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

an error entropy minimization algorithm for supervised training of nonlinear adaptive systems the paper investigates error entropy minimization in adaptive systems training we prove the equivalence between minimization of error s renyi 1970 entropy of order spl alpha and minimization of a csiszar 1981 distance measure between the densities of desired and system outputs a nonparametric estimator for renyi s entropy is presented and it is shown that the global minimum of this estimator is the same as the actual entropy the performance of the error entropy minimization criterion is compared with mean square error minimization in the short term prediction of a chaotic time series and in nonlinear system identification

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

stochastic blind equalization based on pdf fitting using parzen estimator this work presents a new blind equalization approach that aims to force the probability density function pdf at the equalizer output to match the known constellation pdf quadratic distance between pdf s is used as the cost function to be minimized the proposed method relies on the parzen window method to estimate the data pdf and is implemented by a stochastic gradient descent algorithm the kernel size of the parzen estimator allows a dual mode switch or a soft switch between blind and decision directed equalization the proposed method converges faster than the constant modulus algorithm cma working at the symbol rate with a similar computational burden and reduces the residual error of the cma in multilevel modulations at the same time a comparison with the most common blind techniques is presented

robust surface reconstruction via laplace beltrami eigen projection and boundary deformation in medical shape analysis a critical problem is reconstructing a smooth surface of correct topology from a binary mask that typically has spurious features due to segmentation artifacts the challenge is the robust removal of these outliers without affecting the accuracy of other parts of the boundary in this paper we propose a novel approach for this problem based on the laplace beltrami lb eigen projection and properly designed boundary deformations using the metric distortion during the lb eigen projection our method automatically detects the location of outliers and feeds this information to a well composed and topology preserving deformation by iterating between these two steps of outlier detection and boundary deformation we can robustly filter out the outliers without moving the smooth part of the boundary the final surface is the eigen projection of the filtered mask boundary that has the correct topology desired accuracy and smoothness in our experiments we illustrate the robustness of our method on different input masks of the same structure and compare with the popular spharm tool and the topology preserving level set method to show that our method can reconstruct accurate surface representations without introducing artificial oscillations we also successfully validate our method on a large data set of more than 900 hippocampal masks and demonstrate that the reconstructed surfaces retain volume information accurately

convergence analysis of self adaptive equalizers a theoretical analysis of self adaptive equalization for data transmission is carried out starting from known convergence results for the corresponding trained adaptive filter the development relies on a suitable ergodicity model for the sequence of observations at the output of the transmission channel thanks to the boundedness of the decision function used for data recovery it can be proved that the algorithm is bounded strong convergence results can be reached when a perfect noiseless equalizer exists the algorithm will converge to it if the eye pattern is initially open otherwise convergence may take place towards certain other stationary points of the algorithm for which domains of attraction have been defined some of them will result in a poor error rate the case of a noisy channel exhibits limit points for the algorithm that differ from those of the classical trained algorithm the stronger the noise the greater the difference is one of the principal results of this study is the proof of the stability of the usual decision feedback algorithms once the learning period is over

robust blind learning algorithm for nonlinear equalization using input decision information in this paper we propose a new blind learning algorithm namely the benveniste goursat input output decision bg iod to enhance the convergence performance of neural network based equalizers for nonlinear channel equalization in contrast to conventional blind learning algorithms where only the output of the equalizer is employed for updating system parameters the bg iod exploits a new type of extra information the input decision information obtained from the input of the equalizer to mitigate the influence of the nonlinear equalizer structure on parameters learning thereby leading to improved convergence performance we prove that with the input decision information a desirable convergence capability that the output symbol error rate ser is always less than the input ser if the input ser is below a threshold can be achieved then the bg soft switching technique is employed to combine the merits of both input and output decision information where the former is used to guarantee ser convergence and the latter is to improve ser performance simulation results show that the proposed algorithm outperforms conventional blind learning algorithms such as stochastic quadratic distance and dual mode constant modulus algorithm in terms of both convergence performance and ser performance for nonlinear equalization

generalized set valued variational inclusions in banach spaces in this paper a generalization and improvement of noor s theorem for generalized set valued variational inclusion in real banach spaces is studied the results presented in this paper generalize improve and unify a number of recent results

automated 3 d extraction of inner and outer surfaces of cerebral cortex from mri automatic computer processing of large multidimensional images such as those produced by magnetic resonance imaging mri is greatly aided by deformable models which are used to extract identify and quantify specific neuroanatomic structures a general method of deforming polyhedra is presented here with two novel features first explicit prevention of self intersecting surface geometries is provided unlike conventional deformable models which use regularization constraints to discourage but not necessarily prevent such behavior second deformation of multiple surfaces with intersurface proximity constraints allows each surface to help guide other surfaces into place using model based constraints such as expected thickness of an anatomic surface these two features are used advantageously to identify automatically the total surface of the outer and inner boundaries of cerebral cortical gray matter from normal human mr images accurately locating the depths of the sulci even where noise and partial volume artifacts in the image obscure the visibility of sulci the extracted surfaces are enforced to be simple two dimensional manifolds having the topology of a sphere even though the data may have topological holes this automatic 3 d cortex segmentation technique has been applied to 150 normal subjects simultaneously extracting both the gray white and gray cerebrospinal fluid interface from each individual the collection of surfaces has been used to create a spatial map of the mean and standard deviation for the location and the thickness of cortical gray matter three alternative criteria for defining cortical thickness at each cortical location were developed and compared these results are shown to corroborate published postmortem and in vivo measurements of cortical thickness

segmentation of brain mr images through a hidden markov random field model and the expectation maximization algorithm the finite mixture fm model is the most commonly used model for statistical segmentation of brain magnetic resonance mr images because of its simple mathematical form and the piecewise constant nature of ideal brain mr images however being a histogram based model the fm has an intrinsic limitation no spatial information is taken into account this causes the fm model to work only on well defined images with low levels of noise unfortunately this is often not the the case due to artifacts such as partial volume effect and bias field distortion under these conditions fm model based methods produce unreliable results here the authors propose a novel hidden markov random field hmrf model which is a stochastic process generated by a mrf whose state sequence cannot be observed directly but which can be indirectly estimated through observations mathematically it can be shown that the fm model is a degenerate version of the hmrf model the advantage of the hmrf model derives from the way in which the spatial information is encoded through the mutual influences of neighboring sites although mrf modeling has been employed in mr image segmentation by other researchers most reported methods are limited to using mrf as a general prior in an fm model based approach to fit the hmrf model an em algorithm is used the authors show that by incorporating both the hmrf model and the em algorithm into a hmrf em framework an accurate and robust segmentation can be achieved more importantly the hmrf em framework can easily be combined with other techniques as an example the authors show how the bias field correction algorithm of guillemaud and brady 1997 can be incorporated into this framework to achieve a three dimensional fully automated approach for brain mr image segmentation

dual mode type algorithms for blind equalization adaptive channel equalization accomplished without resorting to a training sequence is known as blind equalization the godard algorithm and the generalized sato algorithm are two widely referenced algorithms for blind equalization of a qam system these algorithms exhibit very slow convergence rates when compared to algorithms employed in conventional data aided equalization schemes in order to speed up the convergence process these algorithms may be switched over to a decision directed equalization scheme once the error level is reasonably low the authors present a scheme which is capable of operating in two modes blind equalization mode and a mode similar to the decision directed equalization mode in this proposed scheme the dominant mode of operation changes from the blind equalization mode at higher error levels to the mode similar to the decision directed equalization mode at lower error levels manual switch over to the decision directed mode from the blind equalization mode or vice versa is not necessary since transitions between the two modes take place smoothly and automatically

on the almost sure rate of convergence of linear stochastic approximation algorithms the almost sure rate of convergence of linear stochastic approximation algorithms is analyzed as the main result it is demonstrated that their almost sure rate of convergence is equivalent to the almost sure rate of convergence of the averages of their input data sequences as opposed to most of the existing results on the rate of convergence of stochastic approximation which cover only algorithms with the noise decomposable as the sum of a martingale difference vanishing and telescoping sequence the main results of this paper hold under assumptions not requiring the input data sequences to admit any particular decomposition although no decomposition of the input data sequences is required the results on the almost sure rate of convergence of linear stochastic approximation algorithms obtained in this correspondence are as tight as the rate of convergence in the law of iterated logarithm moreover the main result of this correspondence yields the law of iterated logarithm for linear stochastic approximation if the law of iterated logarithm holds for the input data sequences the obtained general results are illustrated with two nontrivial examples where the input data sequences are strongly mixing strictly stationary random processes or functions of a uniformly ergodic markov chain these results are also applied to the analysis of least mean square lms algorithms

distance weighted discrimination high dimension low sample size statistical analysis is becoming increasingly important in a wide range of applied contexts in such situations the popular support vector machine suffers from data piling at the margin which can diminish generalizability this leads naturally to the development of distance weighted discrimination which is based on second order cone programming a modern computationally intensive optimization method

the resolution of the gibbs phenomenon for spherical harmonics spherical harmonics have been important tools for solving geophysical and astrophysical problems methods have been developed to effectively implement spherical harmonic expansion approximations however the gibbs phenomenon was already observed by weyl for spherical harmonic expansion approximations to functions with discontinuities causing undesirable oscillations over the entire sphere recently methods for removing the gibbs phenomenon for one dimensional discontinuous functions have been successfully developed by gottlieb and shu they proved that the knowledge of the first n expansion coefficients either fourier or gegenbauer of a piecewise analytic function f x is enough to recover an exponentially convergent approximation to the point values of f x in any subinterval in which the function is analytic here we take a similar approach proving that knowledge of the first n spherical harmonic coefficients yield an exponentially convergent approximation to a spherical piecewise smooth function f in any subinterval 1 2 0 2 where the function is analytic thus we entirely overcome the gibbs phenomenon

a theoretical characterization of nonlinear distortion effects in ofdm systems this paper presents a theoretical characterization of nonlinear distortion effects in orthogonal frequency division multiplexing ofdm transmission systems in the theoretical framework developed it is shown that the effects on the decision variables of the in band distortion introduced by a bandpass memoryless nonlinearity can be described by means of a complex gain and an additive gaussian term with zero mean and suitable variance analytical expressions for gain and variance are given the conditions which allow this description are emphasized and discussed as a consequence a completely analytical procedure to evaluate error probability is also obtained and illustrated using ofdm discrete multitone modulation dmt systems with rectangular pulse shaping for the soft envelope limiter nonlinearity a closed form is derived a comparison with simulation results is carried out to verify the accuracy of this method

estimating linear cortical magnification in human primary visual cortex via dynamic programming abstract human primary visual cortex is organized retinotopically with adjacent locations in cortex representing adjacent locations on the retina the spatial sampling in cortex is highly nonuniform the amount of cortex devoted to a unit area of retina decreases with increasing retinal eccentricity this sampling property can be quantified by the linear cortical magnification factor which is expressed in terms of millimeters of cortex per degree of visual angle in this paper we present a new method using dynamic programming and fmri retinotopic eccentricity mapping to estimate the linear cortical magnification factor in human primary visual cortex v1 we localized cortical activity while subjects viewed each of seven stationary contrast reversing radial checkerboard rings of equal thickness that tiled the visual field from 1 62 to 12 96 degrees of eccentricity imaging data from all epochs of each ring were contrasted with data from fixation epochs on a subject by subject basis the resulting t statistic maps were then superimposed on a local coordinate system constructed from the gray white matter boundary surface of each individual subject s occipital lobe separately for each ring smoothed maps of functional activity on the cortical surface were constructed using orthonormal bases of the laplace beltrami operator that incorporate the geometry of the cortical surface this allowed us to stably track the ridge of maximum activation due to each ring via dynamic programming optimization over all possible paths on the cortical surface we estimated the linear cortical magnification factor by calculating geodesic distances between activation ridges on the cortical surface in a population of five normal subjects the reliability of these estimates was assessed by comparing results based on data from one quadrant to those based on data from the full hemifield along with a split half reliability analysis

hyperbolic harmonic mapping for surface registration automatic computation of surface correspondence via harmonic map is an active research field in computer vision computer graphics and computational geometry it may help document and understand physical and biological phenomena and also has broad applications in biometrics medical imaging and motion capture industries although numerous studies have been devoted to harmonic map research limited progress has been made to compute a diffeomorphic harmonic map on general topology surfaces with landmark constraints this work conquers this problem by changing the riemannian metric on the target surface to a hyperbolic metric so that the harmonic mapping is guaranteed to be a diffeomorphism under landmark constraints the computational algorithms are based on ricci flow and nonlinear heat diffusion methods the approach is general and robust we employ our algorithm to study the constrained surface registration problem which applies to both computer vision and medical imaging applications experimental results demonstrate that by changing the riemannian metric the registrations are always diffeomorphic and achieve relatively high performance when evaluated with some popular surface registration evaluation standards

metric optimization for surface analysis in the laplace beltrami embedding space in this paper we present a novel approach for the intrinsic mapping of anatomical surfaces and its application in brain mapping research using the laplace beltrami eigen system we represent each surface with an isometry invariant embedding in a high dimensional space the key idea in our system is that we realize surface deformation in the embedding space via the iterative optimization of a conformal metric without explicitly perturbing the surface or its embedding by minimizing a distance measure in the embedding space with metric optimization our method generates a conformal map directly between surfaces with highly uniform metric distortion and the ability of aligning salient geometric features besides pairwise surface maps we also extend the metric optimization approach for group wise atlas construction and multi atlas cortical label fusion in experimental results we demonstrate the robustness and generality of our method by applying it to map both cortical and hippocampal surfaces in population studies for cortical labeling our method achieves excellent performance in a cross validation experiment with 40 manually labeled surfaces and successfully models localized brain development in a pediatric study of 80 subjects for hippocampal mapping our method produces much more significant results than two popular tools on a multiple sclerosis study of 109 subjects

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

multiscale 3 d shape representation and segmentation using spherical wavelets this paper presents a novel multiscale shape representation and segmentation algorithm based on the spherical wavelet transform this work is motivated by the need to compactly and accurately encode variations at multiple scales in the shape representation in order to drive the segmentation and shape analysis of deep brain structures such as the caudate nucleus or the hippocampus our proposed shape representation can be optimized to compactly encode shape variations in a population at the needed scale and spatial locations enabling the construction of more descriptive nonglobal nonuniform shape probability priors to be included in the segmentation and shape analysis framework in particular this representation addresses the shortcomings of techniques that learn a global shape prior at a single scale of analysis and cannot represent fine local variations in a population of shapes in the presence of a limited dataset specifically our technique defines a multiscale parametric model of surfaces belonging to the same population using a compact set of spherical wavelets targeted to that population we further refine the shape representation by separating into groups wavelet coefficients that describe independent global and or local biological variations in the population using spectral graph partitioning we then learn a prior probability distribution induced over each group to explicitly encode these variations at different scales and spatial locations based on this representation we derive a parametric active surface evolution using the multiscale prior coefficients as parameters for our optimization procedure to naturally include the prior for segmentation additionally the optimization method can be applied in a coarse to fine manner we apply our algorithm to two different brain structures the caudate nucleus and the hippocampus of interest in the study of schizophrenia we show 1 a reconstruction task of a test set to validate the expressiveness of our multiscale prior and 2 a segmentation task in the reconstruction task our results show that for a given training set size our algorithm significantly improves the approximation of shapes in a testing set over the point distribution model which tends to oversmooth data in the segmentation task our validation shows our algorithm is computationally efficient and outperforms the active shape model algorithm by capturing finer shape details

towards a holistic cortical thickness descriptor heat kernel based grey matter morphology signatures in this paper we propose a heat kernel based regional shape descriptor that may be capable of better exploiting volumetric morphological information than other available methods thereby improving statistical power on brain magnetic resonance imaging mri analysis the mechanism of our analysis is driven by the graph spectrum and the heat kernel theory to capture the volumetric geometry information in the constructed tetrahedral meshes in order to capture profound brain grey matter shape changes we first use the volumetric laplace beltrami operator to determine the point pair correspondence between white grey matter and csf grey matter boundary surfaces by computing the streamlines in a tetrahedral mesh secondly we propose multi scale grey matter morphology signatures to describe the transition probability by random walk between the point pairs which reflects the inherent geometric characteristics thirdly a point distribution model is applied to reduce the dimensionality of the grey matter morphology signatures and generate the internal structure features with the sparse linear discriminant analysis we select a concise morphology feature set with improved classification accuracies in our experiments the proposed work outperformed the cortical thickness features computed by freesurfer software in the classification of alzheimer s disease and its prodromal stage i e mild cognitive impairment on publicly available data from the alzheimer s disease neuroimaging initiative the multi scale and physics based volumetric structure feature may bring stronger statistical power than some traditional methods for mri based grey matter morphology analysis

focusr feature oriented correspondence using spectral regularization a method for precise surface matching existing methods for surface matching are limited by the tradeoff between precision and computational efficiency here we present an improved algorithm for dense vertex to vertex correspondence that uses direct matching of features defined on a surface and improves it by using spectral correspondence as a regularization this algorithm has the speed of both feature matching and spectral matching while exhibiting greatly improved precision distance errors of 1 4 percent the method focusr incorporates implicitly such additional features to calculate the correspondence and relies on the smoothness of the lowest frequency harmonics of a graph laplacian to spatially regularize the features in its simplest form focusr is an improved spectral correspondence method that nonrigidly deforms spectral embeddings we provide here a full realization of spectral correspondence where virtually any feature can be used as an additional information using weights on graph edges but also on graph nodes and as extra embedded coordinates as an example the full power of focusr is demonstrated in a real case scenario with the challenging task of brain surface matching across several individuals our results show that combining features and regularizing them in a spectral embedding greatly improves the matching precision to a submillimeter level while performing at much greater speed than existing methods

cortical surface shape analysis based on spherical wavelets in vivo quantification of neuroanatomical shape variations is possible due to recent advances in medical imaging and has proven useful in the study of neuropathology and neurodevelopment in this paper we apply a spherical wavelet transformation to extract shape features of cortical surfaces reconstructed from magnetic resonance images mris of a set of subjects the spherical wavelet transformation can characterize the underlying functions in a local fashion in both space and frequency in contrast to spherical harmonics that have a global basis set we perform principal component analysis pca on these wavelet shape features to study patterns of shape variation within normal population from coarse to fine resolution in addition we study the development of cortical folding in newborns using the gompertz model in the wavelet domain which allows us to characterize the order of development of large scale and finer folding patterns independently given a limited amount of training data we use a regularization framework to estimate the parameters of the gompertz model to improve the prediction performance on new data we develop an efficient method to estimate this regularized gompertz model based on the broyden fletcher goldfarb shannon bfgs approximation promising results are presented using both pca and the folding development model in the wavelet domain the cortical folding development model provides quantitative anatomic information regarding macroscopic cortical folding development and may be of potential use as a biomarker for early diagnosis of neurologic deficits in newborns

weighted fourier series representation and its application to quantifying the amount of gray matter we present a novel weighted fourier series wfs representation for cortical surfaces the wfs representation is a data smoothing technique that provides the explicit smooth functional estimation of unknown cortical boundary as a linear combination of basis functions the basic properties of the representation are investigated in connection with a self adjoint partial differential equation and the traditional spherical harmonic spharm representation to reduce steep computational requirements a new iterative residual fitting irf algorithm is developed its computational and numerical implementation issues are discussed in detail the computer codes are also available at http www stat wisc edu mchung softwares weighted spharm weighted spharm html as an illustration the wfs is applied in quantifying the amount of gray matter in a group of high functioning autistic subjects within the wfs framework cortical thickness and gray matter density are computed and compared

penalized discriminant analysis fisher s linear discriminant analysis lda is a popular data analytic tool for studying the relationship between a set of predictors and a categorical response in this paper we describe a penalized version of lda it is designed for situations in which there are many highly correlated predictors such as those obtained by discretizing a function or the grey scale values of the pixels in a series of images in cases such as these it is natural efficient and sometimes essential to impose a spatial smoothness constraint on the coefficients both for improved prediction performance and interpretability we cast the classification problem into a regression framework via optimal scoring using this our proposal facilitates the use of any penalized regression technique in the classification setting the technique is illustrated with examples in speech recognition and handwritten character recognition

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

sparse discriminant analysis we consider the problem of performing interpretable classification in the high dimensional setting in which the number of features is very large and the number of observations is limited this setting has been studied extensively in the chemometrics literature and more recently has become commonplace in biological and medical applications in this setting a traditional approach involves performing feature selection before classification we propose sparse discriminant analysis a method for performing linear discriminant analysis with a sparseness criterion imposed such that classification and feature selection are performed simultaneously sparse discriminant analysis is based on the optimal scoring interpretation of linear discriminant analysis and can be extended to perform sparse discrimination via mixtures of gaussians if boundaries between classes are nonlinear or if subgroups are present within each class our proposal also provides low dimensional views of the discriminative directions

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