3d topology preserving flows for viewpoint based cortical unfolding we present a variational method for unfolding of the cortex based on a user chosen point of view as an alternative to more traditional global flattening methods which incur more distortion around the region of interest our approach involves three novel contributions the first is an energy function and its corresponding gradient flow to measure the average visibility of a region of interest of a surface with respect to a given viewpoint the second is an additional energy function and flow designed to preserve the 3d topology of the evolving surface the third is a method that dramatically improves the computational speed of the 3d topology preservation approach by creating a tree structure of the 3d surface and using a recursion technique experiments results show that the proposed approach can successfully unfold highly convoluted surfaces such as the cortex while preserving their topology during the evolution

projection methods coupled to level set interface techniques in this work we consider hydrodynamic problems with cold flame propagation by merging a second order projection method for viscous navier stokes equations with modern techniques for computing the motion of interfaces propagating with curvature dependent speeds this is part of our efforts to try and approximate the solution of a simplified model of turbulent combustion results are given for a simple model of a flame burning in driven cavities and shear layers

how general is the vale maurelli simulation approach the vale maurelli vm approach to generating non normal multivariate data involves the use of fleishman polynomials applied to an underlying gaussian random vector this method has been extensively used in monte carlo studies during the last three decades to investigate the finite sample performance of estimators under non gaussian conditions the validity of conclusions drawn from these studies clearly depends on the range of distributions obtainable with the vm method we deduce the distribution and the copula for a vector generated by a generalized vm transformation and show that it is fundamentally linked to the underlying gaussian distribution and copula in the process we derive the distribution of the fleishman polynomial in full generality while data generated with the vm approach appears to be highly non normal its truly multivariate properties are close to the gaussian case a monte carlo study illustrates that generating data with a different copula than that implied by the vm approach severely weakens the performance of normal theory based ml estimates

a remark on computing distance functions we propose a new method for the reconstruction of the signed distance function in the context of level set methods the new method is a modification of the algorithm which makes use of the pde equation for the distance function introduced by m sussman p smereka and s osher 1994 j comput phys 119 146 it is based mainly on the use of a truly upwind discretization near the interface comparison with the previous algorithm shows a definite improvement when used with a first order upwind scheme the method provides first order accuracy for the signed distance function in the whole computational domain and second order accuracy in the location of the interface a second order version of the method is also presented

self repelling snakes for topology preserving segmentation models the implicit framework of the level set method has several advantages when tracking propagating fronts indeed the evolving contour is embedded in a higher dimensional level set function and its evolution can be phrased in terms of a eulerian formulation the ability of this intrinsic method to handle topological changes merging and breaking makes it useful in a wide range of applications fluid mechanics computer vision and particularly in image segmentation the main subject of this paper nevertheless in some applications this topological flexibility turns out to be undesirable for instance when the shape to be detected has a known topology or when the resulting shape must be homeomorphic to the initial one the necessity of designing topology preserving processes arises in medical imaging for example in the human cortex reconstruction it is known that the human cortex has a spherical topology so throughout the reconstruction process this topological feature must be preserved therefore we propose in this paper a segmentation model based on an implicit level set formulation and on the geodesic active contours in which a topological constraint is enforced

parametric surface diffeomorphometry for low dimensional embeddings of dense segmentations and imagery in the field of computational anatomy biological form including our focus neuroanatomy is studied quantitatively through the action of the diffeomorphism group on example anatomies a technique called diffeomorphometry here we design an algorithm within this framework to pass from dense objects common in neuromaging studies binary segmentations structural images to a sparse representation defined on the surface boundaries of anatomical structures and embedded into the low dimensional coordinates of a parametric model our main new contribution is to introduce an expanded group action to simultaneously deform surfaces through direct mapping of points as well as images through functional composition with the inverse this allows us to index the diffeomorphisms with respect to two dimensional surface geometries like subcortical gray matter structures but explicitly map onto cost functions determined by noisy 3 dimensional measurements we consider models generated from empirical covariance of training data as well as bandlimited laplace beltrami eigenfunction models when no such data is available we show applications to noisy or anomalous segmentations and other typical problems in neuroimaging studies we reproduce statistical results detecting changes in alzheimer s disease despite dimensionality reduction lastly we apply our algorithm to the common problem of segmenting subcortical structures from t1 mr 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

fundamental properties of infinite trees infinite trees naturally arise in the formalization and the study of the semantics of programming languages this paper investigates some of their combinatorial and algebraic properties that are especially relevant to semantics

simulating multivariate nonnormal distributions a method for generating multivariate nonnormal distributions with specified intercorrelations and marginal means variances skews and kurtoses is proposed as an example the method is applied to the generation of simulated scores on three psychological tests administered to a single group of individuals

a moving grid framework for geometric deformable models geometric deformable models based on the level set method have become very popular in the last decade to overcome an inherent limitation in accuracy while maintaining computational efficiency adaptive grid techniques using local grid refinement have been developed for use with these models this strategy however requires a very complex data structure yields large numbers of contour points and is inconsistent with the implementation of topology preserving geometric deformable models tgdms in this paper we investigate the use of an alternative adaptive grid technique called the moving grid method with geometric deformable models in addition to the development of a consistent moving grid geometric deformable model framework our main contributions include the introduction of a new grid nondegeneracy constraint the design of a new grid adaptation criterion and the development of novel numerical methods and an efficient implementation scheme the overall method is simpler to implement than using grid refinement requiring no large complex hierarchical data structures it also offers an extra benefit of automatically reducing the number of contour vertices in the final results after presenting the algorithm we demonstrate its performance using both simulated and real images

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

a modified chan vese model and its theoretical proof we present a modified chan vese functional and give its theoretical proof by using the geometric heat flow method to all the euler lagrange equations a system of evolution equations in level set formulation is derived we study the existence of solution to this system by schauder fixed point theorem and the implicit function theorem in banach space this variational formulation can detect interior and exterior boundaries of desired object s in color images

a level set method for simulating island coarsening modeling of microstructural evolution during thin film deposition requires a knowledge of several key activation energies surface diffusion island edge atom diffusion adatom migration over descending step edges etc these and other parameters must be known as a function of crystal orientation in order to generate values for these parameters we have developed a numerical simulation in tandem with physical experiments by tuning the simulation to the results from experiments we can extract and verify approximate values for these parameters the numerical method we use is based upon the level set method our model is a continuum model in directions parallel to the crystal facet and resolves each discrete atomic layer in the normal direction the model includes surface diffusion step edge dynamics and attachment detachment rates all of which may depend upon the local geometry of the step edge the velocity field for advancing the island edges in the level set framework is generated by computing the equilibrium adatom density on the flat terraces resulting in laplace s equation with mixed boundary conditions at the step edges we have turned to the finite element method for solving this equation which results in very good agreement with analytically known solutions and with experiment

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

transport equation with boundary conditions for free surface localization during the filling stage of an injection moulding process which consists in casting a melt polymer in order to manufacture plastic pieces the free interface between polymer and air has to be precisely described we set this interface as a zero level set of an unknown function this function satisfies a transport equation with boundary conditions where the velocity field has few regularity properties

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

evolution implementation and application of level set and fast marching methods for advancing fronts abstract a variety of numerical techniques are available for tracking moving interfaces in this review we concentrate on techniques that result from the link between the partial differential equations that describe moving interfaces and numerical schemes designed for approximating the solutions to hyperbolic conservation laws this link gives rise to computational techniques for tracking moving interfaces in two and three space dimensions under complex speed laws we discuss the evolution of these techniques the fundamental numerical approximations involved implementation details and applications in particular we review some work on three aspects of materials sciences semiconductor process simulations seismic processing and optimal structural topology design

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

a level set model for image classification we present a supervised classification model based on a variational approach this model is devoted to find an optimal partition composed of homogeneous classes with regular interfaces the originality of the proposed approach concerns the definition of a partition by the use of level sets each set of regions and boundaries associated to a class is defined by a unique level set function we use as many level sets as different classes and all these level sets are moving together thanks to forces which interact in order to get an optimal partition we show how these forces can be defined through the minimization of a unique fonctional the coupled partial differential equations pde related to the minimization of the functional are considered through a dynamical scheme given an initial interface set zero level set the different terms of the pde s are governing the motion of interfaces such that at convergence we get an optimal partition as defined above each interface is guided by internal forces regularity of the interface and external ones data term no vacuum no regions overlapping several experiments were conducted on both synthetic and real images

geometric active deformable models in shape modeling this paper analyzes the problem of shape modeling using the principle of active geometric deformable models while the basic modeling technique already exists in the literature we highlight many of its drawbacks and discuss their source and steps to overcome them we propose a new stopping criterion to address the stopping problem we also propose to apply a level set algorithm to implement the active geometric deformable models thereby handling topology changes automatically to alleviate the numerical problems associated with the implementation of the level set algorithm we propose a new adaptive multigrid narrow band algorithm all the proposed new changes have been illustrated with experiments with synthetic images and medical images

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

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

a lagrangian particle level set method we present a novel particle level set method for capturing interfaces the level set equation is solved in a lagrangian frame using particles that carry the level set information a key aspect of the method involves a consistent remeshing procedure for the regularization of the particle locations when the particle map gets distorted by the advection field the lagrangian description of the level set method is inherently adaptive and exact in the case of solid body motions the efficiency and accuracy of the method is demonstrated in several benchmark problems in two and three dimensions involving pure advection and curvature induced motion of the interface the simplicity of the particle description is shown to be well suited for real time simulations of surfaces involving cutting and reconnection as in virtual surgery environments

nonlinear multiscale analysis of three dimensional echocardiographic sequences introduces a new model for multiscale analysis of space time echocardiographic sequences the proposed nonlinear partial differential equation representing the multiscale analysis filters the sequence while keeping the space time coherent structures it combines the ideas of regularized perona malik anisotropic diffusion and the galilean invariant movie multiscale analysis of alvarez et al arch rat mech anal vol 123 p 200 57 1993 a numerical method for solving the proposed partial differential equation is suggested and its stability is shown computational results on synthesized and real sequences are provided a qualitative and quantitative evaluation of the accuracy of the method is presented

curve evolution implementation of the mumford shah functional for image segmentation denoising interpolation and magnification we first address the problem of simultaneous image segmentation and smoothing by approaching the mumford shah 1989 paradigm from a curve evolution perspective in particular we let a set of deformable contours define the boundaries between regions in an image where we model the data via piecewise smooth functions and employ a gradient flow to evolve these contours each gradient step involves solving an optimal estimation problem for the data within each region connecting curve evolution and the mumford shah functional with the theory of boundary value stochastic processes the resulting active contour model offers a tractable implementation of the original mumford shah model i e without resorting to elliptic approximations which have traditionally been favored for greater ease in implementation to simultaneously segment and smoothly reconstruct the data within a given image in a coupled manner various implementations of this algorithm are introduced to increase its speed of convergence we also outline a hierarchical implementation of this algorithm to handle important image features such as triple points and other multiple junctions next by generalizing the data fidelity term of the original mumford shah functional to incorporate a spatially varying penalty we extend our method to problems in which data quality varies across the image and to images in which sets of pixel measurements are missing this more general model leads us to a novel pde based approach for simultaneous image magnification segmentation and smoothing thereby extending the traditional applications of the mumford shah functional which only considers simultaneous segmentation and smoothing

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

on the laplace beltrami operator and brain surface flattening in this paper using certain conformal mappings from uniformization theory the authors give an explicit method for flattening the brain surface in a way which preserves angles from a triangulated surface representation of the cortex the authors indicate how the procedure may be implemented using finite elements further they show how the geometry of the brain surface may be studied using this approach

implementing continuous scale morphology via curve evolution abstract a new approach to digital implementation of continuous scale mathematical morphology is presented the approach is based on discretization of evolution equations associated with continuous multiscale morphological operations those equations and their corresponding numerical implementation can be derived either directly from mathematical morphology definitions or from curve evolution theory the advantages of the proposed approach over the classical discrete morphology are demonstrated

marangoni driven singularities via mean curvature flow in this work it is demonstrated that the existence and topology of the recently observed interfacial singularities driven by marangoni effects can be deduced using mean curvature flow theory extended to account for variations of interfacial tension this suggests that some of the physical mechanisms underlying the formation of these interfacial singularities may originate from be modeled by the surface tension flow the proposed approach also determines the conditions on the surface tension material behavior under which singularities may form as well as the asymptotic behavior near singularities besides the application to marangoni driven singularities the offered generalization of the mean curvature flow is also of independent geometric interest

equivalences and transformations of regular systems applications to recursive program schemes and grammars abstract this work presents a unified theory of recursive program schemes context free grammars grammars on arbitrary algebraic structures and in fact recursive definitions of all kind by means of regular systems the equivalences of regular systems associated with either all their solutions or their least solutions in all domains of appropriate type satisfying a set of algebraic laws expressed by equations are systematically investigated and characterized in some cases in terms of system transformations by folding unfolding and rewriting according to the equational algebraic laws grammars are better characterized in terms of polynomial systems which are regular systems involving the operation of set union and the same questions are raised for them we also examine conditions insuring the uniqueness of the solution of a regular or of a polynomial system this theory applies to grammars of many kinds which generate trees graphs etc we formulate some classical transformations of context free grammars in terms of correct transformations which only use folding unfolding and algebraic laws and we immediately obtain their correctness

a stable manifold theorem for the curve shortening equation on presente une famille de solutions homothetiques de l equation pour une courbe planaire x kn et on demontre l existence de varietes non lineaires stables et instables autour de telles solutions

stationary self propagating fronts in potential flow abstract we analyze the problem of stationary self propagating fronts in potential flow the issues of local existence and uniqueness for solutions of the ode describing stationary fronts multiplicity of solutions and linearized stability of a stationary front as a solution of the hyperbolic evolution equation are addressed the results are illustrated in the case of the dual source system which is a simple model of a combustion system in which local extinction may arise model extensions for combustion applications are presented

a general structural equation model with dichotomous ordered categorical and continuous latent variable indicators a structural equation model is proposed with a generalized measurement part allowing for dichotomous and ordered categorical variables indicators in addition to continuous ones a computationally feasible three stage estimator is proposed for any combination of observed variable types this approach provides large sample chi square tests of fit and standard errors of estimates for situations not previously covered two multiple indicator modeling examples are given one is a simultaneous analysis of two groups with a structural equation model underlying skewed likert variables the second is a longitudinal model with a structural model for multivariate probit regressions

repeatedly smooting discrete scale space evolution and dominant point detection in this paper the fourier analysis is used to derive the properties of an evolving curve an arbitrary kernel repeatedly smoothing akrs evolution of a curve is then introduced it is shown that when the repeated number is large the akrs evolution of a curve is an approximately discrete implementation of the scale based evolution of this curve in the euclidean space as a special case an exponential repeatedly smoothing is proposed to implement the scale based evolution it is shown that in addition to its simple implementation and its desired approximation to a gaussian kernel an exponential function is a function that when it is selected as a repeatedly smoothing kernel the motion both magnitude and direction of a point on a curve from the i 1 th to ith instant is equal to the curvature of the ith smoothed curve at this point finally a perimeter controlled evolution method is proposed to extract dominant points it is shown experimentally that the proposed method is robust to noise object rotation and object changes in sizes

dream 2 s deformable regions driven by an eulerian accurate minimization method for image and video segmentation this paper deals with image and video segmentation using active contours we propose a general form for the energy functional related to region based active contours we compute the associated evolution equation using shape derivation tools and accounting for the evolving region based terms then we apply this general framework to compute the evolution equation from functionals that include various statistical measures of homogeneity for the region to be segmented experimental results show that the determinant of the covariance matrix appears to be a very relevant tool for segmentation of homogeneous color regions as an example it has been successfully applied to face segmentation in real video sequences

algorithms for computing motion by mean curvature we propose a finite element algorithm for computing the motion of a surface moving by mean curvature the algorithm uses the level set formulation so that changes in topology of the surface can be accommodated stability is deduced by showing that the discrete solutions satisfy both l infty and w 1 1 bounds existence of discrete solutions and connections with brakke flows are established some numerical examples and application to related problems such as the phase field equations are also presented

global regularizing flows with topology preservation for active contours and polygons active contour and active polygon models have been used widely for image segmentation in some applications the topology of the object s to be detected from an image is known a priori despite a complex unknown geometry and it is important that the active contour or polygon maintain the desired topology in this work we construct a novel geometric flow that can be added to image based evolutions of active contours and polygons in order to preserve the topology of the initial contour or polygon we emphasize that unlike other methods for topology preservation the proposed geometric flow continually adjusts the geometry of the original evolution in a gradual and graceful manner so as to prevent a topology change long before the curve or polygon becomes close to topology change the flow also serves as a global regularity term for the evolving contour and has smoothness properties similar to curvature flow these properties of gradually adjusting the original flow and global regularization prevent geometrical inaccuracies common with simple discrete topology preservation schemes the proposed topology preserving geometric flow is the gradient flow arising from an energy that is based on electrostatic principles the evolution of a single point on the contour depends on all other points of the contour which is different from traditional curve evolutions in the computer vision literature

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

a topology preserving level set method for shape optimization the classical level set method which represents the boundary of the unknown geometry as the zero level set of a function has been shown to be very effective in solving shape optimization problems the present work addresses the issue of using a level set representation when there are simple geometrical and topological constraints we propose a logarithmic barrier penalty which acts to enforce the constraints leading to an approximate solution to shape design problems

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

a problem with discretizing vale maurelli in simulation studies previous influential simulation studies investigate the effect of underlying non normality in ordinal data using the vale maurelli vm simulation method we show that discretized data stemming from the vm method with a prescribed target covariance matrix are usually numerically equal to data stemming from discretizing a multivariate normal vector this normal vector has however a different covariance matrix than the target it follows that these simulation studies have in fact studied data stemming from normal data with a possibly misspecified covariance structure this observation affects the interpretation of previous simulation studies

the vector distance functions we present a novel method for representing and evolving objects of arbitrary dimension the method called the vector distance function vdf method uses the vector that connects any point in space to its closest point on the object it can deal with smooth manifolds with and without boundaries and with shapes of different dimensions it can be used to evolve such objects according to a variety of motions including mean curvature if discontinuous velocity fields are allowed the dimension of the objects can change the evolution method that we propose guarantees that we stay in the class of vdf s and therefore that the intrinsic properties of the underlying shapes such as their dimension curvatures can be read off easily from the vdf and its spatial derivatives at each time instant the main disadvantage of the method is its redundancy the size of the representation is always that of the ambient space even though the object we are representing may be of a much lower dimension this disadvantage is also one of its strengths since it buys us flexibility

nonparametric statistical snake based on the minimum stochastic complexity we propose a nonparametric statistical snake technique that is based on the minimization of the stochastic complexity minimum description length principle the probability distributions of the gray levels in the different regions of the image are described with step functions with parameters that are estimated the segmentation is thus obtained by minimizing a criterion that does not include any parameter to be tuned by the user we illustrate the robustness of this technique on various types of images with level set and polygonal contour models the efficiency of this approach is also analyzed in comparison with parametric statistical techniques

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

a short time existence uniqueness result for a nonlocal topology preserving segmentation model motivated by a prior applied work of vese and the second author dedicated to segmentation under topological constraints we derive a slightly modified model phrased as a functional minimization problem and propose to study it from a theoretical viewpoint the mathematical model leads to a second order nonlinear pde with a singularity at nabla u 0 and containing a nonlocal term a suitable setting is thus the one of the viscosity solution theory and in this framework we establish a short time existence uniqueness result as well as a lipschitz regularity result for the solution

type inference with recursive types syntax and semantics abstract in this paper we study type inference systems for calculus with a recursion operator over types the main syntactical properties notably the existence of principal type schemes are proved to hold when recursive types are viewed as finite notations for infinite regular type expressions representing their infinite unfoldings exploiting the approximation structure of a model for the untyped language of terms types are interpreted as limits of sequences of their approximations we show that the interpretation is essentially unique and that two types have equal interpretation if and only if their infinite unfoldings are identical finally a completeness theorem is proved to hold w r t the specific model we consider for a natural infinitary extension of the type inference system

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

a topology preserving level set method for geometric deformable models active contour and surface models also known as deformable models are powerful image segmentation techniques geometric deformable models implemented using level set methods have advantages over parametric models due to their intrinsic behavior parameterization independence and ease of implementation however a long claimed advantage of geometric deformable models the ability to automatically handle topology changes turns out to be a liability in applications where the object to be segmented has a known topology that must be preserved we present a new class of geometric deformable models designed using a novel topology preserving level set method which achieves topology preservation by applying the simple point concept from digital topology these new models maintain the other advantages of standard geometric deformable models including subpixel accuracy and production of nonintersecting curves or surfaces moreover since the topology preserving constraint is enforced efficiently through local computations the resulting algorithm incurs only nominal computational overhead over standard geometric deformable models several experiments on simulated and real data are provided to demonstrate the performance of this new deformable model algorithm

goodman and kruskal s gamma coefficient for ordinalized bivariate normal distributions we consider a bivariate normal distribution with linear correlation formula see text whose random components are discretized according to two assigned sets of thresholds on the resulting bivariate ordinal random variable one can compute goodman and kruskal s gamma coefficient formula see text which is a common measure of ordinal association given the known analytical monotonic relationship between pearson s formula see text and kendall s rank correlation formula see text for the bivariate normal distribution and since in the continuous case kendall s formula see text coincides with goodman and kruskal s formula see text the change of this association measure before and after discretization is worth studying we consider several experimental settings obtained by varying the two sets of thresholds or equivalently the marginal distributions of the final ordinal variables this study confirming previous findings shows how the gamma coefficient is always larger in absolute value than kendall s rank correlation this discrepancy lessens when the number of categories increases or given the same number of categories when using equally probable categories based on these results a proposal is suggested to build a bivariate ordinal variable with assigned margins and goodman and kruskal s formula see text by ordinalizing a bivariate normal distribution illustrative examples employing artificial and real data are provided

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

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

variational principles surface evolution pdes level set methods and the stereo problem we present a novel geometric approach for solving the stereo problem for an arbitrary number of images spl ges 2 it is based upon the definition of a variational principle that must be satisfied by the surfaces of the objects in the scene and their images the euler lagrange equations that are deduced from the variational principle provide a set of partial differential equations pdes that are used to deform an initial set of surfaces which then move toward the objects to be detected the level set implementation of these pdes potentially provides an efficient and robust way of achieving the surface evolution and to deal automatically with changes in the surface topology during the deformations i e to deal with multiple objects results of an implementation of our theory also dealing with occlusion and visibility are presented on synthetic and real images

a high order projection method for tracking fluid interfaces in variable density incompressible flows we present a numerical method for computing solutions of the incompressible euler or navier stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties the method is based on a second order projection method for variable density flows using an approximate projection formulation the boundary between the fluids is tracked with a second order volume of fluid interface tracking algorithm we present results for viscious rayleigh taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm we also present computational results for the rayleigh taylor instability in air helium and for bubbles and drops in an air water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts

algorithms for determining relative star height and star height eggan l963 introduced the notion of the star height to each regular expression which is a nonnegative integer denoting the nestedness of star operators in this expression the star height of a regular language is the minimum of the star height of regular expressions denoting this language we remark here that to each regular language there exist generally infinitely many regular expressions denoting this language eggan l963 showed that for each nonnegative integer k there exists a regular language of star height k and posed the problem of determining the star height of any regular language dejean and schutzenberger 1966 showed that for each nonnegative integer k there exists a regular language of star height k over the two letter alphabet mcnaughton 1967 presented an algorithm for determining the loop complexity i e the star height of any regular language whose syntactic monoid is a group cohen 1970 1971 and cohen and brzozowski 1970 investigated many properties of star height some of which provide algorithms for determining the star height of any regular language of certain reset free type hashiguchi and honda 1979 presented an algorithm for determining the star height of any reset free language and any reset language hashiguchi 1982b presented an algorithm for deciding whether or not an arbitrary language is of star height one to obtain this result it uses the limitedness theorem on finite automata with distance functions in short d automata in hashiguci 1982a 1983

topological well composedness and glamorous glue a digital gluing algorithm for topologically constrained front propagation we propose a new approach to front propagation algorithms based on a topological variant of well composedness which contrasts with previous methods based on simple point detection this provides for a theoretical justification based on the digital jordan separation theorem for digitally gluing evolved well composed objects separated by well composed curves or surfaces additionally our framework can be extended to more relaxed topologically constrained algorithms based on multisimple points for both methods this framework has the additional benefit of obviating the requirement for both a user specified connectivity and a topologically consistent marching cubes squares algorithm in meshing the resulting segmentation

level sets of viscosity solutions some applications to fronts and rendez vous problems the authors treat some applications of hamilton jacobi equations to the study of a flame front propagation model and the rendez vows problem the solution of both problems requires the determination of the level sets of the viscosity solution for the corresponding equation in the flame front propagation model described here it is assumed that the evolution is driven by a vector field satisfying a transversality condition at time t 0 the evolution in the normal direction with variable velocity c x geq 0 is considered as a special case this approach is constructive permitting the numerical solution of such problems

algorithms for determining relative inclusion star height and inclusion abstract let c r 1 r m be a finite class of regular languages over a finite alphabet let b1 bm be an alphabet and be the substitution from into such that bi ri for all i let r 10 r 20 be two regular languages the relative inclusion star height h r r 10 r 20 c of r 10 r 20 w r t c is the minimum star height of regular languages l such that r10 l r20 this paper proves the existence of an algorithm for determining relative inclusion star height

evolution of plane curves driven by a nonlinear function of curvature and anisotropy in this paper we study evolution of plane curves satisfying a geometric equation v beta k nu where v is the normal velocity and k and nu are the curvature and tangential angle of a plane curve gamma we follow the direct approach and we analyze the so called intrinsic heat equation governing the motion of plane curves obeying such a geometric equation the intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional alpha we show how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities from an analytical point of view we present some new results on short time existence of a regular family of evolving curves in the degenerate case when beta k nu gamma nu k m 0 m le 2 and the governing system of equations includes a nontrivial tangential velocity functional

shapes shocks and deformations i the components of two dimensional shape and the reaction diffusion space we undertake to develop a general theory of two dimensional shape by elucidating several principles which any such theory should meet the principles are organized around two basic intuitions first if a boundary were changed only slightly then in general its shape would change only slightly this leads us to propose an operational theory of shape based on incremental contour deformations the second intuition is that not all contours are shapes but rather only those that can enclose physical material a theory of contour deformation is derived from these principles based on abstract conservation principles and hamilton jacobi theory these principles are based on the work of sethian 1985a c the osher sethian 1988 level set formulation the classical shock theory of lax 1971 1973 as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of gage hamilton grayson 1986 1989 the result is a characterization of the computational elements of shape deformations parts bends and seeds which show where to place the components of a shape the theory unifies many of the diverse aspects of shapes and leads to a space of shapes the reaction diffusion space which places shapes within a neighborhood of similar ones such similarity relationships underlie descriptions suitable for recognition

limited information estimation and testing of discretized multivariate normal structural models discretized multivariate normal structural models are often estimated using multistage estimation procedures the asymptotic properties of parameter estimates standard errors and tests of structural restrictions on thresholds and polychoric correlations are well known it was not clear how to assess the overall discrepancy between the contingency table and the model for these estimators it is shown that the overall discrepancy can be decomposed into a distributional discrepancy and a structural discrepancy a test of the overall model specification is proposed as well as a test of the distributional specification i e discretized multivariate normality also the small sample performance of overall distributional and structural tests as well as of parameter estimates and standard errors is investigated under conditions of correct model specification and also under mild structural and or distributional misspecification it is found that relatively small samples are needed for parameter estimates standard errors and structural tests larger samples are needed for the distributional and overall tests furthermore parameter estimates standard errors and structural tests are surprisingly robust to distributional misspecification

what is the best causal scale space for three dimensional images we study the unique affine invariant morphological scale space in three dimensions we discuss its properties and show that it improves the dynamic shape model we explain the algorithms and display the first numerical experiments

new possibilities with sobolev active contours recently the sobolev metric was introduced to define gradient flows of various geometric active contour energies it was shown that the sobolev metric outperforms the traditional metric for the same energy in many cases such as for tracking where the coarse scale changes of the contour are important some interesting properties of sobolev gradient flows include that they stabilize certain unstable traditional flows and the order of the evolution pdes are reduced when compared with traditional gradient flows of the same energies in this paper we explore new possibilities for active contours made possible by sobolev metrics the sobolev method allows one to implement new energy based active contour models that were not otherwise considered because the traditional minimizing method render them ill posed or numerically infeasible in particular we exploit the stabilizing and the order reducing properties of sobolev gradients to implement the gradient descent of these new energies we give examples of this class of energies which include some simple geometric priors and new edge based energies we also show that these energies can be quite useful for segmentation and tracking we also show that the gradient flows using the traditional metric are either ill posed or numerically difficult to implement and then show that the flows can be implemented in a stable and numerically feasible manner using the sobolev gradient

3 d traveltime computation using the fast marching method we present a fast algorithm for solving the eikonal equation in three dimensions based on the fast marching method the algorithm is of the order o n log n where n is the total number of grid points in the computational domain the algorithm can be used in any orthogonal coordinate system and globally constructs the solution to the eikonal equation for each point in the coordinate domain the method is unconditionally stable and constructs solutions consistent with the exact solution for arbitrarily large gradient jumps in velocity in addition the method resolves any overturning propagation wavefronts we begin with the mathematical foundation for solving the eikonal equation using the fast marching method and follow with the numerical details we then show examples of traveltime propagation through the seg eage salt model using point source and planewave initial conditions and analyze the error in constant velocity media the algorithm allows for any shape of the initial wavefront while a point source is the most commonly used initial condition initial plane waves can be used for controlled illumination or for downward continuation of the traveltime field from one depth to another or from a topographic depth surface to another the algorithm presented here is designed for computing first arrival traveltimes nonetheless since it exploits the fast marching method for solving the eikonal equation we believe it is the fastest of all possible consistent schemes to compute first arrivals

the solutions of two star height problems for regular trees abstract regular trees can be defined by two types of rational expressions for these two types we solve the star height problem i e we show how to construct a rational expression of minimal star height from the minimal graph of the given tree i e the analogue of the minimal deterministic automation for regular languages in one case the minimal starheight is the rank in the sense of eggan of the minimal graph there corresponds a characterization of the star height of a prefix free regular language w r t rational expressions of a special kind called deterministic as the rank of its minimal deterministic automaton considered as a graph

global minimum for active contour models a minimal path approach a new boundary detection approach for shape modeling is presented it detects the global minimum of an active contour model s energy between two end points initialization is made easier and the curve is not trapped at a local minimum by spurious edges we modify the snake energy by including the internal regularization term in the external potential term our method is based on finding a path of minimal length in a riemannian metric we then make use of a new efficient numerical method to find this shortest path it is shown that the proposed energy though based only on a potential integrated along the curve imposes a regularization effect like snakes we explore the relation between the maximum curvature along the resulting contour and the potential generated from the image the method is capable to close contours given only one point on the objects boundary by using a topology based saddle search routine we show examples of our method applied to real aerial and medical images

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

numerical schemes for the hamilton jacobi and level set equations on triangulated domains borrowing from techniques developed for conservation law equations numerical schemes which discretize the hamilton jacobi h j level set and eikonal equations on triangulated domains are presented the first scheme is a provably monotone discretization for the h j equations unfortunately the basic scheme lacks lipschitz continuity of the numerical hamiltonian by employing a virtual edge flipping technique local lipschitz continuity of the numerical flux is restored on acute triangulations next schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous hamiltonians these schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local lipschitz continuity of the numerical hamiltonian finally a class of petrov galerkin approximations is considered these schemes are stabilized via a least squares bilinear form the petrov galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy discretization of the level set equation also requires the numerical approximation of a mean curvature term a simple mass lumped galerkin approximation is presented in section 6 and analyzed using maximum principle analysis the use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples these numerical examples include discretizations of convex and nonconvex forms of the h j equation the eikonal equation and the level set equation

a fixed grid sharp interface method for bubble dynamics and phase change abstract a numerical method has been developed for direct simulation of bubble dynamics with large liquid to vapor density ratio and phase change the numerical techniques are based on a fixed grid finite volume method capable of treating the interface as a sharp discontinuity the unsteady axisymmetric navier stokes equations and energy equation in both liquid and vapor phases are computed the mass momentum and energy conditions are explicitly matched at the phase boundary to determine the interface shape and movement the cubic b spline is used in conjunction with a fairing algorithm to yield smooth and accurate information of curvatures nondimensional parameters including reynolds weber and jakob numbers are varied to offer insight into the physical and numerical characteristics of the bubble dynamics based on the present sharp interface approach bubble dynamics for density ratio of 1600 or higher with and without phase change can be successfully computed