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

statistics on diffeomorphisms via tangent space representations in this paper we present a linear setting for statistical analysis of shape and an optimization approach based on a recent derivation of a conservation of momentum law 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 along the geodesic is completely determined by the momentum at the origin through geodesic shooting equations the space of initial momentum provides a linear representation of the nonlinear diffeomorphic shape space in which linear statistical analysis can be applied specializing to the landmark matching problem of computational anatomy we derive an algorithm for solving the variational problem with respect to the initial momentum and demonstrate principal component analysis pca in this setting with three dimensional face and hippocampus databases

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

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

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

principal component based diffeomorphic surface mapping we present a new diffeomorphic surface mapping algorithm under the framework of large deformation diffeomorphic metric mapping lddmm unlike existing lddmm approaches this new algorithm reduces the complexity of the estimation of diffeomorphic transformations by incorporating a shape prior in which a nonlinear diffeomorphic shape space is represented by a linear space of initial momenta of diffeomorphic geodesic flows from a fixed template in addition for the first time the diffeomorphic mapping is formulated within a decision theoretic scheme based on bayesian modeling in which an empirical shape prior is characterized by a low dimensional gaussian distribution on initial momentum this is achieved using principal component analysis pca to construct the eigenspace of the initial momentum a likelihood function is formulated as the conditional probability of observing surfaces given any particular value of the initial momentum which is modeled as a random field of vector valued measures characterizing the geometry of surfaces we define the diffeomorphic mapping as a problem that maximizes a posterior distribution of the initial momentum given observable surfaces over the eigenspace of the initial momentum we demonstrate the stability of the initial momentum eigenspace when altering training samples using a bootstrapping method we then validate the mapping accuracy and show robustness to outliers whose shape variation is not incorporated into the shape prior

technical communique robust adaptive control for nonlinear uncertain systems a direct robust adaptive control framework for nonlinear uncertain systems with constant linearly parameterized uncertainty and nonlinear state dependent uncertainty is developed the proposed framework is lyapunov based and guarantees partial asymptotic robust stability of the closed loop system that is asymptotic robust stability with respect to part of the closed loop system states associated with the plant finally a numerical example is provided to demonstrate the efficacy of the proposed approach

large deformation multiresolution diffeomorphic metric mapping for multiresolution cortical surfaces a coarse to fine approach brain surface registration is an important tool for characterizing cortical anatomical variations and understanding their roles in normal cortical development and psychiatric diseases however surface registration remains challenging due to complicated cortical anatomy and its large differences across individuals in this paper we propose a fast coarse to fine algorithm for surface registration by adapting the large diffeomorphic deformation metric mapping lddmm framework for surface mapping and show improvements in speed and accuracy via a multiresolution analysis of surface meshes and the construction of multiresolution diffeomorphic transformations the proposed method constructs a family of multiresolution meshes that are used as natural sparse priors of the cortical morphology at varying resolutions these meshes act as anchor points where the parameterization of multiresolution deformation vector fields can be supported allowing the construction of a bundle of multiresolution deformation fields each originating from a different resolution using a coarse to fine approach we show a potential reduction in computation cost along with improvements in sulcal alignment when compared with lddmm surface mapping

mixed feedforward feedback based adaptive fuzzy control for a class of mimo nonlinear systems this paper proposes a mixed feedforward feedback ffb based adaptive fuzzy controller design for a class of multiple input multiple output mimo uncertain nonlinear systems by integrating both feedforward and feedback compensation we introduce the ffb based fuzzy controller composed of a feedforward fuzzy compensator and a robust error feedback compensator to achieve a forward compensation of uncertainties the feedforward fuzzy compensator takes the desired commands as premise variables of fuzzy rules and adaptively adjusts the consequent part from an error measure meanwhile the feedback controller part is constructed based on hinfin control techniques and nonlinear damping design then the attenuation of both disturbances and estimated fuzzy parametric errors is guaranteed from a linear matrix inequality lmi based gain design the main advantages are i a simpler architecture for implementation is provided and ii the typical boundedness of assumption on fuzzy universal approximation errors is not required finally an inverted pendulum system and a two link robot are taken as application examples to show the expected performance

applying tensor based morphometry to parametric surfaces can improve mri based disease diagnosis many methods have been proposed for computer assisted diagnostic classification full tensor information and machine learning with 3d maps derived from brain images may help detect subtle differences or classify subjects into different groups here we develop a new approach to apply tensor based morphometry to parametric surface models for diagnostic classification we use this approach to identify cortical surface features for use in diagnostic classifiers first with holomorphic 1 forms we compute an efficient and accurate conformal mapping from a multiply connected mesh to the so called slit domain next the surface parameterization approach provides a natural way to register anatomical surfaces across subjects using a constrained harmonic map to analyze anatomical differences we then analyze the full riemannian surface metric tensors which retain multivariate information on local surface geometry as the number of voxels in a 3d image is large sparse learning is a promising method to select a subset of imaging features and to improve classification accuracy focusing on vertices with greatest effect sizes we train a diagnostic classifier using the surface features selected by an l1 norm based sparse learning method stability selection is applied to validate the selected feature sets we tested the algorithm on mri derived cortical surfaces from 42 subjects with genetically confirmed williams syndrome and 40 age matched controls multivariate statistics on the local tensors gave greater effect sizes for detecting group differences relative to other tbm based statistics including analysis of the jacobian determinant and the largest eigenvalue of the surface metric our method also gave reasonable classification results relative to the jacobian determinant the pair of eigenvalues of the jacobian matrix and volume features this analysis pipeline may boost the power of morphometry studies and may assist with image based classification

geodesic estimation for large deformation anatomical shape averaging and interpolation the goal of this research is to promote variational methods for anatomical averaging that operate within the space of the underlying image registration problem this approach is effective when using the large deformation viscous framework where linear averaging is not valid or in the elastic case the theory behind this novel atlas building algorithm is similar to the traditional pairwise registration problem but with single image forces replaced by average forces these group forces drive an average transport ordinary differential equation allowing one to estimate the geodesic that moves an image toward the mean shape configuration this model gives large deformation atlases that are optimal with respect to the shape manifold as defined by the data and the image registration assumptions we use the techniques in the large deformation context here but they also pertain to small deformation atlas construction furthermore a natural inherently inverse consistent image registration is gained for free as is a tool for constant arc length geodesic shape interpolation the geodesic atlas creation algorithm is quantitatively compared to the euclidean anatomical average to elucidate the need for optimized atlases the procedures generate improved average representations of highly variable anatomy from distinct populations

diffeomorphic brain registration under exhaustive sulcal constraints the alignment and normalization of individual brain structures is a prerequisite for group level analyses of structural and functional neuroimaging data the techniques currently available are either based on volume and or surface attributes with limited insight regarding the consistent alignment of anatomical landmarks across individuals this article details a global geometric approach that performs the alignment of the exhaustive sulcal imprints cortical folding patterns across individuals this diffeomorphic sulcal based cortical disco technique proceeds to the automatic extraction identification and simplification of sulcal features from t1 weighted magnetic resonance image mri series these features are then used as control measures for fully 3 d diffeomorphic deformations quantitative and qualitative evaluations show that disco correctly aligns the sulcal folds and gray and white matter volumes across individuals the comparison with a recent iconic diffeomorphic approach dartel highlights how the absence of explicit cortical landmarks may lead to the misalignment of cortical sulci we also feature disco in the automatic design of an empirical sulcal template from group data we also demonstrate how disco can efficiently be combined with an image based deformation dartel to further improve the consistency and accuracy of alignment performances finally we illustrate how the optimized alignment of cortical folds across subjects improves sensitivity in the detection of functional activations in a group level analysis of neuroimaging data

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

whole brain diffeomorphic metric mapping via integration of sulcal and gyral curves cortical surfaces and images this paper introduces a novel large deformation diffeomorphic metric mapping algorithm for whole brain registration where sulcal and gyral curves cortical surfaces and intensity images are simultaneously carried from one subject to another through a flow of diffeomorphisms to the best of our knowledge this is the first time that the diffeomorphic metric from one brain to another is derived in a shape space of intensity images and point sets such as curves and surfaces in a unified manner we describe the euler lagrange equation associated with this algorithm with respect to momentum a linear transformation of the velocity vector field of the diffeomorphic flow the numerical implementation for solving this variational problem which involves large scale kernel convolution in an irregular grid is made feasible by introducing a class of computationally friendly kernels we apply this algorithm to align magnetic resonance brain data our whole brain mapping results show that our algorithm outperforms the image based lddmm algorithm in terms of the mapping accuracy of gyral sulcal curves sulcal regions and cortical and subcortical segmentation moreover our algorithm provides better whole brain alignment than combined volumetric and surface registration postelnicu et al 2009 and hierarchical attribute matching mechanism for elastic registration hammer shen and davatzikos 2002 in terms of cortical and subcortical volume segmentation

large deformation diffeomorphic metric curve mapping we present a matching criterion for curves and integrate it into the large deformation diffeomorphic metric mapping lddmm scheme for computing an optimal transformation between two curves embedded in euclidean space d curves are first represented as vector valued measures which incorporate both location and the first order geometric structure of the curves then a hilbert space structure is imposed on the measures to build the norm for quantifying the closeness between two curves we describe a discretized version of this in which discrete sequences of points along the curve are represented by vector valued functionals this gives a convenient and practical way to define a matching functional for curves we derive and implement the curve matching in the large deformation framework and demonstrate mapping results of curves in 2 and 3 behaviors of the curve mapping are discussed using 2d curves the applications to shape classification is shown and experiments with 3d curves extracted from brain cortical surfaces are presented

design of a parallel distributed fuzzy lqr controller for the twin rotor multi input multi output system in this paper a helicopter like twin rotor multi input multi output system trms is decoupled and is fuzzy takagi sugeno modeled with the complex nonlinear functions simplified as the propositional combination of linear functions the design procedures of the fuzzy takagi sugeno model of trms are detailed based on the derived fuzzy takagi sugeno model parallel distributed fuzzy lqr controller are designed to control the positions of the pitch and yaw angles in trms the stability of the trms system with the proposed fuzzy controllers is discussed moreover simulation results are included to indicate the effectiveness of the presented parallel distributed fuzzy lqr controllers for the trms

synthesized design of a fuzzy logic controller for an underactuated unicycle in this paper we propose synthesized design of a fuzzy logic controller flc for control of an underactuated unicycle system the flc objective is velocity control of the wheel while keeping the pendulum upright which is an unstable equilibrium the synthesized design consists of three phases first the flc structures including the number of rules membership functions inference and parametric relations are chosen based on heuristic knowledge about the unicycle second on the basis of a linearized model and linear feedback the flc output parameters are determined quantitatively for stabilization of the unicycle third the flc output parameters are tuned using an iterative learning tuning ilt algorithm which minimizes an objective function that specifies the desired unicycle performance the rationale for the synthesized flc design is full utilization of the available information which is achieved by combining model based and model free designs and hence improves the flc performance we minimize the number of flc rules and fuzzy labels six rules are used for regulation or setpoint tasks whereas 10 rules are used with extra integral control to eliminate steady state errors induced by system uncertainties and disturbances only two fuzzy labels are adopted for each fuzzy variable the ilt process consists of two phases exploration for stabilization and exploitation for better performance the effectiveness of the proposed flc is validated using intensive simulations and comparisons

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

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

an approach to fuzzy control of nonlinear systems stability and design issues presents a design methodology for stabilization of a class of nonlinear systems first the authors represent a nonlinear plant with a takagi sugeno fuzzy model then a model based fuzzy controller design utilizing the concept of the so called parallel distributed compensation is employed the main idea of the controller design is to derive each control rule so as to compensate each rule of a fuzzy system the design procedure is conceptually simple and natural moreover the stability analysis and control design problems can be reduced to linear matrix inequality lmi problems therefore they can be solved efficiently in practice by convex programming techniques for lmis the design methodology is illustrated by application to the problem of balancing and swing up of an inverted pendulum on a cart

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

analysis and design of fuzzy controller and fuzzy observer this paper addresses the analysis and design of a fuzzy controller and a fuzzy observer on the basis of the takagi sugeno t s fuzzy model the main contribution of the paper is the development of the separation property that is the fuzzy controller and the fuzzy observer can be independently designed a numerical simulation and an experiment on an inverted pendulum system are described to illustrate the performance of the fuzzy controller and the fuzzy observer

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

brain surface conformal parameterization with the ricci flow in brain mapping research parameterized 3 d surface models are of great interest for statistical comparisons of anatomy surface based registration and signal processing here we introduce the theories of continuous and discrete surface ricci flow which can create riemannian metrics on surfaces with arbitrary topologies with user defined gaussian curvatures the resulting conformal parameterizations have no singularities and they are intrinsic and stable first we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves secondly we conformally parameterize each cortical surface to a parameter domain with a user designed gaussian curvature arrangement in the parameter domain a shape index based on conformal invariants is computed and inter subject cortical surface matching is performed by solving a constrained harmonic map we illustrate various target curvature arrangements and demonstrate the stability of the method using longitudinal data to map statistical differences in cortical morphometry we studied brain asymmetry in 14 healthy control subjects we used a manifold version of hotelling s t2 test applied to the jacobian matrices of the surface parameterizations a permutation test along with the cumulative distribution of p values were used to estimate the overall statistical significance of differences the results show our algorithm s power to detect subtle group differences in cortical surfaces

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

fuzzy regulators and fuzzy observers relaxed stability conditions and lmi based designs this paper presents new relaxed stability conditions and lmi linear matrix inequality based designs for both continuous and discrete fuzzy control systems they are applied to design problems of fuzzy regulators and fuzzy observers first takagi and sugeno s fuzzy models and some stability results are recalled to design fuzzy regulators and fuzzy observers nonlinear systems are represented by takagi sugeno s ts fuzzy models the concept of parallel distributed compensation is employed to design fuzzy regulators and fuzzy observers from the ts fuzzy models new stability conditions are obtained by relaxing the stability conditions derived in previous papers lmi based design procedures for fuzzy regulators and fuzzy observers are constructed using the parallel distributed compensation and the relaxed stability conditions other lmi s with respect to decay rate and constraints on control input and output are also derived and utilized in the design procedures design examples for nonlinear systems demonstrate the utility of the relaxed stability conditions and the lmi based design procedures

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

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

locally linear diffeomorphic metric embedding lldme for surface based anatomical shape modeling this paper presents the algorithm locally linear diffeomorphic metric embedding lldme for constructing efficient and compact representations of surface based brain shapes whose variations are characterized using large deformation diffeomorphic metric mapping lddmm our hypothesis is that the shape variations in the infinite dimensional diffeomorphic metric space can be captured by a low dimensional space to do so traditional locally linear embedding lle that reconstructs a data point from its neighbors in euclidean space is extended to lldme that requires interpolating a shape from its neighbors in the infinite dimensional diffeomorphic metric space this is made possible through the conservation law of momentum derived from lddmm it indicates that initial momentum a linear transformation of the initial velocity of diffeomorphic flows at a fixed template shape determines the geodesic connecting the template to a subject s shape in the diffeomorphic metric space and becomes the shape signature of an individual subject this leads to the compact linear representation of the nonlinear diffeomorphisms in terms of the initial momentum since the initial momentum is in a linear space a shape can be approximated by a linear combination of its neighbors in the diffeomorphic metric space in addition we provide efficient computations for the metric distance between two shapes through the first order approximation of the geodesic using the initial momentum as well as for the reconstruction of a shape given its low dimensional euclidean coordinates using the geodesic shooting with the initial momentum as the initial condition experiments are performed on the hippocampal shapes of 302 normal subjects across the whole life span 18 94years compared with principal component analysis and isomap lldme provides the most compact and efficient representation of the age related hippocampal shapes even though the hippocampal volumes among young adults are as variable as those in older adults lldme disentangles the hippocampal local shape variation from the hippocampal size and thus reveals the nonlinear relationship of the hippocampal morphometry with age

stability analysis and design of fuzzy control systems abstract the stability analysis and the design technique of fuzzy control systems using fuzzy block diagrams are discussed first we show the concept of fuzzy blocks and consider the connection problems of fuzzy blocks diagrams we derive some theorems and corollaries with respect to two basic types of connections of fuzzy blocks in order to preserve some properties in a connection of fuzzy blocks continuous piecewise polynomial membership functions are defined secondly a sufficient condition which guarantees the stability of fuzzy systems is obtained in terms of lyapunov s direct method we give an important fact based on this condition thirdly we propose a new design technique of a fuzzy controller the fuzzy block diagrams and the stability analysis are applied to the design problems of a model based fuzzy controller

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

optimal mass transport for shape matching and comparison surface based 3d shape analysis plays a fundamental role in computer vision and medical imaging this work proposes to use optimal mass transport map for shape matching and comparison focusing on two important applications including surface registration and shape space the computation of the optimal mass transport map is based on monge brenier theory in comparison to the conventional method based on monge kantorovich theory this method significantly improves the efficiency by reducing computational complexity from o n 2 to o n for surface registration problem one commonly used approach is to use conformal map to convert the shapes into some canonical space although conformal mappings have small angle distortions they may introduce large area distortions which are likely to cause numerical instability thus resulting failures of shape analysis this work proposes to compose the conformal map with the optimal mass transport map to get the unique area preserving map which is intrinsic to the riemannian metric unique and diffeomorphic for shape space study this work introduces a novel riemannian framework conformal wasserstein shape space by combing conformal geometry and optimal mass transport theory in our work all metric surfaces with the disk topology are mapped to the unit planar disk by a conformal mapping which pushes the area element on the surface to a probability measure on the disk the optimal mass transport provides a map from the shape space of all topological disks with metrics to the wasserstein space of the disk and the pullback wasserstein metric equips the shape space with a riemannian metric we validate our work by numerous experiments and comparisons with prior approaches and the experimental results demonstrate the efficiency and efficacy of our proposed approach

the partition function and hecke operators abstract the theory of congruences for the partition function p n depends heavily on the properties of half integral weight hecke operators the subject has been complicated by the absence of closed formulas for the hecke images p z t l 2 where p z is the relevant modular generating function we obtain such formulas using euler s pentagonal number theorem and the denominator formula for the monster lie algebra as a corollary we obtain congruences for certain powers of ramanujan s delta function

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

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

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