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

compensating q effects in viscoelastic media by adjoint based least squares reverse time migration low values of p and s wave quality factors qp and qs result in strong intrinsic seismic attenuation in both seismic modeling and imaging we use a linearized waveform inversion approach by generalizing least squares reverse time migration lsrtm for viscoelastic media q lsrtm to compensate the attenuation loss for both p and s images we use the first order particle velocity stress and memory variable equations with explicit q in the formulations based on the generalized standard linear solid gsls as the forward modeling operator the linearized two way viscoelastic modeling operator is obtained with modulus perturbations introduced for the relaxed p and s moduli the viscoelastic adjoint operator and the p and s imaging conditions for modulus perturbations are derived using the adjoint state method and an augmented lagrangian functional q lsrtm solves the viscoelastic linearized modeling operator for generating synthetic data and the adjoint operator is used for back propagating the data re

multiscale adaptive regression models for neuroimaging data neuroimaging studies aim to analyze imaging data with complex spatial patterns in a large number of locations called voxels on a two dimensional 2d surface or in a 3d volume conventional analyses of imaging data include two sequential steps spatially smoothing imaging data and then independently fitting a statistical model at each voxel however conventional analyses suffer from the same amount of smoothing throughout the whole image the arbitrary choice of smoothing extent and low statistical power in detecting spatial patterns we propose a multiscale adaptive regression model marm to integrate the propagation separation ps approach polzehl and spokoiny 2000 2006 with statistical modeling at each voxel for spatial and adaptive analysis of neuroimaging data from multiple subjects marm has three features being spatial being hierarchical and being adaptive we use a multiscale adaptive estimation and testing procedure maet to utilize imaging observations from the neighboring voxels of the current voxel to adaptively calculate parameter estimates and test statistics theoretically we establish consistency and asymptotic normality of the adaptive parameter estimates and the asymptotic distribution of the adaptive test statistics our simulation studies and real data analysis confirm that marm significantly outperforms conventional analyses of imaging data

modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional laplacians we evaluated a time domain wave equation for modeling acoustic wave propagation in attenuating media the wave equation was derived from kjartansson s constant q constitutive stress strain relation in combination with the mass and momentum conservation equations our wave equation expressed by a second order temporal derivative and two fractional laplacian operators described very nearly constant q attenuation and dispersion effects the advantage of using our formulation of two fractional laplacians over the traditional fractional time derivative approach was the avoidance of time history memory variables and thus it offered more economic computations in numerical simulations we formulated the first order constitutive equations with the perfectly matched layer absorbing boundaries the temporal derivative was calculated with a staggered grid finite difference approach the fractional laplacians are calculated in the spatial frequency domain using a fourier pseudospectral implementation we validated our numerical results through comparisons with theoretical constant q attenuation and dispersion solutions field measurements from the pierre shale and results from 2d viscoacoustic analytical modeling for the homogeneous pierre shale we also evaluated different formulations to show separated amplitude loss and dispersion effects on wavefields furthermore we generalized our rigorous formulation for homogeneous media to an approximate equation for viscoacoustic waves in heterogeneous media we then investigated the accuracy of numerical modeling in attenuating media with different q values and its stability in largecontrast heterogeneous media finally we tested the applicability of our time domain formulation in a heterogeneous medium with high attenuation

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

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

a primal sketch of the cortex mean curvature a morphogenesis based approach to study the variability of the folding patterns in this paper we propose a new representation of the cortical surface that may be used to study the cortex folding process and to recover some putative stable anatomical landmarks called sulcal roots usually buried in the depth of adult brains this representation is a primal sketch derived from a scale space computed for the mean curvature of the cortical surface this scale space stems from a diffusion equation geodesic to the cortical surface the primal sketch is made up of objects defined from mean curvature minima and saddle points the resulting sketch aims first at highlighting significant elementary cortical folds second at representing the fold merging process during brain growth the relevance of the framework is illustrated by the study of central sulcus sulcal roots from antenatal to adult age some results are proposed for ten different brains some preliminary results are also provided for superior temporal sulcus

q compensated least squares reverse time migration using low rank one step wave extrapolation abstractattenuation of seismic waves needs to be taken into account to improve the accuracy of seismic imaging in viscoacoustic media reverse time migration rtm can be performed with q compensation which is also known as q rtm least squares rtm lsrtm has also been shown to be able to compensate for attenuation through linearized inversion however seismic attenuation may significantly slow down the convergence rate of the least squares iterative inversion process without proper preconditioning we have found that incorporating attenuation compensation into lsrtm can improve the speed of convergence in attenuating media obtaining high quality images within the first few iterations based on the low rank one step seismic modeling operator in viscoacoustic media we have derived its adjoint operator using nonstationary filtering theory the proposed forward and adjoint operators can be efficiently applied to propagate viscoacoustic waves and to implement attenuation compensation recognizing that

aggregation functions construction methods conjunctive disjunctive and mixed classes in this second part of our state of the art overview on aggregation theory based again on our recent monograph on aggregation functions we focus on several construction methods for aggregation functions and on special classes of aggregation functions covering the well known conjunctive disjunctive and mixed aggregation functions some fields of applications are included

one step extrapolation method for reverse time migration we have proposed a new method a one step extrapolation algorithm to solve the acoustic wave equation by introducing a square root operator the two way wave equation can be formulated as a first order partial differential equation in time which is similar to the one way wave equation to solve the new wave equation we used a stable explicit extrapolation method in the time direction and handled lateral velocity variations in the space and wavenumber domains unlike the conventional explicit finite difference schemes the new method does not suffer from numerical instability or numerical dispersion problems it can be used to design cost effective and high quality reverse time migration or modeling code

aggregation operators and commuting commuting is an important property in any two step information merging procedure where the results should not depend on the order in which the single steps are performed we investigate the property of commuting for aggregation operators in connection with their relationship to bisymmetry in case of bisymmetric aggregation operators we show a sufficient condition ensuring that two operators commute while for bisymmetric aggregation operators with neutral element we even provide a full characterization of commuting n ary operators by means of unary distributive functions the case of associative operations especially uninorms is considered in detail

using a deformable surface model to obtain a shape representation of the cortex this paper examines the problem of obtaining a mathematical representation of the outer cortex of the human brain which is a key problem in several applications including morphological analysis of the brain and spatial normalization and registration of brain images a parameterization of the outer cortex is first obtained using a deformable surface algorithm which motivated by the structure of the cortex is constructed to find the central layer of thick surfaces based on this parameterization a hierarchical representation of the outer cortical structure is proposed through its depth map and its curvature maps at various scales various experiments on magnetic resonance data are presented

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

a constant fractional order viscoelastic wave equation and its numerical simulation scheme abstractefficient modeling schemes currently exist to handle the spatially variable order fractional laplacians in the fractional laplacian viscoacoustic wave equation the simplest approach is to change the spatially variable order fractional laplacians into a linear combination of several constant fractional order laplacians we generalize the constant fractional order scheme to a spatially variable fractional order viscoelastic wave equation and develop an almost equivalent constant fractional order viscoelastic wave equation our constant fractional order scheme avoids the simulation error introduced by directly averaging the spatially varying fractional order thus our scheme simulates seismic wave propagation in viscoelastic media with sharp q contrasts well the fast fourier transform is used in the approximation of the fractional laplacians which improves the spectral accuracy in space several simulation examples are performed to verify that the numerical solution of a homogeneous q model obtai

on fuzzy convexity and parametric fuzzy optimization abstract in this paper some different types of convexity and generalized convexity of fuzzy sets are defined and their properties are studied a general fuzzy nonlinear programming problem is formulated with application of the concept of convexity

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

aggregation functions means this two part state of the art overview on aggregation theory summarizes the essential information concerning aggregation issues an overview of aggregation properties is given including the basic classification on aggregation functions in this first part the stress is put on means i e averaging aggregation functions both with fixed arity n ary means and with multiple arities extended means

high dimensional spatial normalization of diffusion tensor images improves the detection of white matter differences an example study using amyotrophic lateral sclerosis spatial normalization of diffusion tensor images plays a key role in voxel based analysis of white matter wm group differences currently it has been achieved using low dimensional registration methods in the large majority of clinical studies this paper aims to motivate the use of high dimensional normalization approaches by generating evidence of their impact on the findings of such studies using an ongoing amyotrophic lateral sclerosis als study we evaluated three normalization methods representing the current range of available approaches low dimensional normalization using the fractional anisotropy fa high dimensional normalization using the fa and high dimensional normalization using full tensor information each method was assessed in terms of its ability to detect significant differences between als patients and controls our findings suggest that inadequate normalization with low dimensional approaches can result in insufficient removal of shape differences which in turn can confound fa differences in a complex manner and that utilizing high dimensional normalization can both significantly minimize the confounding effect of shape differences to fa differences and provide a more complete description of wm differences in terms of both size and tissue architecture differences we also found that high dimensional approaches by leveraging full tensor features instead of tensor derived indices can further improve the alignment of wm tracts

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

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

a generalization of the fourier pseudospectral method the fourier pseudospectral ps method is generalized to the case of derivatives of nonnatural order fractional derivatives and irrational powers of the differential operators the generalization is straightforward because the calculation of the spatial derivatives with the fast fourier transform is performed in the wavenumber domain where the operator is an irrational power of the wavenumber modeling constant q propagation with this approach is highly efficient because it does not require memory variables or additional spatial derivatives the classical acoustic wave equation is modified by including those with a space fractional laplacian which describes wave propagation with attenuation and velocity dispersion in particular the example considers three versions of the uniform density wave equation based on fractional powers of the laplacian and fractional spatial derivatives

shape smoothing using medial axis properties a shape smoothing algorithm is presented which uses properties of the medial axis to define segments of the shape border and their prominence relative to the local radius of the shape prominence values are used to classify axes as either major or minor and minor axes are then deleted an augmented medial axis transform is also defined using an approximate euclidean distance transform and medial axis interpolation and linking

adjusted exponentially tilted likelihood with applications to brain morphology in this article we develop a nonparametric method called adjusted exponentially tilted et likelihood and apply it to the analysis of morphometric measures the adjusted exponential tilting estimator is shown to have the same first order asymptotic properties as that of the original et likelihood the adjusted et likelihood ratio statistic is applied to test linear hypotheses of unknown parameters such as the associations of brain measures e g cortical and subcortical surfaces with covariates of interest such as age gender and gene simulation studies show that the adjusted exponential tilted likelihood ratio statistic performs as well as the t test when the imaging data are symmetrically distributed while it is superior when the imaging data have skewed distribution we demonstrate the application of our new statistical methods to the detection of statistically significant differences in the morphology of the hippocampus between two schizophrenia groups and healthy subjects

cortical thickness analysis examined through power analysis and a population simulation we have previously developed a procedure for measuring the thickness of cerebral cortex over the whole brain using 3 d mri data and a fully automated surface extraction asp algorithm this paper examines the precision of this algorithm its optimal performance parameters and the sensitivity of the method to subtle focal changes in cortical thickness the precision of cortical thickness measurements was studied using a simulated population study and single subject reproducibility metrics cortical thickness was shown to be a reliable method reaching a sensitivity probability of a true positive of 0 93 six different cortical thickness metrics were compared the simplest and most precise method measures the distance between corresponding vertices from the white matter to the gray matter surface given two groups of 25 subjects a 0 6 mm 15 change in thickness can be recovered after blurring with a 3 d gaussian kernel full width half max 30 mm smoothing across the 2 d surface manifold also improves precision in this experiment the optimal kernel size was 30 mm

target oriented wave equation least squares migration inversion with phase encoded hessian prestack depth migration produces blurred images resulting from limited acquisition apertures complexities in the velocity model and band limited characteristics of seismic waves this distortion can be partially corrected using the model space least squares migration inversion approach where a target oriented wave equation hessian operator is computed explicitly and then inverse filtering is applied iteratively to deblur or invert for the reflectivity however one difficulty is the cost of computing the explicit hessian operator which requires storing a large number of green s functions making it challenging for large scale applications a new method to compute the hessian operator for the wave equation based least squares migration inversion problem modifies the original explicit hessian formula enabling efficient computation of this operator an advantage is that the method eliminates disk storage of green s functions the modifications however also introduce undesired crosstalk artifacts two different phase encoding schemes plane wave phase encoding and random phase encoding suppress the crosstalk when the randomly phase encoded hessian operator is applied to the sigsbee2a synthetic data set an improved subsalt image with more balanced amplitudes is obtained

least squares reverse time migration with an angle dependent weighting factor least squares reverse time migration lsrtm produces higher quality images than conventional rtm however directly using the standard gradient formula the inverted images suffer from low wavenumber noise using a simple high pass filter on the gradient can alleviate the effect of the low wavenumber noise but owing to the illumination issue the amplitudes are not balanced and in the deep part they are often weak these two issues can be mitigated by the iterative approach but it needs more iterations we introduced an angle dependent weighting factor to weight the gradient of lsrtm to suppress the low wavenumber noise and also to emphasize the gradient in the deep part an optimal step length for the l2 norm objective function is also presented to scale the gradient to the right order two numerical examples performed with the data synthesized on the sigsbee2a and marmousi models indicate that when using this weighted gradient combined with the preconditioned l bfgs algorithm with the optimal step length only a few iterations can achieve satisfying results

a statistical analysis of brain morphology using wild bootstrapping methods for the analysis of brain morphology including voxel based morphology and surface based morphometries have been used to detect associations between brain structure and covariates of interest such as diagnosis severity of disease age iq and genotype the statistical analysis of morphometric measures usually involves two statistical procedures 1 invoking a statistical model at each voxel or point on the surface of the brain or brain subregion followed by mapping test statistics e g t test or their associated p values at each of those voxels 2 correction for the multiple statistical tests conducted across all voxels on the surface of the brain region under investigation we propose the use of new statistical methods for each of these procedures we first use a heteroscedastic linear model to test the associations between the morphological measures at each voxel on the surface of the specified subregion e g cortical or subcortical surfaces and the covariates of interest moreover we develop a robust test procedure that is based on a resampling method called wild bootstrapping this procedure assesses the statistical significance of the associations between a measure of given brain structure and the covariates of interest the value of this robust test procedure lies in its computationally simplicity and in its applicability to a wide range of imaging data including data from both anatomical and functional magnetic resonance imaging fmri simulation studies demonstrate that this robust test procedure can accurately control the family wise error rate we demonstrate the application of this robust test procedure to the detection of statistically significant differences in the morphology of the hippocampus over time across gender groups in a large sample of healthy subjects

determining the optimal level of smoothing in cortical thickness analysis a hierarchical approach based on sequential statistical thresholding article i nfo the extent of smoothing applied to cortical thickness maps critically influences sensitivity anatomical precision and resolution of statistical change detection theoretically it could be optimized by increasing the trade off between vertex wise sensitivity and specificity across several levels of smoothing but to date neither parametric nor nonparametric methods are able to control the error at the vertex level if the null hypothesis is rejected after smoothing of cortical thickness maps to overcome these drawbacks we applied sequential statistical thresholding based on a simple hierarchical model this methodology aims at controlling erroneous detections firstly at the level of clusters over smoothed statistical maps and secondly at the vertex level over unsmoothed statistical maps by applying an adaptive false discovery rate fdr procedure to clusters previously detected the superior performance of the proposed methodology over other conventional procedures was demonstrated in simulation studies as expected only the hierarchical method yielded a predictable false discovery proportion near the predefined fdr q value for any smoothing level at the same time as being as sensitive as the others at the optimal setting it was therefore the only method able to approximate the optimal size of spatial smoothing when the true change was assumed unknown the hierarchical method was further validated in a cross sectional study comparing moderate alzheimer s disease ad patients with healthy elderly subjects results suggest that the extent of cortical thinning reported in previous ad studies might be artificially inflated by the choice of inadequate smoothing in these cases interpretation should be based on the location of local maxima of suprathreshold regions rather than on the spatial extent of the detected signal in the statistical parametric map

detecting sparse signals in random fields with an application to brain mapping brain mapping data have been modeled as gaussian random fields and local increases in mean are detected by local maxima of a random field of test statistics derived from these data accurate p values for local maxima are readily available for isotropic data based on the expected euler characteristic of the excursion set of the test statistic random field in this article we give a simple method for dealing with nonisotropic data our approach has connections to the model of sampson and guttorp for nonisotropy in which there exists an unknown mapping of the support of the data to a space in which the random fields are isotropic heuristic justification for our approach comes from the nash embedding theorem formally we show that our method gives consistent unbiased estimators for the true p values based on new results of taylor and adler for random fields on manifolds which replace the euclidean metric by the variogram the results are used to detect gender differences in the cortical thickness of the b

hamilton jacobi skeleton on cortical surfaces in this paper we propose a new method to construct graphical representations of cortical folding patterns by computing skeletons on triangulated cortical surfaces in our approach a cortical surface is first partitioned into sulcal and gyral regions via the solution of a variational problem using graph cuts which can guarantee global optimality after that we extend the method of hamilton jacobi skeleton to subsets of triangulated surfaces together with a geometrically intuitive pruning process that can trade off between skeleton complexity and the completeness of representing folding patterns compared with previous work that uses skeletons of 3 d volumes to represent sulcal patterns the skeletons on cortical surfaces can be easily decomposed into branches and provide a simpler way to construct graphical representations of cortical morphometry in our experiments we demonstrate our method on two different cortical surface models its ability of capturing major sulcal patterns and its application to compute skeletons of gyral regions

spatial regression models over two dimensional manifolds we propose a regression model for data spatially distributed over general two dimensional riemannian manifolds this is a generalized additive model with a roughness penalty term involving a differential operator computed over the non planar domain by virtue of a semiparametric framework the model allows inclusion of space varying covariate information estimation can be performed by conformally parameterizing the non planar domain and then generalizing existing models for penalized spatial regression over planar domains the conformal coordinates and the estimation problem are both computed with a finite element approach

theory of nonstationary linear filtering in the fourier domain with application to time variant filtering a general linear theory describes the extension of the convolutional method to nonstationary processes this theory can apply any linear nonstationary filter with arbitrary time and frequency variation in the time fourier or mixed domains the filter application equations and the expressions to move the filter between domains are all ordinary fourier transforms or generalized convolutional integrals nonstationary transforms such as the wavelet transform are not required there are many possible applications of this theory including the one way propagation of waves through complex media time migration normal moveout removal time variant filtering and forward and inverse q filtering two complementary nonstationary filters are developed by generalizing the stationary convolution integral the first called nonstationary convolution corresponds to the linear superposition of scaled impulse responses of a nonstationary filter the second called nonstationary combination does not correspond to such a superposition but is shown to be a linear process capable of achieving arbitrarily abrupt temporal variations in the output frequency spectrum both extensions have stationary convolution as a limiting form and in the discrete case can be formulated as matrix operations fourier transformation shows that both filter types are nonstationary filter integrals in the fourier domain as well this result is a generalization of the convolution theorem for stationary signals because as the filter becomes stationary in one domain the integral in the other domain collapses to a scalar multiplication for discrete signals stationary filters are a matrix multiplication of the input signal spectrum by a diagonal spectral matrix while nonstationary filters require off diagonal terms for quasi stationary filters a computational advantage is obtained by computing only the significant terms near the diagonal unlike stationary theory a mixed domain of time and frequency is also possible in this context the nonstationary filter is applied simultaneously with the transform from time to frequency or the reverse nonstationary convolution becomes a generalized forward fourier integral and nonstationary combination is a generalized inverse fourier integral

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

model based deformable surface finding for medical images describes a new global shape parameterization for smoothly deformable three dimensional 3 d objects such as those found in biomedical images whose diversity and irregularity make them difficult to represent in terms of fixed features or parts this representation is used for geometric surface matching to 3 d medical image data such as from magnetic resonance imaging mri the parameterization decomposes the surface into sinusoidal basis functions four types of surfaces are modeled tori open surfaces closed surfaces and tubes this parameterization allows a wide variety of smooth surfaces to be described with a small number of parameters extrinsic model based information is incorporated by introducing prior probabilities on the parameters surface finding is formulated as an optimization problem results of the method applied to synthetic images and 3 d medical images of the heart and brain are presented

continuous medial representation of brain structures using the biharmonic pde a new approach for constructing deformable continuous medial models for anatomical structures is presented medial models describe geometrical objects by first specifying the skeleton of the object and then deriving the boundary surface corresponding to the skeleton however an arbitrary specification of a skeleton will not be valid unless a certain set of sufficient conditions is satisfied the most challenging of these is the non linear equality constraint that must hold along the boundaries of the manifolds forming the skeleton the main contribution of this paper is to leverage the biharmonic partial differential equation as a mapping from a codimension 0 subset of euclidean space to the space of skeletons that satisfy the equality constraint the pde supports robust numerical solution on freeform triangular meshes providing additional flexibility for shape modeling the approach is evaluated by generating continuous medial models for a large dataset of hippocampus shapes generalizations to modeling more complex shapes and to representing branching skeletons are demonstrated

modeling of a constant q methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique linear anelastic phenomena in wave propagation problems can be well modeled through a viscoelastic mechanical model consisting of standard linear solids in this paper we present a method for modeling of constant q as a function of frequency based on an explicit closed formula for calculation of the parameter fields several standard linear solids connected in parallel can be tuned through a single parameter to yield an excellent constant q approximation the proposed method enables substantial savings in computations and memory requirements experiments show that the new method also yields higher accuracy in the modeling of q than e g the pade approximant method

multivariate tensor based morphometry on surfaces application to mapping ventricular abnormalities in hiv aids here we developed a new method called multivariate tensor based surface morphometry tbm and applied it to study lateral ventricular surface differences associated with hiv aids using concepts from differential geometry and the theory of differential forms we created mathematical structures known as holomorphic one forms to obtain an efficient and accurate conformal parameterization of the lateral ventricular surfaces in the brain the new meshing approach also provides a natural way to register anatomical surfaces across subjects and improves on prior methods as it handles surfaces that branch and join at complex 3d junctions to analyze anatomical differences we computed new statistics from the riemannian surface metrics these retain multivariate information on local surface geometry we applied this framework to analyze lateral ventricular surface morphometry in 3d mri data from 11 subjects with hiv aids and 8 healthy controls our method detected a 3d profile of surface abnormalities even in this small sample multivariate statistics on the local tensors gave better effect sizes for detecting group differences relative to other tbm based methods including analysis of the jacobian determinant the largest and smallest eigenvalues of the surface metric and the pair of eigenvalues of the jacobian matrix the resulting analysis pipeline may improve the power of surface based morphometry studies of the brain

detection of focal changes in human cortical thickness spherical wavelets versus gaussian smoothing subtle but progressive variations in human cortical thickness have been associated with the initial phases of prevalent neurological and psychiatric conditions but slight changes in cortical thickness at preclinical stages are typically masked by effects of the gaussian kernel smoothing on the cortical surface shape descriptors here we present the first study aimed at detecting changes in human cortical thickness maps by applying soft thresholding to multiresolution spherical wavelet coefficients in order to make gaussian and wavelet smoothing methods comparable the trade off between sensitivity and specificity was optimized to detect simulated thickness changes in various cortical areas of healthy elderly subjects results revealed a better sensitivity specificity trade off when using wavelet based methods as compared to gaussian smoothing in both the whole neocortex p 10 7 and cortical region based statistical analyses p 10 9 which was mainly due to the higher specificity obtained with the wavelet approach the lower smoothing introduced by wavelets and their adaptive properties may account for the enhanced specificity and sensitivity when compared with gaussian spatial filters these results strongly support the use of spherical wavelet methods to detect subtle variations in cortical thickness maps which may be crucial in better understanding the course of neuronal loss in normal aging and in finding early markers of cortical degeneration

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

a width independent fast thinning algorithm the skeleton of a digital figure can often be regarded as a convenient alternative to the figure itself it is useful both to diminish drastically the amount of data to be handled and to simplify the computational procedures required for description and classification purposes thinning a digital figure down to its skeleton is a time consuming process when conventional sequential computers are employed the procedure we propose allows one to speed up the thinning transformation and to get a well shaped skeleton after cleaning of the input picture has been performed the pixels of the figure are labeled according to their distance from the background and a set whose pixels are symmetrically placed with respect to distinct contour parts of the figure is found this set is then given a linear structure by applying topology preserving removal operations finally a pruning step regarding branches not relevant in the framework of the problem domain completes the process the resulting skeleton is a labeled set of pixels which is shown to possess all the required properties particularly those concerning connectedness topology and shape moreover the original figure can almost completely be recovered by means of a reverse distance transformation only a fixed and small number of sequential passes through the picture is necessary to achieve the goal the computational effort is rather modest and the use of the proposed algorithm turns out to be more advantageous the greater the width of the figure to be thinned

simulating the grassfire transform using an active contour model a method for shape description of planar objects that integrates both region and boundary features is presented the method is an implementation of a 2d dynamic grassfire that relies on a distance surface on which elastic contours minimize an energy function the method is based on an active contour model numerous implementation aspects of the shape description method were optimized a euclidean metric was used for optimal accuracy and the active contour model permits bypassing some of the discretization limitations inherent in using a digital grid noise filtering was performed on the basis of both contour feature measures and region measures that is curvature extremum significance and ridge support respectively to obtain robust shape descriptors other improvements and variations of the algorithmic implementation are proposed

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

elastic model based segmentation of 3 d neuroradiological data sets this paper presents a new technique for the automatic model based segmentation of three dimensional 3 d objects from volumetric image data the development closely follows the seminal work of taylor and cootes on active shape models but is based on a hierarchical parametric object description rather than a point distribution model the segmentation system includes both the building of statistical models and the automatic segmentation of new image data sets via a restricted elastic deformation of shape models geometric models are derived from a sample set of image data which have been segmented by experts the surfaces of these binary objects are converted into parametric surface representations which are normalized to get an invariant object centered coordinate system surface representations are expanded into series of spherical harmonics which provide parametric descriptions of object shapes it is shown that invariant object surface parametrization provides a good approximation to automatically determine object homology in terms of sets of corresponding sets of surface points gray level information near object boundaries is represented by 1 d intensity profiles normal to the surface considering automatic segmentation of brain structures as their driving application the authors choice of coordinates for object alignment was the well accepted stereotactic coordinate system major variation of object shapes around the mean shape also referred to as shape eigenmodes are calculated in shape parameter space rather than the feature space of point coordinates segmentation makes use of the object shape statistics by restricting possible elastic deformations into the range of the training shapes the mean shapes are initialized in a new data set by specifying the landmarks of the stereotactic coordinate system the model elastically deforms driven by the displacement forces across the object s surface which are generated by matching local intensity profiles elastical deformations are limited by setting bounds for the maximum variations in eigenmode space the technique has been applied to automatically segment left and right hippocampus thalamus putamen and globus pallidus from volumetric magnetic resonance scans taken from schizophrenia studies the results have been validated by comparison of automatic segmentation with the results obtained by interactive expert segmentation

disconnected skeleton shape at its absolute scale we present a new skeletal representation along with a matching framework to address the deformable shape recognition problem the disconnectedness arises as a result of excessive regularization that we use to describe a shape at an attainably coarse scale our motivation is to rely on stable properties the shape instead of inaccurately measured secondary details the new representation does not suffer from the common instability problems of the traditional connected skeletons and the matching process gives quite successful results on a diverse database of 2d shapes an important difference of our approach from the conventional use of skeleton is that we replace the local coordinate frame with a global euclidean frame supported by additional mechanisms to handle articulations and local boundary deformations as a result we can produce descriptions that are sensitive to any combination of changes in scale position orientation and articulation as well as invariant ones

q least squares reverse time migration with viscoacoustic deblurring filters abstractviscoacoustic least squares reverse time migration also denoted as q lsrtm linearly inverts for the subsurface reflectivity model from lossy data compared with conventional migration methods it can compensate for the amplitude loss in the migrated images due to strong subsurface attenuation and can produce reflectors that are accurately positioned in depth however the adjoint q propagators used for backward propagating the residual data are also attenuative thus the inverted images from q lsrtm with a small number of iterations are often observed to have lower resolution when compared with the benchmark acoustic lsrtm images from acoustic data to increase the resolution and accelerate the convergence of q lsrtm we used viscoacoustic deblurring filters as a preconditioner for q lsrtm these filters can be estimated by matching a simulated migration image to its reference reflectivity model numerical tests on synthetic and field data demonstrate that q lsrtm combined with viscoacoustic de

viscoelastic finite difference modeling real earth media disperse and attenuate propagating mechanical waves this anelastic behavior can be described well by a viscoelastic model we have developed a finite difference simulator to model wave propagation in viscoelastic media the finite difference method was chosen in favor of other methods for several reasons finite difference codes are more portable than for example pseudospectral codes moreover finite difference schemes provide a convenient environment in which to define complicated boundaries a staggered scheme of second order accuracy in time and fourth order accuracy in space appears to be optimally efficient because of intrinsic dispersion no fixed grid points per wavelength rule can be given instead we present tables which enable a choice of grid parameters for a given level of accuracy since the scheme models energy absorption natural and efficient absorbing boundaries may be implemented merely by changing the parameters near the grid boundary the viscoelastic scheme is

migration of seismic data by phase shift plus interpolation under the horizontally layered velocity assumption migration is defined by a set of independent ordinary differential equations in the wavenumber frequency domain the wave components are extrapolated downward by rotating their phases this paper shows that one can generalize the concepts of the phase shift method to media having lateral velocity variations the wave extrapolation procedure consists of two steps in the first step the wave field is extrapolated by the phase shift method using l laterally uniform velocity fields the intermediate result is l reference wave fields in the second step the actual wave field is computed by interpolation from the reference wave fields the phase shift plus interpolation pspi method is unconditionally stable and lends itself conveniently to migration of three dimensional data the performance of the methods is demonstrated on synthetic examples the pspi migration results are then compared with those obtained from a finite difference method

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

continuous medial representation for anatomical structures the m rep approach pioneered by pizer 2003 is a powerful morphological tool that makes it possible to employ features derived from medial loci skeletons in shape analysis this paper extends the medial representation paradigm into the continuous realm modeling skeletons and boundaries of three dimensional objects as continuous parametric manifolds while also maintaining the proper geometric relationship between these manifolds the parametric representation of the boundary medial relationship makes it possible to fit shape based coordinate systems to the interiors of objects providing a framework for combined statistical analysis of shape and appearance our approach leverages the idea of inverse skeletonization where the skeleton of an object is defined first and the object s boundary is derived analytically from the skeleton this paper derives a set of sufficient conditions ensuring that inverse skeletonization is well posed for single manifold skeletons and formulates a partial differential equation whose solutions satisfy the sufficient conditions an efficient variational algorithm for deformable template modeling using the continuous medial representation is described and used to fit a template to the hippocampus in 87 subjects from a schizophrenia study with sub voxel accuracy and 95 mean overlap

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

multiscale medial loci and their properties blum s medial axes have great strengths in principle in intuitively describing object shape in terms of a quasi hierarchy of figures but it is well known that derived from a boundary they are damagingly sensitive to detail in that boundary the development of notions of spatial scale has led to some definitions of multiscale medial axes different from the blum medial axis that considerably overcame the weakness three major multiscale medial axes have been proposed iteratively pruned trees of voronoi edges ogniewicz 1993 szekely 1996 naf 1996 shock loci of reaction diffusion equations kimia et al 1995 siddiqi and kimia 1996 and height ridges of medialness cores fritsch et al 1994 morse et al 1993 pizer et al 1998 these are different from the blum medial axis and each has different mathematical properties of generic branching and ending properties singular transitions and geometry of implied boundary and they have different strengths and weaknesses for computing object descriptions from images or from object boundaries these mathematical properties and computational abilities are laid out and compared and contrasted in this paper

determining the geometry of boundaries of objects from medial data we consider a region in r2 or r3 with generic smooth boundary b and blum medial axis m on which is defined a multivalued radial vector field u from points x on m to the points of tangency of the sphere at x with b we introduce a radial shape operator srad and an edge shape operator se which measure how u bends along m these are not traditional differential geometric shape operators nonetheless we derive all local differential geometric invariants of b from these operators this allows us to define from m u a geometric medial map on m which corresponds via a radial map from m to b to the differential geometric properties of b the geometric medial map also includes a description of the relative geometry of b this is defined using the relative critical set of the radius function r on m this set consists of a network of curves on m which describe where b is thickest and thinnest it is computed using the covariant derivative of the tangential component of the unit radial vector field we further determine how these invariants are related to the differential geometric invariants of m and how these invariants change under deforming diffeomorphisms of m

signal adapted tight frames on graphs the analysis of signals on complex topologies modeled by graphs is a topic of increasing importance decompositions play a crucial role in the representation and processing of such information here we propose a new tight frame design that is adapted to a class of signals on a graph the construction starts from a prototype meyer type system of kernels with uniform subbands the ensemble energy spectral density is then defined for a given set of signals defined on the graph the prototype design is then warped such that the resulting subbands capture the same amount of energy for the signal class this approach accounts at the same time for graph topology and signal features the proposed frames are constructed for three different graph signal sets and are compared with non signal adapted frames vertex localization of a set of resulting atoms is studied the frames are then used to decompose a set of real graph signals and are also used in a setting of signal denoising the results illustrate the superiority of the designed signal adapted frames over frames blind to signal characteristics in representing data and in denoising

on the intrinsic reconstruction of shape from its symmetries the main question we address is what is the minimal information required to generate closed nonintersecting planar boundaries for this paper we restrict shape to this meaning more precisely we examine whether the medial axis together with dynamics can serve as a language to design shapes and to effect shape changes we represent the medial axis together with a direction of flow along the axis as the shock graph and examine the reconstruction of shape along each of the three types of medial axis points a sub 1 sup 2 a sub 1 sup 3 a sub 3 and the associated six types of shock points first we show that the tangent and curvature of the medial axis and the speed and acceleration of the shock with respect to time of propagation are sufficient to determine the boundary tangent and curvature at corresponding points of the boundary this implies that a rather coarse sampling of the symmetry axis its tangent curvature speed and acceleration is sufficient to regenerate accurately a local neighborhood of shape at regular axis points a sub 1 sup 2 second we examine the reconstruction of shape at branch points a sub 1 sup 3 where three regular branches are joined we show that the three pairs of geometry that is curvature and dynamics that is acceleration must satisfy certain constraints finally we derive similar results for the end points of shock branches a sub 3 points these formulas completely specify the local reconstruction of a shape from its shock graph or medial axis and the conditions required to form a coherent shape from the medial axis

accelerating full waveform inversion with attenuation compensation abstractthe calculation of the gradient in full waveform inversion fwi usually involves crosscorrelating the forward propagated source wavefield and the back propagated data residual wavefield at each time step in the real earth propagating waves are typically attenuated due to the viscoelasticity which results in an attenuated gradient for fwi replacing the attenuated true gradient with a q compensated gradient can accelerate the convergence rate of the inversion process we have used a phase dispersion and an amplitude loss decoupled constant q wave equation to formulate a viscoacoustic fwi we used this wave equation to generate a q compensated gradient which recovers amplitudes while preserving the correct kinematics we construct an exact adjoint operator in a discretized form using the low rank wave extrapolation technique and we implement the gradient compensation by reversing the sign of the amplitude loss term in the forward and adjoint operators this leads to a q dependent gradient prec

l fuzzy numbers and their properties in this paper the notions of l fuzzy convex sets and l fuzzy numbers are introduced where l is a completely distributive lattice the notions of 0 1 fuzzy convex sets and 0 1 fuzzy numbers are generalized furthermore their properties and characterizations are presented in terms of cut sets of l fuzzy sets

altered white matter connectivity and network organization in polymicrogyria revealed by individual gyral topology based analysis polymicrogyria pmg is a cortical malformation characterized by multiple small gyri and altered cortical lamination which may be associated with disrupted white matter connectivity however little is known about the topological patterns of white matter networks in pmg we examined structural connectivity and network topology using individual primary gyral pattern based nodes in pmg patients overcoming the limitations of an atlas based approach structural networks were constructed from structural and diffusion magnetic resonance images in 25 typically developing and 14 pmg subjects the connectivity analysis for different fiber groups divided based on gyral topology revealed severely reduced connectivity between neighboring primary gyri short u fibers in pmg which was highly correlated with the regional involvement and extent of abnormal gyral folding the patients also showed significantly reduced connectivity between distant gyri long association fibers and between the two cortical hemispheres in relation to these results gyral node based graph theoretical analysis revealed significantly altered topological organization of the network lower clustering and higher modularity and disrupted network hub architecture in cortical association areas involved in cognitive and language functions in pmg patients furthermore the network segregation in pmg patients decreased with the extent of pmg and the degree of language impairment our approach provides the first detailed findings and interpretations on altered cortical network topology in pmg related to abnormal cortical structure and brain function and shows the potential for an individualized method to characterize network properties and alterations in connections that are associated with malformations of cortical development

the groupwise medial axis transform for fuzzy skeletonization and pruning medial representations of shapes are useful due to their use of an object centered coordinate system that directly captures intuitive notions of shape such as thickness bending and elongation however it is well known that an object s medial axis transform mat is unstable with respect to small perturbations of its boundary this instability results in additional unwanted branches in the skeletons which must be pruned in order to recover the portions of the skeletons arising purely from the uncorrupted shape information almost all approaches to skeleton pruning compute a significance measure for each branch according to some heuristic criteria and then prune the least significant branches first current approaches to branch significance computation can be classified as either local solely using information from a neighborhood surrounding each branch or global using information about the shape as a whole in this paper we propose a third groupwise approach to branch significance computation we develop a groupwise skeletonization framework that yields a fuzzy significance measure for each branch derived from information provided by the group of shapes we call this framework the groupwise medial axis transform g mat we propose and evaluate four groupwise methods for computing branch significance and report superior performance compared to a recent leading method we measure the performance of each pruning algorithm using denoising classification and within class skeleton similarity measures this research has several applications including object retrieval and shape analysis

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

accelerating extended least squares migration with weighted conjugate gradient iteration abstractleast squares migration lsm iteratively achieves a mean square best fit to seismic reflection data provided that a kinematically accurate velocity model is available the subsurface offset extension adds extra degrees of freedom to the model thereby allowing lsm to fit the data even in the event of significant velocity error this type of extension also implies additional computational expense per iteration from crosscorrelating source and receiver wavefields over the subsurface offset and therefore places a premium on rapid convergence we have accelerated the convergence of extended least squares migration by combining the conjugate gradient algorithm with weighted norms in range data and domain model spaces that render the extended born modeling operator approximately unitary we have developed numerical examples that demonstrate that the proposed algorithm dramatically reduces the number of iterations required to achieve a given level of fit or gradient reduction compared with conjuga

a formal classification of 3d medial axis points and their local geometry this paper proposes a novel hypergraph skeletal representation for 3d shape based on a formal derivation of the generic structure of its medial axis by classifying each skeletal point by its order of contact we show that genetically the medial axis consists of five types of points which are then organized into sheets curves and points 1 sheets manifolds with boundary which are the locus of bitangent spheres with regular tangency a sub 1 sup 2 a sub k sup n notation means n distinct k fold tangencies of the sphere of contact as explained in the text two types of curves 2 the intersection curve of three sheets and the locus of centers of tritangent spheres a sub 1 sup 3 and 3 the boundary of sheets which are the locus of centers of spheres whose radius equals the larger principal curvature i e higher order contact a sub 3 points and two types of points 4 centers of quad tangent spheres a sub 1 sup 4 and 5 centers of spheres with one regular tangency and one higher order tangency a sub 1 a sub 3 the geometry of the 3d medial axis thus consists of sheets a sub 1 sup 2 bounded by one type of curve a sub 3 on their free end which corresponds to ridges on the surface and attached to two other sheets at another type of curve a sub 1 sup 3 which support a generalized cylinder description the a sub 3 curves can only end in a sub 1 a sub 3 points where they must meet an a sub 1 sup 3 curve the a sub 1 sup 3 curves meet together in fours at an a sub 1 sup 4 point this formal result leads to a compact representation for 3d shape referred to as the medial axis hypergraph representation consisting of nodes a sub 1 sup 4 and a sub 1 a sub 3 points links between pairs of nodes a sub 1 sup 3 and a sub 3 curves and hyperlinks between groups of links a sub 1 sup 2 sheets the description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes in analogy to 2d we derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph together with the radius function thus this information completely characterizes 3d shape and lays the theoretical foundation for its use in recognition morphing design and manipulation of shapes

hierarchical shape description via the multiresolution symmetric axis transform a method is proposed that produces a shape description in the form of a hierarchy by scale of simple symmetric axis sequences an axis segment that is a child of another has smaller scale and is seen as a branch of its parent the scale value and parent child relationship are induced by following the symmetric axis under successive reduction of resolution the result in two or three dimensions is a figure rather than boundary oriented shape description that has natural segments and is insensitive to noise in the object description the general approach of hierarchy production by following a feature through successive resolution reduction will be presented as will methods of resolution reduction and computer implementation also the relation of this figure based shape description to those based on boundary curvature will be briefly discussed

viscoacoustic modeling and imaging using low rank approximation abstracta constant q wave equation involving fractional laplacians was recently introduced for viscoacoustic modeling and imaging this fractional wave equation has a convenient mixed domain space wavenumber formulation which involves the fractional laplacian operators with a spatially varying power we have applied the low rank approximation to the mixed domain symbol which enables a space variable attenuation specified by the variable fractional power of the laplacians using the proposed approximation scheme we formulated the framework of the q compensated reverse time migration q rtm for attenuation compensation numerical examples using synthetic data demonstrated the improved accuracy of using low rank wave extrapolation with a constant q fractional laplacian wave equation for seismic modeling and migration in attenuating media low rank q rtm applied to viscoacoustic data is capable of producing images comparable in quality with those produced by conventional rtm from acoustic data

artificial shifting of fmri activation localized by volume and surface based analyses spatial smoothing is an important post processing procedure that is used to increase the signal to noise ratio snr of blood oxygenation level dependent signals bold in common functional magnetic resonance imaging fmri applications however recent studies have shown that smoothing artificially shifts probabilistic local maxima of fmri activations in this study we show shifting of the localization of functional centers in hand motor areas of the cerebral cortex by three dimensional isotropic gaussian kernel smoothing or two dimensional heat kernel smoothing in volume and surface based fmri analyses activation maps derived from smoothed echo planar imaging epi data by volume and surface based analyses were assigned to the nodes of individual cortical surface models and local maxima in the primary motor area m1 and the primary somatosensory cortex s1 were compared with those derived from non smoothed risk map analysis which is commonly used in presurgical applications for each analysis the euclidean and geodesic distances between the correlation coefficients of local maxima derived from smoothed and non smoothed epi data were measured the results show that the correlation coefficients derived from the volume and surface based analyses were about 29 4 and 42 9 higher for smoothed than for non smoothed risk map analyses and show minimum shifting of localizations by 12 1 mm and 6 9 mm on average in euclidean distance respectively and about 9 5 mm and 5 7 mm on average in geodesic distance respectively

approximate linearized inversion by optimal scaling of prestack depth migration linearized inversion provides one possible sense of image amplitude correctness an image or bandlimited model perturbation has correct amplitudes if it is an approximate inversion that is if linearized modeling demigration with the image as input reproduces the data approximately the theory of linearized acoustic inverse scattering with slowly varying background or macromodel shows that an approximate inversion may be recovered from the output of prestack depth migration by a combination of scaling and filtering the necessary filter is completely specified by the theory and the scale factor may be estimated via filtering linearized modeling a second migration and the solution of a small auxiliary inverse problem

medial axis transformation of a planar shape the medial axis transformation is a means first proposed by blum to describe a shape in this paper we present a 0 n log n algorithm for computing the medial axis of a planar shape represented by an n edge simple polygon the algorithm is an improvement over most previously known results interms of both efficiency and exactness and has been implemented in fortran some computer plotted output of the program are also shown in the paper

teichmuller shape descriptor and its application to alzheimer s disease study we propose a novel method to apply teichmuller space theory to study the signature of a family of nonintersecting closed 3d curves on a general genus zero closed surface our algorithm provides an efficient method to encode both global surface and local contour shape information the signature teichmuller shape descriptor is computed by surface ricci flow method which is equivalent to solving an elliptic partial differential equation on surfaces and is numerically stable we propose to apply the new signature to analyze abnormalities in brain cortical morphometry experimental results with 3d mri data from alzheimer s disease neuroimaging initiative adni dataset 152 healthy control subjects versus 169 alzheimer s disease ad patients demonstrate the effectiveness of our method and illustrate its potential as a novel surface based cortical morphometry measurement in ad research

multiple sensitivity profiles to diversity and transition structure in non stationary input recent formalizations suggest that the human brain codes for the degree of order in the environment and utilizes this knowledge to optimize perception and performance in the immediate future however the neural bases of how the brain spontaneously codes for order are poorly understood it has been shown that activity in lateral temporal cortex and the hippocampus is linearly correlated with the order of short visual series under tasks requiring attention to the input and when series order is invariant over time here we examined if sensitivity to order is manifested in both linear and non linear bold response profiles quantified the degree to which order sensitive regions operate as a functional network and evaluated these questions using a paradigm in which performance of the ongoing task could be completed without any attention to the stimulus whose order was manipulated participants listened to a 10 minute sequence of tones characterized by non stationary order and fmri identified cortical regions sensitive to time varying statistical features of this input activity in perisylvian regions was negatively correlated with input diversity quantified via shannon s entropy activity in ventral premotor lateral temporal and insular regions was correlated linearly parabolically or via a step function with the strength of transition constraints in the series quantified via markov entropy granger causality analysis revealed that order sensitive regions form a functional network with regions showing non linear responses to order associated with more afferent connectivity than those showing linear responses these findings identify networks that spontaneously code and respond to diverse aspects of order via multiple response profiles and that play a central role in generating and gating predictive neural activity

tensor based cortical surface morphometry via weighted spherical harmonic representation we present a new tensor based morphometric framework that quantifies cortical shape variations using a local area element the local area element is computed from the riemannian metric tensors which are obtained from the smooth functional parametrization of a cortical mesh for the smooth parametrization we have developed a novel weighted spherical harmonic spharm representation which generalizes the traditional spharm as a special case for a specific choice of weights the weighted spharm is shown to be the least squares approximation to the solution of an isotropic heat diffusion on a unit sphere the main aims of this paper are to present the weighted spharm and to show how it can be used in the tensor based morphometry as an illustration the methodology has been applied in the problem of detecting abnormal cortical regions in the group of high functioning autistic subjects

hierarchic voronoi skeletons abstract robust and time efficient skeletonization of a planar shape which is connectivity preserving and based on euclidean metrics can be achieved by first regularizing the voronoi diagram vd of a shape s boundary points i e by removal of noise sensitive parts of the tessellation and then by establishing a hierarchic organization of skeleton constituents each component of the vd is attributed with a measure of prominence which exhibits the expected invariance under geometric transformations and noise the second processing step a hierarchic clustering of skeleton branches leads to a multiresolution representation of the skeleton termed skeleton pyramid

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

generalized tensor based morphometry of hiv aids using multivariate statistics on deformation tensors this paper investigates the performance of a new multivariate method for tensor based morphometry tbm statistics on riemannian manifolds are developed that exploit the full information in deformation tensor fields in tbm multiple brain images are warped to a common neuroanatomical template via 3 d nonlinear registration the resulting deformation fields are analyzed statistically to identify group differences in anatomy rather than study the jacobian determinant volume expansion factor of these deformations as is common we retain the full deformation tensors and apply a manifold version of hotelling s test to them in a log euclidean domain in 2 d and 3 d magnetic resonance imaging mri data from 26 hiv aids patients and 14 matched healthy subjects we compared multivariate tensor analysis versus univariate tests of simpler tensor derived indices the jacobian determinant the trace geodesic anisotropy and eigenvalues of the deformation tensor and the angle of rotation of its eigenvectors we detected consistent but more extensive patterns of structural abnormalities with multivariate tests on the full tensor manifold their improved power was established by analyzing cumulative value plots using false discovery rate fdr methods appropriately controlling for false positives this increased detection sensitivity may empower drug trials and large scale studies of disease that use tensor based morphometry

segmentation registration and measurement of shape variation via image object shape a model of object shape by nets of medial and boundary primitives is justified as richly capturing multiple aspects of shape and yet requiring representation space and image analysis work proportional to the number of primitives metrics are described that compute an object representation s prior probability of local geometry by reflecting variabilities in the net s node and link parameter values and that compute a likelihood function measuring the degree of match of an image to that object representation a paradigm for image analysis of deforming such a model to optimize a posteriori probability is described and this paradigm is shown to be usable as a uniform approach for object definition object based registration between images of the same or different imaging modalities and measurement of shape variation of an abnormal anatomical object compared with a normal anatomical object examples of applications of these methods in radiotherapy surgery and psychiatry are given

aggregation of convex intuitionistic fuzzy sets aggregation of intuitionistic fuzzy sets is studied from the point of view of preserving various kinds of convexity we focus on aggregation functions for intuitionistic fuzzy sets these functions correspond to simultaneous separate aggregations of the membership as well as of the nonmembership indicators it is performed by means of the so called representable functions sufficient and necessary conditions are analyzed in order to guarantee that the composition of two intuitionistic fuzzy sets preserves convexity

wavefield extrapolation by nonstationary phase shift the phase shift method of wavefield extrapolation applies a phase shift in the fourier domain to deduce a scalar wavefield at one depth level given its value at another the phase shift operator varies with frequency and wavenumber and assumes constant velocity across the extrapolation step we use nonstationary filter theory to generalize this method to nonstationary phase shift nsps which allows the phase shift to vary laterally depending upon the local propagation velocity for comparison we derive an analytic form for the popular phase shift plus interpolation pspi method in the limit of an exhaustive set of reference velocities nsps and this limiting form of pspi can be written as generalized fourier integrals which reduce to ordinary phase shift in the constant velocity limit in the x domain these processes are the transpose of each other however only nsps has the physical interpretation of forming the scaled linear superposition of laterally variable impulse responses i e huygen

skeletal shape correspondence through entropy we present a novel approach for improving the shape statistics of medical image objects by generating correspondence of skeletal points each object s interior is modeled by an s rep i e by a sampled folded two sided skeletal sheet with spoke vectors proceeding from the skeletal sheet to the boundary the skeleton is divided into three parts the up side the down side and the fold curve the spokes on each part are treated separately and using spoke interpolation are shifted along that skeleton in each training sample so as to tighten the probability distribution on those spokes geometric properties while sampling the object interior regularly as with the surface boundary based correspondence method of cates et al entropy is used to measure both the probability distribution tightness and the sampling regularity here of the spokes geometric properties evaluation on synthetic and real world lateral ventricle and hippocampus data sets demonstrate improvement in the performance of statistics using the resulting probability distributions this improvement is greater than that achieved by an entropy based correspondence method on the boundary points

automatic and robust computation of 3d medial models incorporating object variability this paper presents a novel processing scheme for the automatic and robust computation of a medial shape model which represents an object population with shape variability the sensitivity of medial descriptions to object variations and small boundary perturbations are fundamental problems of any skeletonization technique these problems are approached with the computation of a model with common medial branching topology and grid sampling this model is then used for a medial shape description of individual objects via a constrained model fit the process starts from parametric 3d boundary representations with existing point to point homology between objects the voronoi skeleton of each sampled object boundary is partitioned into non branching medial sheets and simplified by a novel pruning algorithm using a volumetric contribution criterion using the surface homology medial sheets are combined to form a common medial branching topology finally the medial sheets are sampled and represented as meshes of medial primitives results on populations of up to 184 biological objects clearly demonstrate that the common medial branching topology can be described by a small number of medial sheets and that even a coarse sampling leads to a close approximation of individual objects

a skeletonization algorithm by maxima tracking on euclidean distance transform abstract a simple and efficient algorithm using the maxima tracking approach on euclidean distance transform to detect skeleton points is presented the advantages of the skeleton obtained are 1 connectivity preservation 2 single pixel in width and 3 its locations as close as to the most symmetrical axes besides the condition of the least slope change of skeleton is used to ensure the fairness of the digital medial axes with the least effort the algorithm can be modified to eliminate non significant short skeletal branches originating from the object contour while the critical shape informative medial axes are preserved

a parameterization based numerical method for isotropic and anisotropic diffusion smoothing on non flat surfaces neuroimaging data such as 3d maps of cortical thickness or neural activation can often be analyzed more informatively with respect to the cortical surface rather than the entire volume of the brain any cortical surface based analysis should be carried out using computations in the intrinsic geometry of the surface rather than using the metric of the ambient 3d space we present parameterization based numerical methods for performing isotropic and anisotropic filtering on triangulated surface geometries in contrast to existing fem based methods for triangulated geometries our approach accounts for the metric of the surface in order to discretize and numerically compute the isotropic and anisotropic geometric operators we first parameterize the surface using a p harmonic mapping we then use this parameterization as our computational domain and account for the surface metric while carrying out isotropic and anisotropic filtering to validate our method we compare our numerical results to the analytical expression for isotropic diffusion on a spherical surface we apply these methods to smoothing of mean curvature maps on the cortical surface a step commonly required for analysis of gyrification or for registering surface based maps across subjects

aggregation operators preserving quasiconvexity quasiconvexity of a fuzzy set is the necessary and sufficient condition for its cuts to be convex we study the class of those two variable aggregation operators that preserve quasiconvexity on a bounded lattice i e a m n is quasiconvex for quasiconvex lattice valued fuzzy sets m n the class of all such aggregation operators is characterized by a lattice identity that they have to fulfill in case of a unit interval we show the construction of aggregation operators preserving quasiconvexity from a pair of real valued functions on the unit interval as a consequence we get that the intersection of quasiconvex fuzzy sets is quasiconvex if and only if the intersection is based on the minimum triangular norm