an o n log log n time algorithm for triangulating a simple polygon given a simple n vertex polygon the triangulation problem is to partition the interior of the polygon into n 2 triangles by adding n 3 nonintersecting diagonals we propose an o n log log n time algorithm for this problem improving on the previously best bound of o n log n and showing that triangulation is not as hard as sorting improved algorithms for several other computational geometry problems including testing whether a polygon is simple follow from our result

cortical thickness analysis in autism with heat kernel smoothing we present a novel data smoothing and analysis framework for cortical thickness data defined on the brain cortical manifold gaussian kernel smoothing which weights neighboring observations according to their 3d euclidean distance has been widely used in 3d brain images to increase the signal to noise ratio when the observations lie on a convoluted brain surface however it is more natural to assign the weights based on the geodesic distance along the surface we therefore develop a framework for geodesic distance based kernel smoothing and statistical analysis on the cortical manifolds as an illustration we apply our methods in detecting the regions of abnormal cortical thickness in 16 high functioning autistic children via random field based multiple comparison correction that utilizes the new smoothing technique

cortical surface based analysis ii inflation flattening and a surface based coordinate system the surface of the human cerebral cortex is a highly folded sheet with the majority of its surface area buried within folds as such it is a difficult domain for computational as well as visualization purposes we have therefore designed a set of procedures for modifying the representation of the cortical surface to i inflate it so that activity buried inside sulci may be visualized ii cut and flatten an entire hemisphere and iii transform a hemisphere into a simple parameterizable surface such as a sphere for the purpose of establishing a surface based coordinate system r 1999 academic press

a fictitious domain method with mixed finite elements for elastodynamics we consider in this paper the wave scattering problem by an object with neumann boundary conditions in an anisotropic elastic body to obtain an efficient numerical method permitting the use of regular grids we follow a fictitious domain approach coupled with a first order velocity stress formulation for elastodynamics we first observe that the method does not always converge when the q 1 rm div q 0 finite element is used in particular the method converges for some scattering object geometries but not for others note that the convergence of the q 1 rm div q 0 finite element method was shown in e becache p joly and c tsogka siam j numer anal 39 2002 pp 2109 2132 for the elastodynamic problem in the absence of a scattering object i e without the coupling of the mixed finite elements with the fictitious domain method therefore we propose here a modification of the q 1 rm div q 0 element following the approach in e becache j rodriguez and c tsogka on the convergence of the fictitious domain method for wave equation problems technical report 5802 inria 2006 where the simpler acoustic case was considered to study the numerical properties of the new element we carry out a dispersion analysis several numerical simulations as well as a numerical convergence analysis show that the proposed method provides a good approximate solution

intrinsic and extrinsic analysis in computational anatomy we present intrinsic and extrinsic methods for studying anatomical coordinates in order to perform statistical inference on random physiological signals f across clinical populations in both intrinsic and extrinsic methods we introduce generalized partition functions of the coordinates psi x x epsilon m which are used to construct a random field of f on m as statistical model in the intrinsic analysis such partition functions are built intrinsically for individual anatomical coordinate based on courant s theorem on nodal analysis via self adjoint linear elliptic differential operators in contrast the extrinsic method needs only one set of partition functions for a template coordinate system and then applies to each anatomical coordinate system via diffeomorphic transformation for illustration we apply both intrinsic and extrinsic methods to a clinical study cortical thickness variation of the left cingulate gyrus in schizophrenia both methods show that the left cingulate gyrus tends to become thinner in schizophrenia relative to the healthy control population however the intrinsic method increases the statistical power

linear kinetic theory of hard sphere fluids a new general formalism for a linear kinetic theory is proposed it is based on a diagrammatic analysis of the linear integral operators containing equilibrium correlations the diagrams are similar to the mayer graphs but possess two kinds of root points this new diagrammatic technique is used to analyse the bbgky hierarchy for a hard sphere fluid as a result a new hierarchy of evolution equations is obtained in which the enskog like renormalization of the binary collisions has been explicitly carried out thus the previous ideas of resibois and lebowitz have been fully implemented the final results are shown to be equivalent to those of van beijeren possible further applications of the proposed formalism are indicated and will be the subject of following publications

distributed lagrange multipliers based on fictitious domain method for second order elliptic problems in this article we further investigate the solution of linear second order elliptic boundary value problems by distributed lagrange multipliers based fictitious domain methods the following issues are addressed i derivation of the fictitious domain formulations ii finite element approximation iii iterative solution of the resulting finite dimensional problems of the saddle point type by preconditioned conjugate gradient and lanczos algorithms

quantitative evaluation of three cortical surface flattening methods during the past decade several computational approaches have been proposed for the task of mapping highly convoluted surfaces of the human brain to simpler geometric objects such as a sphere or a topological disc we report the results of a quantitative comparison of freesurfer circlepack and lscm with respect to measurements of geometric distortion and computational speed our results indicate that freesurfer performs best with respect to a global measurement of metric distortion whereas lscm performs best with respect to angular distortion and best in all but one case with a local measurement of metric distortion freesurfer provides more homogeneous distribution of metric distortion across the whole cortex than circlepack and lscm lscm is the most computationally efficient algorithm for generating spherical maps while circlepack is extremely fast for generating planar maps from patches

time sequence diffeomorphic metric mapping and parallel transport track time dependent shape changes abstract serial mri human brain scans have facilitated the detection of brain development and of the earliest signs of neuropsychiatric and neurodegenerative diseases monitoring disease progression and resolving drug effects in clinical trials for preventing or slowing the rate of brain degeneration to track anatomical shape changes in serial images we introduce new point based time sequence large deformation diffeomorphic metric mapping ts lddmm to infer the time flow of within subject geometric shape changes that carry known observations through a period its euler lagrange equation is generalized for anatomies whose shapes are characterized by point sets such as landmarks curves and surfaces the time dependent momentum obtained from the ts lddmm encodes within subject shape changes for the purpose of across subject shape comparison we then propose a diffeomorphic analysis framework to translate within subject deformation in a global template without incorporating across subject anatomical variations via parallel transport technique the analysis involves the retraction of the within subject time dependent momentum along the ts lddmm trajectory from each time to the baseline the translation of the momentum in a global template and the reconstruction of the ts lddmm trajectory starting from the global template

glauber dynamics of the sk model theory and simulations in the high temperature phase the low temperature glauber dynamics of the sherrington kirkpatrick spin glass model is studied using a combination of theory and computer simulations the theoretical approach follows the spirit of mori s continuous fraction expansion the predicted amplitude of the long time decay of the time dependent equilibrium spin spin correlation function agrees well with the glauber dynamics simulation results the theory predicts a universal dynamic exponent of in the low temperature phase the simulations cannot distinguish between a universal exponent and a temperature dependent one

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

hippocampal shape modeling based on a progressive template surface deformation and its verification accurately recovering the hippocampal shapes against rough and noisy segmentations is as challenging as achieving good anatomical correspondence between the individual shapes to address these issues we propose a mesh to volume registration approach characterized by a progressive model deformation our model implements flexible weighting scheme for model rigidity under a multi level neighborhood for vertex connectivity this method induces a large to small scale deformation of a template surface to build the pairwise correspondence by minimizing geometric distortion while robustly restoring the individuals shape characteristics we evaluated the proposed method s 1 accuracy and robustness in smooth surface reconstruction 2 sensitivity in detecting significant shape differences between healthy control and disease groups mild cognitive impairment and alzheimer s disease 3 robustness in constructing the anatomical correspondence between individual shape models and 4 applicability in identifying subtle shape changes in relation to cognitive abilities in a healthy population we compared the performance of the proposed method with other well known methods spharm pdm shapeworks and lddmm volume registration with template injection using various metrics of shape similarity surface roughness volume and shape deformity the experimental results showed that the proposed method generated smooth surfaces with less volume differences and better shape similarity to input volumes than others the statistical analyses with clinical variables also showed that it was sensitive in detecting subtle shape changes of hippocampus

fictitious domain methods for the numerical solution of two dimensional scattering problems fictitious domain methods for the numerical solution of two dimensional scattering problems are considered the original exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a nonreflecting boundary condition on the artificial boundary first order second order and exact nonreflecting boundary conditions are tested on rectangular and circular boundaries the finite element discretizations of the corresponding approximate boundary value problems are performed using locally fitted meshes and the discrete equations are solved with fictitious domain methods a special finite element method using nonmatching meshes is considered this method uses the macro hybrid formulation based on domain decomposition to couple polar and cartesian coordinate systems a special preconditioner based on fictitious domains is introduced for the arising algebraic saddle point system such that the subspace of constraints becomes invariant with respect to the preconditioned iterative procedure the performance of the new method is compared to the fictitious domain methods both with respect to accuracy and computational cost

approach to equilibrium for a class of random quantum models of infinite range we consider random generalizations of a quantum model of infinite range introduced by emch and radin the generalizations allow a neat extension from the class l 1 of absolutely summable lattice potentials to the optimal class l 2 of square summable potentials first considered by khanin and sinai and generalised by van enter and van hemmen the approach to equilibrium in the case of a gaussian distribution is proved to be faster than for a bernoulli distribution for both short range and long range lattice potentials while exponential decay to equilibrium is excluded in the nonrandom l 1 case it is proved to occur for both short and long range potentials for gaussian distributions and for potentials of class l 2 in the bernoulli case open problems are discussed

distributed lagrange multiplier methods for incompressible viscous flow around moving rigid bodies abstract in this article we discuss the application of a distributed lagrange multiplier based fictitious domain method to the numerical simulation of incompressible viscous flow modelled by the navier stokes equations around moving bodies we suppose the rigid bodies motion known a priori the solution method combines finite element approximations time discretization by operator splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method numerical experiment results for two dimensional flow around a moving disk are presented

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

a boundary difference method for electromagnetic scattering problems with perfect conductors and corners boundary integral equations bie are widely used in applied electromagnetics with the boundary element method bem typically employed for numerical solution the paper extends the recently proposed boundary difference method bdm an alternative to bem to electromagnetic scattering problems with perfect conductors and with corners bdm shares several key features and advantages of bem but avoids the singular kernels inherent in bie convergence rates of the method established via numerical experiments are in agreement with the theoretical predictions

combining anatomical manifold information via diffeomorphic metric mappings for studying cortical thinning of the cingulate gyrus in schizophrenia spatial normalization is a crucial step in assessing patterns of neuroanatomical structure and function associated with health and disease errors that occur during spatial normalization can influence hypothesis testing due to the dimensionalities of mapping algorithms and anatomical manifolds landmarks curves surfaces volumes used to drive the mapping algorithms the primary aim of this paper is to improve statistical inference using multiple anatomical manifolds and large deformation diffeomorphic metric mapping lddmm algorithms we propose that combining information generated by the various manifolds and algorithms improves the reliability of hypothesis testing we used this unified approach to assess variation in the thickness of the cingulate gyrus in subjects with schizophrenia and healthy comparison subjects three different lddmm algorithms for mapping landmarks curves and triangulated meshes were used to transform thickness maps of the cingulate surfaces into an atlas coordinate system we then tested for group differences by combining the information from the three types of anatomical manifolds and lddmm mapping algorithms the unified approach provided reliable statistical results and eliminated ambiguous results due to surface mismatches subjects with schizophrenia had non uniform cortical thinning over the left and right cingulate gyri especially in the anterior portion as compared to

quantitative evaluation of lddmm freesurfer and caret for cortical surface mapping cortical surface mapping has been widely used to compensate for individual variability of cortical shape and topology in anatomical and functional studies while many surface mapping methods were proposed based on landmarks curves spherical or native cortical coordinates few studies have extensively and quantitatively evaluated surface mapping methods across different methodologies in this study we compared five cortical surface mapping algorithms including large deformation diffeomorphic metric mapping lddmm for curves lddmm curve for surfaces lddmm surface multi manifold lddmm mm lddmm freesurfer and caret using 40 mri scans and 10 simulated datasets we computed curve variation errors and surface alignment consistency for assessing the mapping accuracy of local cortical features e g gyral sulcal curves and sulcal regions and the curvature correlation for measuring the mapping accuracy in terms of overall cortical shape in addition the simulated datasets facilitated the investigation of mapping error distribution over the cortical surface when the mm lddmm freesurfer and caret mapping algorithms were applied our results revealed that the lddmm curve mm lddmm and caret approaches best aligned the local curve features with their own curves the mm lddmm approach was also found to be the best in aligning the local regions and cortical folding patterns e g curvature as compared to the other mapping approaches the simulation experiment showed that the mm lddmm mapping yielded less local and global deformation errors than the caret and freesurfer mappings

observer independent method for microstructural parcellation of cerebral cortex a quantitative approach to cytoarchitectonics we describe a new observer independent procedure for identifying boundaries between cortical areas the method is useful for images obtained from sections which provide microstructural information on the cortical laminar pattern e g nissl myelin or immunohistochemically stained sections or receptor autoradiographs the laminar pattern is represented by profile curves extending from the cortical surface to the white matter boundary these profiles are constructed from digitized images digitization is based on the grey level index nissl or densitometry myelin immunohistochemistry receptor autoradiography the shapes of neighboring profiles are compared by calculating their distances according to feature vectors extracted from the profiles profiles derived from a homogeneous area can be expected to be similar in shape and hence show low distance values between each other maximum distances can be found between profiles which lie on opposite sides of a structural boundary the mahalanobis distance was found to be more sensitive and to yield greater spatial resolution than other distance measures such as the euclidean distance cell stained sections of the human neocortex were analyzed the method not only verified boundaries which had been defined by visual inspection it also revealed new ones which had not been detected visually the procedure offers an important supplement to the traditional methods based on visual inspection which for the first time is based on quantitative data and therefore offers a new level of reproducibility and observer independence anatomical atlases based on this procedure thus provide a new tool for the interpretation of structural data obtained from functional imaging techniques

cortical cartography using the discrete conformal approach of circle packings cortical flattening algorithms are becoming more widely used to assist in visualizing the convoluted cortical gray matter sheet of the brain metric based approaches are the most common but suffer from high distortions conformal or angle based algorithms are supported by a comprehensive mathematical theory the conformal approach that uses circle packings is versatile in the manipulation and display of results in addition it offers some new and interesting metrics that may be useful in neuroscientific analysis and are not available through numerical partial differential equation conformal methods in this paper we begin with a brief description of cortical flat mapping from data acquisition to map displays including a brief review of past flat mapping approaches we then describe the mathematics of conformal geometry and key elements of conformal mapping we introduce the mechanics of circle packing and discuss its connections with conformal geometry using a triangulated surface representing a cortical hemisphere we illustrate several manipulations available using circle packing methods and describe the associated ensemble conformal features ecfs we conclude by discussing current and potential uses of conformal methods in neuroscience and computational anatomy

dynamic programming generation of boundaries of local coordinatized submanifolds in the neocortex application to the planum temporale dynamic programming is used to define boundaries of cortical submanifolds with focus on the planum temporale pt of the superior temporal gyrus stg which has been implicated in a variety of neuropsychiatric disorders to this end automated methods are used to generate the pt manifold from 10 high resolution mri subvolumes roi masks encompassing the stg a procedure to define the subvolume roi masks from original mri brain scans is developed bayesian segmentation is then used to segment the subvolumes into cerebrospinal fluid gray matter gm and white matter wm 3d isocontouring using the intensity value at which there is equal probability of gm and wm is used to reconstruct the triangulated graph representing the stg cortical surface enabling principal curvature at each point on the graph to be computed dynamic programming is used to delineate the pt manifold by tracking principal curves from the retro insular end of the heschl s gyrus hg to the stg along the posterior stg up to the start of the ramus and back to the retro insular end of the hg a coordinate system is then defined on the pt manifold the origin is defined by the retro insular end of the hg and the y axis passes through the point on the posterior stg where the ramus begins automated labeling of gm in the stg is robust with l 1 distances between bayesian and manual segmentation in the range 0 001 0 12 n 20 pt reconstruction is also robust with 90 of the vertices of the reconstructed pt within about 1 voxel n 20 from semiautomated contours finally the reliability index based on interrater intraclass correlation for the surface area derived from repeated reconstructions is 0 96 for the left pt and 0 94 for the right pt thus demonstrating the robustness of dynamic programming in defining a coordinate system on the pt it provides a method with potential significance in the study of neuropsychiatric disorders

near isometric flattening of brain surfaces flattened representations of brain surfaces are often used to visualize and analyze spatial patterns of structural organization and functional activity here we present a set of rigorous criteria and accompanying test cases with which to evaluate flattening algorithms that attempt to preserve shortest path distances on the original surface we also introduce a novel flattening algorithm that is the first to satisfy all of these criteria and demonstrate its ability to produce accurate flat maps of human and macaque visual cortex using this algorithm we have recently obtained results showing a remarkable unexpected degree of consistency in the shape and topographic structure of visual cortical areas within humans and macaques as well as between these two species

quasi isometric parameterization for texture mapping in this paper we present a new 3d triangular mesh parameterization method that is computationally efficient and yields minimized distance errors the method has four steps firstly multidimensional scaling mds is used to flatten each submesh consisting of one vertex and its direct neighbours on the 3d triangular mesh secondly an optimal method is used to compute the linear reconstructing weights of each vertex with respect to its neighbours thirdly a spectral decomposition method is used to obtain initial 2d parameterization coordinates fourthly the initial coordinates are rotated and scaled to minimize the distance errors it is demonstrated that this method can be used for texture mapping analyses and examples show the effectiveness of this parameterization method compared with alternatives

the emerging discipline of computational functional anatomy computational functional anatomy cfa is the study of functional and physiological response variables in anatomical coordinates for this we focus on two things i the construction of bijections via diffeomorphisms between the coordinatized manifolds of human anatomy and ii the transfer group action and parallel transport of functional information into anatomical atlases via these bijections we review advances in the unification of the bijective comparison of anatomical submanifolds via point sets including points curves and surface triangulations as well as dense imagery we examine the transfer via these bijections of functional response variables into anatomical coordinates via group action on scalars and matrices in dti as well as parallel transport of metric information across multiple templates which preserves the inner product

a multi resolution scheme for distortion minimizing mapping between human subcortical structures based on geodesic construction on riemannian manifolds in this paper we deal with a subcortical surface registration problem subcortical structures including hippocampi and caudates have a small number of salient features such as heads and tails unlike cortical surfaces therefore it is hard if not impossible to perform subcortical surface registration with only such features it is also non trivial for neuroanatomical experts to select landmarks consistently for subcortical surfaces of different subjects we therefore present a landmark free approach for subcortical surface registration by measuring the amount of mesh distortion between subcortical surfaces assuming that the surfaces are represented by meshes the input meshes can be constructed using any surface modeling tool available in the public domain since our registration method is independent of a surface modeling process given the source and target surfaces together with their representing meshes the vertex positions of the source mesh are iteratively displaced while preserving the underlying surface shape in order to minimize the distortion to the target mesh by representing each surface mesh as a point on a high dimensional riemannian manifold we define a distance metric on the manifold that measures the amount of distortion from a given source mesh to the target mesh based on the notion of isometry while penalizing triangle flipping under this metric we reduce the distortion minimization problem to the problem of constructing a geodesic curve from the moving source point to the fixed target point on the manifold while satisfying the shape preserving constraint we adopt a multi resolution framework to solve the problem for distortion minimizing mapping between the source and target meshes we validate our registration scheme through several experiments distance metric comparison visual validation using real data robustness test to mesh variations feature alignment using anatomic landmarks consistency with previous clinical findings and comparison with a surface based registration method lddmm surface

a finite element capacitance matrix method for exterior helmholtz problems we introduce an algorithm for the efficient numerical solution of exterior boundary value problems for the helmholtz equation the problem is reformulated as an equivalent one on a bounded domain using an exact non local boundary condition on a circular artificial boundary an fft based fast helmholtz solver is then derived for a finite element discretization on an annular domain the exterior problem for domains of general shape are treated using an imbedding or capacitance matrix method the imbedding is achieved in such a way that the resulting capacitance matrix has a favorable spectral distribution leading to mesh independent convergence rates when krylov subspace methods are used to solve the capacitance matrix equation

convergence of discrete laplace beltrami operators over surfaces the convergence property of the discrete laplace beltrami operator is the foundationof convergence analysis of the numerical simulation process of some geometric partial differential equations which involve the operator the aim of this paper is to review several already used discrete laplace beltrami operators over triangulated surface and study numerically as well as theoretically their convergent behavior we show that none of them is convergent in general but two of them proposed by desbrun et al and meyer et al are convergent in a special case we point out that this special case is very important in the numerical simulation of geometric partial differential equations

multi structure network shape analysis via normal surface momentum maps article i nfo we present a shape analysis pipeline for the assessment of anatomical variations in subcortical networks in mr images the shape analysis pipeline injects the global shape properties of the cfa subcortical template into the subcortical parcellations generated from freesurfer via large deformation diffeomorphic metric mapping lddmm examples are shown for this injection in several subcortical structures whose raw mr images were sampled from the database of open access series of imaging studies oasis the shape analysis is performed on random field representation of the template surface momentum maps that encode the shape variation of subcortical structure targets of each individual subject relative to the template the momentum maps have the optimum property that they are supported only on the boundary of the subcortical structures with the direction normal to the subcortical nuclei boundary thereby reducing the dimension of shape variation significantly a two level statistical model was built on these momentum maps to assess anatomical connectivity among the subcortical structures on the basis of similar surface deformation compression or expansion results in the study of healthy aging on the hippocampus amygdala network indicate the anatomical connectivity between the basolateral complex of the amygdala and the subiculum of the hippocampus on the basis of shape compression in healthy elders relative to young adults

spherical demons fast diffeomorphic landmark free surface registration we present the spherical demons algorithm for registering two spherical images by exploiting spherical vector spline interpolation theory we show that a large class of regularizors for the modified demons objective function can be efficiently approximated on the sphere using iterative smoothing based on one parameter subgroups of diffeomorphisms the resulting registration is diffeomorphic and fast the spherical demons algorithm can also be modified to register a given spherical image to a probabilistic atlas we demonstrate two variants of the algorithm corresponding to warping the atlas or warping the subject registration of a cortical surface mesh to an atlas mesh both with more than 160 k nodes requires less than 5 min when warping the atlas and less than 3 min when warping the subject on a xeon 3 2 ghz single processor machine this is comparable to the fastest nondiffeomorphic landmark free surface registration algorithms furthermore the accuracy of our method compares favorably to the popular freesurfer registration algorithm we validate the technique in two different applications that use registration to transfer segmentation labels onto a new image 1 parcellation of in vivo cortical surfaces and 2 brodmann area localization in ex vivo cortical surfaces

diffeomorphic metric surface mapping in subregion of the superior temporal gyrus this paper describes the application of large deformation diffeomorphic metric mapping to cortical surfaces based on the shape and geometric properties of subregions of the superior temporal gyrus in the human brain the anatomical surfaces of the cortex are represented as triangulated meshes the diffeomorphic matching algorithm is implemented by defining a norm between the triangulated meshes based on the algorithms of vaillant and glaunes the diffeomorphic correspondence is defined as a flow of the extrinsic three dimensional coordinates containing the cortical surface that registers the initial and target geometry by minimizing the norm the methods are demonstrated in 40 high resolution mri cortical surfaces of planum temporale pt constructed from subsets of the superior temporal gyrus stg the effectiveness of the algorithm is demonstrated via the euclidean positional distance distance of normal vectors and curvature before and after the surface matching as well as the comparison with a landmark matching algorithm the results demonstrate that both the positional and shape variability of the anatomical configurations are being represented by the diffeomorphic maps

a fictitious domain method for dirichlet problem and applications abstract in this article we discuss the solution of the dirichlet problem for a class of elliptic operators by a lagrange multiplier fictitious domain method this approach allows the use of regular grids and therefore of fast specialized solvers for problems on complicated geometries the resulting saddle point system can be solved by an uzawa conjugate gradient algorithm in the case of two dimensional problems a quasi optimal preconditioner has been found by fourier analysis and numerical experiments confirm its nice scaling properties the resulting methodology is applied to a nonlinear time dependent problem namely the flow of a viscous plastic medium in a cylindrical pipe showing the potential of this methodology for some classes of nonlinear problems

kinetic theory approach to the sk spin glass model with glauber dynamics a new method to analyse glauber dynamics of the sherrington kirkpatrick sk spin glass model is presented the method is based on ideas used in the classical kinetic theory of fluids it is applied to study spin correlations in the high temperature phase of the sk model at zero external field the zeroth order theory is equivalent to a disorder dependent local equilibrium approximation its predictions agree well with computer simulation results the first order theory involves coupled evolution equations for the spin correlations and the dynamic excess parts of the local field distributions it qualitatively accounts for the error made in the zeroth approximation

an operator splitting scheme with a distributed lagrange multiplier based fictitious domain method for wave propagation problems we propose a novel fictitious domain method based on a distributed lagrange multiplier technique for the solution of the time dependent problem of scattering by an obstacle we study discretizations that include a fully conforming approach as well as mixed finite element formulations utilizing the lowest order nedelec edge elements in 2d on rectangular grids we also present a symmetrized operator splitting scheme for the scattering problem which decouples the operator that propagates the wave from the operator that enforces the dirichlet condition on the boundary of an obstacle a new perfectly matched layer pml model is developed to model the unbounded problem of interest this model is based on a formulation of the wave equation as a system of first order equations and uses a change of variables approach that has been developed to construct pml s for maxwell s equations we present an analysis of our fictitious domain approach for a one dimensional wave problem based on calculations of reflection coefficients we demonstrate the advantages of our fictitious domain approach over the staircase approximation of the finite difference method we also demonstrate some important properties of the distributed multiplier approach that are not shared by a boundary multiplier fictitious domain approach for the same problem numerical results for two dimensional wave problems that validate the effectiveness of the different methods are presented

multi manifold diffeomorphic metric mapping for aligning cortical hemispheric surfaces cortical surface based analysis has been widely used in anatomical and functional studies because it is geometrically appropriate for the cortex one of the main challenges in the cortical surface based analysis is to optimize the alignment of the cortical hemispheric surfaces across individuals in this paper we introduce a multi manifold large deformation diffeomorphic metric mapping mm lddmm algorithm that allows simultaneously carrying the cortical hemispheric surface and its sulcal curves from one to the other through a flow of diffeomorphisms we present an algorithm based on recent derivation of a law of momentum conservation for the geodesics of diffeomorphic flow once a template is fixed the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point on curves and surfaces along the geodesic is completely determined by the momentum at the origin we solve for trajectories geodesics of the kinetic energy by computing its variation with respect to the initial momentum and by applying a gradient descent scheme the mm lddmm algorithm optimizes the initial momenta encoding the anatomical variation of each individual relative to a common coordinate system in a linear space which provides a natural scheme for shape deformation average and template or atlas generation we applied the mm lddmm algorithm for constructing the templates for the cortical surface and 14 sulcal curves of each hemisphere using a group of 40 subjects the estimated template shape reflects regions which are highly variable across these subjects compared with existing single manifold lddmm algorithms such as the lddmm curve mapping and the lddmm surface mapping the mm lddmm mapping provides better results in terms of surface to surface distances in five predefined regions

exact geodesics and shortest paths on polyhedral surfaces we present two algorithms for computing distances along convex and non convex polyhedral surfaces the first algorithm computes exact minimal geodesic distances and the second algorithm combines these distances to compute exact shortest path distances along the surface both algorithms have been extended to compute the exact minimal geodesic paths and shortest paths these algorithms have been implemented and validated on surfaces for which the correct solutions are known in order to verify the accuracy and to measure the run time performance which is cubic or less for each algorithm the exact distance computations carried out by these algorithms are feasible for large scale surfaces containing tens of thousands of vertices and are a necessary component of near isometric surface flattening methods that accurately transform curved manifolds into flat representations

a fictitious domain method for external incompressible viscous flow modeled by navier stokes equations abstract in this article we discuss a fictitious domain method for the numerical solutions of three dimensional elliptic problems with dirichlet boundary conditions and also of the navier stokes equations modeling incompressible viscous flow the methodology for the navier stokes equations described here takes a systematic advantage of time discretization by operator splitting in order to treat separately advection imbedding and incompressibility due to the decoupling fast elliptic solvers can be used to treat the incompressibility condition even if the original problem is taking place on a nonregular geometry the resulting methodology is applied to two dimensional unsteady external incompressible viscous flow problems and three dimensional stokes problems

on finite element domain imbedding methods finite element domain imbedding methods for the helmholtz equation with neumann and dirichlet boundary conditions on nonrectangular domains in two space dimensions are considered a survey and comparison of known methods are given and a number of improvements suggested results of numerical experiments are also presented

a capacitance matrix method for dirichlet problem on polygon region an efficient algorithm for the solution of linear equations arising in a finite element method for the dirichlet problem is given the cost of the algorithm is proportional ton 2log2 n n 1 h where the cost of solving the capacitance matrix equations isnlog2 n on regular grids andn 3 2log2 n on irregular ones

surface fluid registration of conformal representation application to detect disease burden and genetic influence on hippocampus in this paper we develop a new automated surface registration system based on surface conformal parameterization by holomorphic 1 forms inverse consistent surface fluid registration and multivariate tensor based morphometry mtbm first we conformally map a surface onto a planar rectangle space with holomorphic 1 forms second we compute surface conformal representation by combining its local conformal factor and mean curvature and linearly scale the dynamic range of the conformal representation to form the feature image of the surface third we align the feature image with a chosen template image via the fluid image registration algorithm which has been extended into the curvilinear coordinates to adjust for the distortion introduced by surface parameterization the inverse consistent image registration algorithm is also incorporated in the system to jointly estimate the forward and inverse transformations between the study and template images this alignment induces a corresponding deformation on the surface we tested the system on alzheimer s disease neuroimaging initiative adni baseline dataset to study ad symptoms on hippocampus in our system by modeling a hippocampus as a 3d parametric surface we nonlinearly registered each surface with a selected template surface then we used mtbm to analyze the morphometry difference between diagnostic groups experimental results show that the new system has better performance than two publicly available subcortical surface registration tools first and spharm we also analyzed the genetic influence of the apolipoprotein e 4 allele apoe4 which is considered as the most prevalent risk factor for ad our work successfully detected statistically significant difference between apoe4 carriers and non carriers in both patients of mild cognitive impairment mci and healthy control subjects the results show evidence that the apoe genotype may be associated with accelerated brain atrophy so that our work provides a new mri analysis tool that may help presymptomatic ad research

automated sulcal segmentation using watersheds on the cortical surface the human cortical surface is a highly complex folded structure sulci the spaces between the folds define location on the cortex and provide a parcellation into anatomically distinct areas a topic that has recently received increased attention is the segmentation of these sulci from magnetic resonance images with most work focusing on extracting either the sulcal spaces between the folds or curve representations of sulci unlike these methods we propose a technique that extracts actual regions of the cortical surface that surround sulci which we call sulcal regions the method is based on a watershed algorithm applied to a geodesic depth measure on the cortical surface a well known problem with the watershed algorithm is a tendency toward oversegmentation meaning that a single region is segmented as several pieces to address this problem we propose a postprocessing algorithm that merges appropriate segments from the watershed algorithm the sulcal regions are then manually labeled by simply selecting the appropriate regions with a mouse click and a preliminary study of sulcal depth is reported finally a scheme is presented for computing a complete parcellation of the cortical surface

irreducible memory function for dissipative stochastic systems with detailed balance abstract we show quite generally that the memory function for purely dissipative stochastic systems with detailed balance can be reduced further and can be expressed in terms of the irreducible memory function this generalizes the similar finding for the smoluchowski equation the irreducible memory function is intimately connected with the lifetime renormalization and is shown to provide a useful starting point for introducing approximations such as the mode coupling approximation

genus zero surface conformal mapping and its application to brain surface mapping we developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one forms for surfaces with or without boundaries gu and yau 2002 gu and yau 2003 for genus zero surfaces our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map in this paper we apply the algorithm to the cortical surface matching problem we use a mesh structure to represent the brain surface further constraints are added to ensure that the conformal map is unique empirical tests on magnetic resonance imaging mri data show that the mappings preserve angular relationships are stable in mris acquired at different times and are robust to differences in data triangulation and resolution compared with other brain surface conformal mapping algorithms our algorithm is more stable and has good extensibility