difference formulas for the surface laplacian on a triangulated surface abstract different approximating expressions for the surface laplacian operator on a triangulated surface are derived they are evaluated on a triangulated spherical surface for which the analytical expression of the surface laplacian is known it is shown that in order to obtain accurate results due care has to be taken of irregularities present in the triangulation grid if this is done the approximation will equal the performance of an expression based on least squares which can be derived next the different approximations obtained are used as a regularization operator in the solution of an ill posed inverse problem in electrical volume conduction it is shown that in this application a crude approximation to the surface laplacian suffices

a coupled arbitrary lagrangian eulerian and lagrangian method for computation of free surface flows with insoluble surfactants a finite element scheme to compute the dynamics of insoluble surfactant on a deforming free surface is presented the free surface is tracked by the arbitrary lagrangian eulerian ale approach whereas the surfactant concentration transport equation is approximated in a lagrangian manner since boundary resolved moving meshes are used in the ale approach the surface tension which may be a linear or nonlinear function of surfactant concentration equation of state and the marangoni forces can be incorporated directly into the numerical scheme further the laplace beltrami operator technique which reduces one order of differentiation associated with the curvature is used to handle the curvature approximation a number of 3d axisymmetric computations are performed to validate the proposed numerical scheme an excellent surfactant mass conservation without any additional mass correction scheme is obtained the differences in using a linear and a nonlinear equation of state respectively on the flow dynamics of a freely oscillating droplet are demonstrated

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

g dominance reference point based dominance for multiobjective metaheuristics abstract one of the main tools for including decision maker dm preferences in the multiobjective optimization mo literature is the use of reference points and achievement scalarizing functions a p wierzbicki the use of reference objectives in multiobjective optimization in g fandel t gal eds multiple criteria decision making theory and application springer verlag new york 1980 pp 469 486 the core idea in these approaches is converting the original mo problem into a single objective optimization problem through the use of a scalarizing function based on a reference point as a result a single efficient point adapted to the dm s preferences is obtained however a single solution can be less interesting than an approximation of the efficient set around this area as stated for example by deb in k deb j sundar n udaya bhaskara rao s chaudhuri reference point based multiobjective optimization using evolutionary algorithms international journal of computational intelligence research 2 3 2006 273 286 in this paper we propose a variation of the concept of pareto dominance called g dominance which is based on the information included in a reference point and designed to be used with any mo evolutionary method or any mo metaheuristic this concept will let us approximate the efficient set around the area of the most preferred point without using any scalarizing function on the other hand we will show how it can be easily used with any mo evolutionary method or any mo metaheuristic just changing the dominance concept and to exemplify its use we will show some results with some state of the art methods and some test problems

a hybrid finite volume finite element method for bulk surface coupled problems abstract the paper develops a hybrid method for solving a system of advection diffusion equations in a bulk domain coupled to advection diffusion equations on an embedded surface a monotone nonlinear finite volume method for equations posed in the bulk is combined with a trace finite element method for equations posed on the surface in our approach the surface is not fitted by the mesh and is allowed to cut through the background mesh in an arbitrary way moreover a triangulation of the surface into regular shaped elements is not required the background mesh is an octree grid with cubic cells as an example of an application we consider the modeling of contaminant transport in fractured porous media one standard model leads to a coupled system of advection diffusion equations in a bulk matrix and along a surface fracture a series of numerical experiments with both steady and unsteady problems and different embedded geometries illustrate the numerical properties of the hybrid approach the method demonstrates great flexibility in handling curvilinear or branching lower dimensional embedded structures

a general framework for low level vision we introduce a new geometrical framework based on which natural flows for image scale space and enhancement are presented we consider intensity images as surfaces in the x i space the image is thereby a two dimensional 2 d surface in three dimensional 3 d space for gray level images and 2 d surfaces in five dimensions for color images the new formulation unifies many classical schemes and algorithms via a simple scaling of the intensity contrast and results in new and efficient schemes extensions to multidimensional signals become natural and lead to powerful denoising and scale space algorithms

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

a grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion we develop numerical methods for solving partial differential equations pde defined on an evolving interface represented by the grid based particle method gbpm recently proposed in s leung h k zhao a grid based particle method for moving interface problems j comput phys 228 2009 7706 7728 in particular we develop implicit time discretization methods for the advection diffusion equation where the time step is restricted solely by the advection part of the equation we also generalize the gbpm to solve high order geometrical flows including surface diffusion and willmore type flows the resulting algorithm can be easily implemented since the method is based on meshless particles quasi uniformly sampled on the interface furthermore without any computational mesh or triangulation defined on the interface we do not require remeshing or reparametrization in the case of highly distorted motion or when there are topological changes as an interesting application we study locally inextensible flows governed by energy minimization we introduce tension force via a lagrange multiplier determined by the solution to a helmholtz equation defined on the evolving interface extensive numerical examples are also given to demonstrate the efficiency of the proposed approach

implicit brain imaging we describe how implicit surface representations can be used to solve fundamental problems in brain imaging this kind of representation is not only natural following the state of the art segmentation algorithms reported in the literature to extract the different brain tissues but it is also as shown in this paper the most appropriate one from the computational point of view examples are provided for finding constrained special curves on the cortex such as sulcal beds regularizing surface based measures such as cortical thickness and for computing warping fields between surfaces such as the brain cortex all these result from efficiently solving partial differential equations pdes and variational problems on surfaces represented in implicit form the implicit framework avoids the need to construct intermediate mappings between 3 d anatomical surfaces and parametric objects such planes or spheres a complex step that introduces errors and is required by many other cortical processing approaches

a simple level set method for solving stefan problems a simple level set method for solving stefan problems is presented this method can be applied to problems involving dendritic solidification our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance our method is accurate with respect to some exact solutions of the stefan problem results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification

orientation diffusions diffusions are useful for image processing and computer vision because they provide a convenient way of smoothing noisy data analyzing images at multiple scales and enhancing discontinuities a number of diffusions of image brightness have been defined and studied so far they may be applied to scalar and vector valued quantities that are naturally associated with intervals of either the real line or other flat manifolds some quantities of interest in computer vision and other areas of engineering that deal with images are defined on curved manifolds typical examples are orientation and hue that are defined on the circle generalizing brightness diffusions to orientation is not straightforward especially in the case where a discrete implementation is sought an example of what may go wrong is presented a method is proposed to define diffusions of orientation like quantities first a definition in the continuum is discussed then a discrete orientation diffusion is proposed the behavior of such diffusions is explored both analytically and experimentally it is shown how such orientation diffusions contain a nonlinearity that is reminiscent of edge process and anisotropic diffusion a number of open questions are proposed

fast data extrapolating data extrapolating extension technique has important applications in image processing on implicit surfaces and in level set methods the existing data extrapolating techniques are inefficient because they are designed without concerning the specialities of the extrapolating equations besides there exists little work on locating the narrow band after data extrapolating a very important problem in narrow band level set methods in this paper we put forward the general huygens principle and based on the principle we present two efficient data extrapolating algorithms the algorithms can easily locate the narrow band in data extrapolating furthermore we propose a prediction correction version for the data extrapolating algorithms and the corresponding band locating method for a special case where the direct band locating method is hard to apply experiments demonstrate the efficiency of our algorithms and the convenience of the band locating method

a geometry driven optical flow warping for spatial normalization of cortical surfaces spatial normalization is frequently used to map data to a standard coordinate system by removing intersubject morphological differences thereby allowing for group analysis to be carried out the work presented in this paper is motivated by the need for an automated cortical surface normalization technique that will automatically identify homologous cortical landmarks and map them to the same coordinates on a standard manifold the geometry of a cortical surface is analyzed using two shape measures that distinguish the sulcal and gyral regions in a multiscale framework a multichannel optical flow warping procedure aligns these shape measures between a reference brain and a subject brain creating the desired normalization the partial differential equation that carries out the warping is implemented in a euclidean framework in order to facilitate a multiresolution strategy thereby permitting large deformations between the two surfaces the technique is demonstrated by aligning 33 normal cortical surfaces and showing both improved structural alignment in manually labeled sulci and improved functional alignment in positron emission tomography data mapped to the surfaces a quantitative comparison between our proposed surface based spatial normalization method and a leading volumetric spatial normalization method is included to show that the surface based spatial normalization performs better in matching homologous cortical anatomies

solving variational problems and partial differential equations mapping into general target manifolds a framework for solving variational problems and partial differential equations that define maps onto a given generic manifold is introduced in this paper we discuss the framework for arbitrary target manifolds while the domain manifold problem was addressed in j comput phys 174 2 2001 759 the key idea is to implicitly represent the target manifold as the level set of a higher dimensional function and then implement the equations in the cartesian coordinate system where this embedding function is defined in the case of variational problems we restrict the search of the minimizing map to the class of maps whose target is the level set of interest in the case of partial differential equations we re write all the equation s geometric characteristics with respect to the embedding function we then obtain a set of equations that while defined on the whole euclidean space are intrinsic to the implicitly defined target manifold and map into it this permits the use of classical numerical techniques in cartesian grids regardless of the geometry of the target manifold the extension to open surfaces and submanifolds is addressed in this paper as well in the latter case the submanifold is defined as the intersection of two higher dimensional hypersurfaces and all the computations are restricted to this intersection examples of the applications of the framework here described include harmonic maps in liquid crystals where the target manifold is a hypersphere probability maps where the target manifold is a hyperplane chroma enhancement texture mapping and general geometric mapping between high dimensional manifolds

convolution generated motion as a link between cellular automata and continuum pattern dynamics cellular automata have been used to model the formation and dynamics of patterns in a variety of chemical biological and ecological systems however for patterns in which sharp interfaces form and propagate automata simulations can exhibit undesirable properties including spurious anisotropy and poor representation of interface curvature effects these simulations are also prohibitively slow when high accuracy is required even in two dimensions also the highly discrete nature of automata models makes theoretical analysis difficult in this paper we present a method for generating interface motions that is similar to the threshold dynamics type cellular automata but based on continuous convolutions rather than discrete sums these convolution generated motions naturally achieve the fine grid limit of the corresponding automata and they are also well suited to numerical and theoretical analysis because of this the desired pattern dynamics can be computed accurately and efficiently using adaptive resolution and fast fourier transform techniques and for a large class of convolutions the limiting interface motion laws can be derived analytically thus convolution generated motion provides a numerically and analytically tractable link between cellular automata models and the smooth features of pattern dynamics this is useful both as a means of describing the continuum limits of automata and as an independent foundation for expressing models for pattern dynamics in this latter role it also has a number of benefits over the traditional reaction diffusion ginzburg landau continuum pde models of pattern formation which yield true moving interfaces only as singular limits we illustrate the power of this approach with convolution generated motion models for pattern dynamics in developmental biology and excitable media

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

diffusion of general data on non flat manifolds viaharmonic maps theory the direction diffusion case in a number of disciplines directional data provides a fundamental source of information a novel framework for isotropic and anisotropic diffusion of directions is presented in this paper the framework can be applied both to denoise directional data and to obtain multiscale representations of it the basic idea is to apply and extend results from the theory of harmonic maps and in particular harmonic maps in liquid crystals this theory deals with the regularization of vectorial data while satisfying the intrinsic unit norm constraint of directional data we show the corresponding variational and partial differential equations formulations for isotropic diffusion obtained from an l 2 norm and edge preserving diffusion obtained from an l norm in general and an l 1 norm in particular in contrast with previous approaches the framework is valid for directions in any dimensions supports non smooth data and gives both isotropic and anisotropic formulations in addition the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general non flat manifolds that is functions between two general manifolds we present a number of theoretical results open questions and examples for gradient vectors optical flow and color images

schwarz solvers and preconditioners for the closest point method the discretization of surface intrinsic elliptic partial differential equations pdes poses interesting challenges not seen in flat spaces the discretization of these pdes typically proceeds by e

an adaptive octree finite element method for pdes posed on surfaces abstract the paper develops a finite element method for partial differential equations posed on hypersurfaces in r n n 2 3 the method uses traces of bulk finite element functions on a surface embedded in a volumetric domain the bulk finite element space is defined on an octree grid which is locally refined or coarsened depending on error indicators and estimated values of the surface curvatures the cartesian structure of the bulk mesh leads to easy and efficient adaptation process while the trace finite element method makes fitting the mesh to the surface unnecessary the number of degrees of freedom involved in computations is consistent with the two dimension nature of surface pdes no parametrization of the surface is required it can be given implicitly by a level set function in practice a variant of the marching cubes method is used to recover the surface with the second order accuracy we prove the optimal order of accuracy for the trace finite element method in h 1 and l 2 surface norms for a problem with smooth solution and quasi uniform mesh refinement experiments with less regular problems demonstrate optimal convergence with respect to the number of degrees of freedom if grid adaptation is based on an appropriate error indicator the paper shows results of numerical experiments for a variety of geometries and problems including advection diffusion equations on surfaces analysis and numerical results of the paper suggest that combination of cartesian adaptive meshes and the unfitted trace finite elements provide simple efficient and reliable tool for numerical treatment of pdes posed on surfaces

diffusion on a curved surface coupled to diffusion in the volume an algorithm is presented for solving a diffusion equation on a curved surface coupled to diffusion in the volume a problem often arising in cell biology it applies to pixilated surfaces obtained from experimental images and performs at low computational cost in the method the laplace beltrami operator is approximated locally by the laplacian on the tangential plane and then a finite volume discretization scheme based on a voronoi decomposition is applied convergence studies show that mass conservation built in the discretization scheme and cancellation of sampling error ensure convergence of the solution in space with an order between 1 and 2 the method is applied to a cell biological problem where a signaling molecule g protein rac cycles between the cytoplasm and cell membrane thus coupling its diffusion in the membrane to that in the cell interior simulations on realistic cell geometry are performed to validate and determine the accuracy of a recently proposed simplified quantitative analysis of fluorescence loss in photobleaching the method is implemented within the virtual cell computational framework freely accessible at www vcell org

color image enhancement via chromaticity diffusion a novel approach for color image denoising is proposed in this paper the algorithm is based on separating the color data into chromaticity and brightness and then processing each one of these components with partial differential equations or diffusion flows in the proposed algorithm each color pixel is considered as an n dimensional vector the vectors direction a unit vector gives the chromaticity while the magnitude represents the pixel brightness the chromaticity is processed with a system of coupled diffusion equations adapted from the theory of harmonic maps in liquid crystals this theory deals with the regularization of vectorial data while satisfying the intrinsic unit norm constraint of directional data such as chromaticity both isotropic and anisotropic diffusion flows are presented for this n dimensional chromaticity diffusion flow the brightness is processed by a scalar median filter or any of the popular and well established anisotropic diffusion flows for scalar image enhancement we present the underlying theory a number of examples and briefly compare with the current literature

l 2 estimates for the evolving surface finite element method in this paper we consider the evolving surface nite element method for the advection and di usion of a conserved scalar quantity on a moving surface in an earlier paper using a suitable variational formulation in time dependent sobolev space we proposed and analysed a nite element method using surface nite elements on evolving triangulated surfaces an optimal order h error bound was proved for linear nite elements in this work we prove the optimal error bound in l t uniformly in time

higher order meshing of implicit geometries part ii approximations on manifolds abstract a new concept for the higher order accurate approximation of partial differential equations on manifolds is proposed where a surface mesh composed by higher order elements is automatically generated based on level set data thereby it enables a completely automatic workflow from the geometric description to the numerical analysis without any user intervention a master level set function defines the shape of the manifold through its zero isosurface which is then restricted to a finite domain by additional level set functions it is ensured that the surface elements are sufficiently continuous and shape regular which is achieved by manipulating the background mesh the numerical results show that optimal convergence rates are obtained with a moderate increase in the condition number compared to handcrafted surface meshes

numerical methods for p harmonic flows and applications to image processing we propose in this paper an alternative approach for computing p harmonic maps and flows instead of solving a constrained minimization problem on sn 1 we solve an unconstrained minimization problem on the entire space of functions this is possible using the projection on the sphere of any arbitrary function then we show how this formulation can be used in practice for problems with both isotropic and anisotropic diffusion with applications to image processing using a new finite difference scheme

a simple and efficient finite difference method for the phase field crystal equation on curved surfaces abstract we present a simple and efficient finite difference method for the phase field crystal pfc equation on curved surfaces embedded in r 3 we employ a narrow band neighborhood of a curved surface that is defined as a zero level set of a signed distance function the pfc equation on the surface is extended to the three dimensional narrow band domain by using the closest point method and applying a pseudo neumann boundary condition we can use the standard seven point discrete laplacian operator instead of the discrete laplace beltrami operator on the surface the pfc equation on the narrow band domain is discretized using an unconditionally stable scheme and the resulting implicit discrete system of equations is solved by using the jacobi iterative method computational results are presented to demonstrate the efficiency and usefulness of the proposed method

region tracking on level sets methods since the work by osher and sethian 1988 on level sets algorithms for numerical shape evolutions this technique has been used for a large number of applications in numerous fields in medical imaging this numerical technique has been successfully used for example in segmentation and cortex unfolding algorithms the migration from a lagrangian implementation to a eulerian one via implicit representations or level sets brought some of the main advantages of the technique i e topology independence and stability this migration means also that the evolution is parametrization free therefore the authors do not know exactly how each part of the shape is deforming and the point wise correspondence is lost in this note they present a technique to numerically track regions on surfaces that are being deformed using the level sets method the basic idea is to represent the region of interest as the intersection of two implicit surfaces and then track its deformation from the deformation of these surfaces this technique then solves one of the main shortcomings of the very useful level sets approach applications include lesion localization in medical images region tracking in functional mri fmri visualization and geometric surface mapping

a localized meshless method for diffusion on folded surfaces partial differential equations pdes on surfaces arise in a variety of application areas including biological systems medical imaging fluid dynamics mathematical physics image processing and computer graphics in this paper we propose a radial basis function rbf discretization of the closest point method the corresponding localized meshless method may be used to approximate diffusion on smooth or folded surfaces our method has the benefit of having an a priori error bound in terms of percentage of the norm of the solution a stable solver is used to avoid the ill conditioning that arises when the radial basis functions rbfs become flat

variational problems and partial differential equations on implicit surfaces a novel framework for solving variational problems and partial differential equations for scalar and vector valued data defined on surfaces is introduced in this paper the key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed cartesian coordinate system using this new embedding function the equations are then both intrinsic to the surface and defined in the embedding space this approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces as is commonly done in the literature we describe the framework and present examples in computer graphics and image processing applications including texture synthesis flow field visualization and image and vector field intrinsic regularization for data defined on 3d surfaces

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

a short time beltrami kernel for smoothing images and manifolds we introduce a short time kernel for the beltrami image enhancing flow the flow is implemented by convolving the image with a space dependent kernel in a similar fashion to the solution of the heat equation by a convolution with a gaussian kernel the kernel is appropriate for smoothing regular flat 2 d images for smoothing images painted on manifolds and for simultaneously smoothing images and the manifolds they are painted on the kernel combines the geometry of the image and that of the manifold into one metric tensor thus enabling a natural unified approach for the manipulation of both additionally the derivation of the kernel gives a better geometrical understanding of the beltrami flow and shows that the bilateral filter is a euclidean approximation of it on a practical level the use of the kernel allows arbitrarily large time steps as opposed to the existing explicit numerical schemes for the beltrami flow in addition the kernel works with equal ease on regular 2 d images and on images painted on parametric or triangulated manifolds we demonstrate the denoising properties of the kernel by applying it to various types of images and manifolds

the implicit closest point method for the numerical solution of partial differential equations on surfaces many applications in the natural and applied sciences require the solutions of partial differential equations pdes on surfaces or more general manifolds the closest point method is a simple and accurate embedding method for numerically approximating pdes on rather general smooth surfaces however the original formulation is designed to use explicit time stepping this may lead to a strict time step restriction for some important pdes such as those involving the laplace beltrami operator or higher order derivative operators to achieve improved stability and efficiency we introduce a new implicit closest point method for surface pdes the method allows for large stable time steps while retaining the principal benefits of the original method in particular it maintains the order of accuracy of the discretization of the underlying embedding pde it works on sharply defined bands without degrading the accuracy of the method and it applies to general smooth surfaces it also is very simple and may be applied to a rather general class of surface pdes convergence studies for the in surface heat equation and a fourth order biharmonic problem are given to illustrate the accuracy of the method we demonstrate the flexibility and generality of the method by treating flows involving diffusion reaction diffusion and fourth order spatial derivatives on a variety of interesting surfaces including surfaces of mixed codimension

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

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

the method of fundamental solutions applied to boundary value problems on the surface of a sphere abstract in this work we propose using the method of fundamental solutions mfs to solve boundary value problems for the helmholtz beltrami equation on a sphere we prove density and convergence results that justify the proposed mfs approximation several numerical examples are considered to illustrate the good performance of the method

simple computation of reaction diffusion processes on point clouds the study of reaction diffusion processes is much more complicated on general curved surfaces than on standard cartesian coordinate spaces here we show how to formulate and solve systems of reaction diffusion equations on surfaces in an extremely simple way using only the standard cartesian form of differential operators and a discrete unorganized point set to represent the surface our method decouples surface geometry from the underlying differential operators as a consequence it becomes possible to formulate and solve rather general reaction diffusion equations on general surfaces without having to consider the complexities of differential geometry or sophisticated numerical analysis to illustrate the generality of the method computations for surface diffusion pattern formation excitable media and bulk surface coupling are provided for a variety of complex point cloud surfaces

partial difference operators on weighted graphs for image processing on surfaces and point clouds partial difference equations pdes and variational methods for image processing on euclidean domains spaces are very well established because they permit to solve a large range of real computer vision problems with the recent advent of many 3d sensors there is a growing interest in transposing and solving pdes on surfaces and point clouds in this paper we propose a simple method to solve such pdes using the framework of pdes on graphs this latter approach enables us to transcribe for surfaces and point clouds many models and algorithms designed for image processing to illustrate our proposal three problems are considered 1 p laplacian restoration and inpainting 2 pdes mathematical morphology and 3 active contours segmentation

brain surface conformal parameterization using riemann surface structure in medical imaging parameterized 3 d surface models are useful for anatomical modeling and visualization statistical comparisons of anatomy and surface based registration and signal processing here we introduce a parameterization method based on riemann surface structure which uses a special curvilinear net structure conformal net to partition the surface into a set of patches that can each be conformally mapped to a parallelogram the resulting surface subdivision and the parameterizations of the components are intrinsic and stable their solutions tend to be smooth functions and the boundary conditions of the dirichlet problem can be enforced conformal parameterization also helps transform partial differential equations pdes that may be defined on 3 d brain surface manifolds to modified pdes on a two dimensional parameter domain since the jacobian matrix of a conformal parameterization is diagonal the modified pde on the parameter domain is readily solved to illustrate our techniques we computed parameterizations for several types of anatomical surfaces in 3 d magnetic resonance imaging scans of the brain including the cerebral cortex hippocampi and lateral ventricles for surfaces that are topologically homeomorphic to each other and have similar geometrical structures we show that the parameterization results are consistent and the subdivided surfaces can be matched to each other finally we present an automatic sulcal landmark location algorithm by solving pdes on cortical surfaces the landmark detection results are used as constraints for building conformal maps between surfaces that also match explicitly defined landmarks

an axiomatic approach to scalar data interpolation on surfaces we discuss possible algorithms for interpolating data given on a set of curves in a surface of 3 we propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations the absolutely minimizing lipschitz extension model amle is singled out and studied in more detail we study the correctness of our numerical approach and we show experiments illustrating the interpolation of data on some simple test surfaces like the sphere and the torus

higher order nonlinear priors for surface reconstruction for surface reconstruction problems with noisy and incomplete range data a bayesian estimation approach can improve the overall quality of the surfaces the bayesian approach to surface estimation relies on a likelihood term which ties the surface estimate to the input data and the prior which ensures surface smoothness or continuity this paper introduces a new high order nonlinear prior for surface reconstruction the proposed prior can smooth complex noisy surfaces while preserving sharp geometric features and it is a natural generalization of edge preserving methods in image processing such as anisotropic diffusion an exact solution would require solving a fourth order partial differential equation pde which can be difficult with conventional numerical techniques our approach is to solve a cascade system of two second order pdes which resembles the original fourth order system this strategy is based on the observation that the generalization of image processing to surfaces entails filtering the surface normals we solve one pde for processing the normals and one for refitting the surface to the normals furthermore we implement the associated surface deformations using level sets hence the algorithm can accommodate very complex shapes with arbitrary and changing topologies this paper gives the mathematical formulation and describes the numerical algorithms we also show results using range and medical data

estimation of planar curves surfaces and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation the author addresses the problem of parametric representation and estimation of complex planar curves in 2 d surfaces in 3 d and nonplanar space curves in 3 d curves and surfaces can be defined either parametrically or implicitly with the latter representation used here a planar curve is the set of zeros of a smooth function of two variables x y a surface is the set of zeros of a smooth function of three variables x y z and a space curve is the intersection of two surfaces which are the set of zeros of two linearly independent smooth functions of three variables x y z for example the surface of a complex object in 3 d can be represented as a subset of a single implicit surface with similar results for planar and space curves it is shown how this unified representation can be used for object recognition object position estimation and segmentation of objects into meaningful subobjects that is the detection of interest regions that are more complex than high curvature regions and hence more useful as features for object recognition

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

smooth functional and structural maps on the neocortex via orthonormal bases of the laplace beltrami operator functional and structural maps such as a curvature cortical thickness and functional magnetic resonance imaging mri maps indexed over the local coordinates of the cortical manifold play an important role in neuropsychiatric studies due to the highly convoluted nature of the cerebral cortex and image quality these functions are generally uninterpretable without proper methods of association and smoothness onto the local coordinate system in this paper we generalized the spline smoothing problem wahba 1990 from a sphere to any arbitrary two dimensional 2 d manifold with boundaries we first seek a numerical solution to orthonormal basis functions of the laplace beltrami lb operator with neumann boundary conditions for a 2 d manifold m then solve the spline smoothing problem in a reproducing kernel hilbert space r k h s of real valued functions on manifold m with kernel constructed from the basis functions the explicit discrete lb representation is derived using the finite element method calculated directly on the manifold coordinates so that finding discrete lb orthonormal basis functions is equivalent to solving an algebraic eigenvalue problem and then smoothed functions in r k h s can be represented as a linear combination of the basis functions we demonstrate numerical solutions of spherical harmonics on a unit sphere and brain orthonormal basis functions on a planum temporale manifold then synthetic data is used to quantify the goodness of the smoothness compared with the ground truth and discuss how many basis functions should be incorporated in the smoothing we present applications of our approach to smoothing sulcal mean curvature cortical thickness and functional statistical maps on submanifolds of the neocortex

a posteriori error estimates for finite volume approximations of elliptic equations on general surfaces abstract in this paper we present a residual based a posteriori error estimate for the finite volume discretization of steady convection diffusion reaction equations defined on surfaces in r 3 which are often implicitly represented as level sets of smooth functions reliability and efficiency of the proposed a posteriori error estimator are rigorously proved numerical experiments are also conducted to verify the theoretical results and demonstrate the robustness of the error estimator

a meshfree generalized finite difference method for surface pdes abstract in this paper we propose a novel meshfree generalized finite difference method gfdm approach to discretize pdes defined on manifolds derivative approximations for the same are done directly on the tangent space in a manner that mimics the procedure followed in volume based meshfree gfdms as a result the proposed method not only does not require a mesh it also does not require an explicit reconstruction of the manifold in contrast to some existing methods it avoids the complexities of dealing with a manifold metric while also avoiding the need to solve a pde in the embedding space a major advantage of this method is that all developments in usual volume based numerical methods can be directly ported over to surfaces using this framework we propose discretizations of the surface gradient operator the surface laplacian and surface diffusion operators possibilities to deal with anisotropic and discontinuous surface properties with large jumps are also introduced and a few practical applications are presented