered grid finite difference sfd methods are widely used in modeling seismic wave propagation and the coefficients of finite difference fd operators can be estimated by minimizing dispersion errors using taylor series expansion te or optimization we developed novel optimal time space domain sfd schemes for acoustic and elastic wave equation modeling in our schemes a fourth order multiextreme value objective function with respect to fd coefficients was involved to yield the globally optimal solution with low computational cost we first used variable substitution to turn our optimization problem into a quadratic convex one and then used least squares ls to derive the optimal sfd coefficients by minimizing the relative error of time space domain dispersion relations over a given frequency range to ensure the robustness of our schemes a constraint condition was imposed that the dispersion error at each frequency point did not exceed a given threshold moreover the hybrid absorbin