Cortical cartography using the discrete conformal approach of circle packings

Automatic landmark detection and registration of brain cortical surfaces via quasi-conformal geometry and convolutional neural networks

In medical imaging, surface registration is extensively used for performing systematic comparisons between anatomical structures, with a prime example being the highly convoluted brain cortical surfaces. To obtain a meaningful registration, a common approach is to identify prominent features on the surfaces and establish a low-distortion mapping between them with the feature correspondence encoded as landmark constraints. Prior registration works have primarily focused on using manually labeled landmarks and solving highly nonlinear optimization problems, which are time-consuming and hence hinder practical applications. In this work, we propose a novel framework for the automatic landmark detection and registration of brain cortical surfaces using quasi-conformal geometry and convolutional neural networks. We first develop a landmark detection network (LD-Net) that allows for the automatic extraction of landmark curves given two prescribed starting and ending points based on the surface geometry. We then utilize the detected landmarks and quasi-conformal theory for achieving the surface registration. Specifically, we develop a coefficient prediction network (CP-Net) for predicting the Beltrami coefficients associated with the desired landmark-based registration and a mapping network called the disk Beltrami solver network (DBS-Net) for generating quasi-conformal mappings from the predicted Beltrami coefficients, with the bijectivity guaranteed by quasi-conformal theory. Experimental results are presented to demonstrate the effectiveness of our proposed framework. Altogether, our work paves a new way for surface-based morphometry and medical shape analysis.

Fast Spherical Mapping of Cortical Surface Meshes Using Deep Unsupervised Learning

Spherical mapping of cortical surface meshes provides a more convenient and accurate space for cortical surface registration and analysis and thus has been widely adopted in neuroimaging field. Conventional approaches typically first inflate and project the original cortical surface mesh onto a sphere to generate an initial spherical mesh which contains large distortions. Then they iteratively reshape the spherical mesh to minimize the metric (distance), area or angle distortions. However, these methods suffer from two major issues: 1) the iterative optimization process is computationally expensive, making them not suitable for large-scale data processing; 2) when metric distortion cannot be further minimized, either area or angle distortion is minimized at the expense of the other, which is not flexible to generate application-specific meshes based on both of them. To address these issues, for the first time, we propose a deep learning-based algorithm to learn the mapping between the original cortical surface and spherical surface meshes. Specifically, we take advantage of the Spherical U-Net model to learn the spherical diffeomorphic deformation field for minimizing the distortions between the icosahedron-reparameterized original surface and spherical surface meshes. The end-to-end unsupervised learning scheme is very flexible to incorporate various optimization objectives. We further integrate it into a coarse-to-fine multi-resolution framework for better correcting fine-scaled distortions. We have validated our method on 800+ cortical surfaces, demonstrating reduced distortions than FreeSurfer (the most popularly used tool), while speeding up the process from 20 min to 5 s.

Quantitative characterization of the human retinotopic map based on quasiconformal mapping

The retinotopic map depicts the cortical neurons' response to visual stimuli on the retina and has contributed significantly to our understanding of human visual system. Although recent advances in high field functional magnetic resonance imaging (fMRI) have made it possible to generate the in vivo retinotopic map with great detail, quantifying the map remains challenging. Existing quantification methods do not preserve surface topology and often introduce large geometric distortions to the map. In this study, we developed a new framework based on computational conformal geometry and quasiconformal Teichmüller theory to quantify the retinotopic map. Specifically, we introduced a general pipeline, consisting of cortical surface conformal parameterization, surface-spline-based cortical activation signal smoothing, and vertex-wise Beltrami coefficient-based map description. After correcting most of the violations of the topological conditions, the result was a "Beltrami coefficient map" (BCM) that rigorously and completely characterizes the retinotopic map by quantifying the local quasiconformal mapping distortion at each visual field location. The BCM provided topological and fully reconstructable retinotopic maps. We successfully applied the new framework to analyze the V1 retinotopic maps from the Human Connectome Project (n=181), the largest state of the art retinotopy dataset currently available. With unprecedented precision, we found that the V1 retinotopic map was quasiconformal and the local mapping distortions were similar across observers. The new framework can be applied to other visual areas and retinotopic maps of individuals with and without eye diseases, and improve our understanding of visual cortical organization in normal and clinical populations.

Computational geometric tools for quantitative comparison of locomotory behavior

A fundamental challenge for behavioral neuroscientists is to accurately quantify (dis)similarities in animal behavior without excluding inherent variability present between individuals. We explored two new applications of curve and shape alignment techniques to address this issue. As a proof-of-concept we applied these methods to compare normal or alarmed behavior in pairs of medaka (Oryzias latipes). The curve alignment method we call Behavioral Distortion Distance (BDD) revealed that alarmed fish display less predictable swimming over time, even if individuals incorporate the same action patterns like immobility, sudden changes in swimming trajectory, or changing their position in the water column. The Conformal Spatiotemporal Distance (CSD) technique on the other hand revealed that, in spite of the unpredictability, alarmed individuals exhibit lower variability in overall swim patterns, possibly accounting for the widely held notion of "stereotypy" in alarm responses. More generally, we propose that these new applications of established computational geometric techniques are useful in combination to represent, compare, and quantify complex behaviors consisting of common action patterns that differ in duration, sequence, or frequency.

LMap: Shape-Preserving Local Mappings for Biomedical Visualization

Visualization of medical organs and biological structures is a challenging task because of their complex geometry and the resultant occlusions. Global spherical and planar mapping techniques simplify the complex geometry and resolve the occlusions to aid in visualization. However, while resolving the occlusions these techniques do not preserve the geometric context, making them less suitable for mission-critical biomedical visualization tasks. In this paper, we present a shape-preserving local mapping technique for resolving occlusions locally while preserving the overall geometric context. More specifically, we present a novel visualization algorithm, LMap, for conformally parameterizing and deforming a selected local region-of-interest (ROI) on an arbitrary surface. The resultant shape-preserving local mappings help to visualize complex surfaces while preserving the overall geometric context. The algorithm is based on the robust and efficient extrinsic Ricci flow technique, and uses the dynamic Ricci flow algorithm to guarantee the existence of a local map for a selected ROI on an arbitrary surface. We show the effectiveness and efficacy of our method in three challenging use cases: (1) multimodal brain visualization, (2) optimal coverage of virtual colonoscopy centerline flythrough, and (3) molecular surface visualization.

4D hyperspherical harmonic (HyperSPHARM) representation of surface anatomy: a holistic treatment of multiple disconnected anatomical structures

Image-based parcellation of the brain often leads to multiple disconnected anatomical structures, which pose significant challenges for analyses of morphological shapes. Existing shape models, such as the widely used spherical harmonic (SPHARM) representation, assume topological invariance, so are unable to simultaneously parameterize multiple disjoint structures. In such a situation, SPHARM has to be applied separately to each individual structure. We present a novel surface parameterization technique using 4D hyperspherical harmonics in representing multiple disjoint objects as a single analytic function, terming it HyperSPHARM. The underlying idea behind HyperSPHARM is to stereographically project an entire collection of disjoint 3D objects onto the 4D hypersphere and subsequently simultaneously parameterize them with the 4D hyperspherical harmonics. Hence, HyperSPHARM allows for a holistic treatment of multiple disjoint objects, unlike SPHARM. In an imaging dataset of healthy adult human brains, we apply HyperSPHARM to the hippocampi and amygdalae. The HyperSPHARM representations are employed as a data smoothing technique, while the HyperSPHARM coefficients are utilized in a support vector machine setting for object classification. HyperSPHARM yields nearly identical results as SPHARM, as will be shown in the paper. Its key advantage over SPHARM lies computationally; HyperSPHARM possess greater computational efficiency than SPHARM because it can parameterize multiple disjoint structures using much fewer basis functions and stereographic projection obviates SPHARM's burdensome surface flattening. In addition, HyperSPHARM can handle any type of topology, unlike SPHARM, whose analysis is confined to topologically invariant structures.

Brain Morphometry on Congenital Hand Deformities based on Teichmüller Space Theory

Congenital Hand Deformities (CHD) are usually occurred between fourth and eighth week after the embryo is formed. Failure of the transformation from arm bud cells to upper limb can lead to an abnormal appearing/functioning upper extremity which is presented at birth. Some causes are linked to genetics while others are affected by the environment, and the rest have remained unknown. CHD patients develop prehension through the use of their hands, which affect the brain as time passes. In recent years, CHD have gain increasing attention and researches have been conducted on CHD, both surgically and psychologically. However, the impacts of CHD on brain structure are not well-understood so far. Here, we propose a novel approach to apply Teichmüller space theory and conformal welding method to study brain morphometry in CHD patients. Conformal welding signature reflects the geometric relations among different functional areas on the cortex surface, which is intrinsic to the Riemannian metric, invariant under conformal deformation, and encodes complete information of the functional area boundaries. The computational algorithm is based on discrete surface Ricci flow, which has theoretic guarantees for the existence and uniqueness of the solutions. In practice, discrete Ricci flow is equivalent to a convex optimization problem, therefore has high numerically stability. In this paper, we compute the signatures of contours on general 3D surfaces with surface Ricci flow method, which encodes both global and local surface contour information. Then we evaluated the signatures of pre-central and post-central gyrus on healthy control and CHD subjects for analyzing brain cortical morphometry. Preliminary experimental results from 3D MRI data of CHD/control data demonstrate the effectiveness of our method. The statistical comparison between left and right brain gives us a better understanding on brain morphometry of subjects with Congenital Hand Deformities, in particular, missing the distal part of the upper limb.

Studying ventricular abnormalities in mild cognitive impairment with hyperbolic Ricci flow and tensor-based morphometry

Mild Cognitive Impairment (MCI) is a transitional stage between normal aging and dementia and people with MCI are at high risk of progression to dementia. MCI is attracting increasing attention, as it offers an opportunity to target the disease process during an early symptomatic stage. Structural magnetic resonance imaging (MRI) measures have been the mainstay of Alzheimer's disease (AD) imaging research, however, ventricular morphometry analysis remains challenging because of its complicated topological structure. Here we describe a novel ventricular morphometry system based on the hyperbolic Ricci flow method and tensor-based morphometry (TBM) statistics. Unlike prior ventricular surface parameterization methods, hyperbolic conformal parameterization is angle-preserving and does not have any singularities. Our system generates a one-to-one diffeomorphic mapping between ventricular surfaces with consistent boundary matching conditions. The TBM statistics encode a great deal of surface deformation information that could be inaccessible or overlooked by other methods. We applied our system to the baseline MRI scans of a set of MCI subjects from the Alzheimer's Disease Neuroimaging Initiative (ADNI: 71 MCI converters vs. 62 MCI stable). Although the combined ventricular area and volume features did not differ between the two groups, our fine-grained surface analysis revealed significant differences in the ventricular regions close to the temporal lobe and posterior cingulate, structures that are affected early in AD. Significant correlations were also detected between ventricular morphometry, neuropsychological measures, and a previously described imaging index based on fluorodeoxyglucose positron emission tomography (FDG-PET) scans. This novel ventricular morphometry method may offer a new and more sensitive approach to study preclinical and early symptomatic stage AD.

Metric optimization for surface analysis in the Laplace-Beltrami embedding space

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

Conformal mapping via metric optimization with application for cortical label fusion

In this paper we develop a novel approach for computing conformal maps between anatomical surfaces with the ability of aligning anatomical features and achieving greatly reduced metric distortion. In contrast to conventional approaches that focused on conformal maps to the sphere or plane, our method computes the conformal map between surfaces in the embedding space formed the intrinsically defined Laplace-Beltrami (LB) eigenfunctions. Utilizing the power of LB eigenfunctions as informative descriptors of global geometry, the conformal maps computed by our method can effectively align anatomical features on cortical surfaces. By computing such feature-aware conformal maps to a group-wisely optimal atlas surface, which is also computed with metric optimization in the LB embedding space, we develop a fully automated system for cortical labeling with the fusion of labels on a large number of atlas surfaces. In our experiments, we build our system with 40 labeled surfaces and demonstrate its excellent performance with leave-one-out cross validation. We also applied the automated labeling system to cortical surfaces reconstructed from MR scans of 50 patients with Alzheimer's disease (AD) and 50 normal controls (NC) to illustrate its robustness and effectiveness in clinical data analysis.

Ricci Flow-based Spherical Parameterization and Surface Registration

This paper presents an improved Euclidean Ricci flow method for spherical parameterization. We subsequently invent a scale space processing built upon Ricci energy to extract robust surface features for accurate surface registration. Since our method is based on the proposed Euclidean Ricci flow, it inherits the properties of Ricci flow such as conformality, robustness and intrinsicalness, facilitating efficient and effective surface mapping. Compared with other surface registration methods using curvature or sulci pattern, our method demonstrates a significant improvement for surface registration. In addition, Ricci energy can capture local differences for surface analysis as shown in the experiments and applications.

Quantitative analysis of structural neuroimaging of mesial temporal lobe epilepsy

Mesial temporal lobe epilepsy (MTLE) is the most common of the surgically remediable drug-resistant epilepsies. MRI is the primary diagnostic tool to detect anatomical abnormalities and, when combined with EEG, can more accurately identify an epileptogenic lesion, which is often hippocampal sclerosis in cases of MTLE. As structural imaging technology has advanced the surgical treatment of MTLE and other lesional epilepsies, so too have the analysis techniques that are used to measure different structural attributes of the brain. These techniques, which are reviewed here and have been used chiefly in basic research of epilepsy and in studies of MTLE, have identified different types and the extent of anatomical abnormalities that can extend beyond the affected hippocampus. These results suggest that structural imaging and sophisticated imaging analysis could provide important information to identify networks capable of generating spontaneous seizures and ultimately help guide surgical therapy that improves postsurgical seizure-freedom outcomes.

Model-driven harmonic parameterization of the cortical surface: HIP-HOP

In the context of inter subject brain surface matching, we present a parameterization of the cortical surface constrained by a model of cortical organization. The parameterization is defined via an harmonic mapping of each hemisphere surface to a rectangular planar domain that integrates a representation of the model. As opposed to previous landmark-based registration methods we do not match folds between individuals but instead optimize the fit between cortical sulci and specific iso-coordinate axis in the model. This strategy overcomes some limitation to sulcus-based registration techniques such as topological variability in sulcal landmarks across subjects. Experiments on 62 subjects with manually traced sulci are presented and compared with the result of the Freesurfer software. The evaluation involves a measure of dispersion of sulci with both angular and area distortions. We show that the model-based strategy can lead to a natural, efficient and very fast (less than 5 min per hemisphere) method for defining inter subjects correspondences. We discuss how this approach also reduces the problems inherent to anatomically defined landmarks and open the way to the investigation of cortical organization through the notion of orientation and alignment of structures across the cortex.

Applying tensor-based morphometry to parametric surfaces can improve MRI-based disease diagnosis

Many methods have been proposed for computer-assisted diagnostic classification. Full tensor information and machine learning with 3D maps derived from brain images may help detect subtle differences or classify subjects into different groups. Here we develop a new approach to apply tensor-based morphometry to parametric surface models for diagnostic classification. We use this approach to identify cortical surface features for use in diagnostic classifiers. First, with holomorphic 1-forms, we compute an efficient and accurate conformal mapping from a multiply connected mesh to the so-called slit domain. Next, the surface parameterization approach provides a natural way to register anatomical surfaces across subjects using a constrained harmonic map. To analyze anatomical differences, we then analyze the full Riemannian surface metric tensors, which retain multivariate information on local surface geometry. As the number of voxels in a 3D image is large, sparse learning is a promising method to select a subset of imaging features and to improve classification accuracy. Focusing on vertices with greatest effect sizes, we train a diagnostic classifier using the surface features selected by an L1-norm based sparse learning method. Stability selection is applied to validate the selected feature sets. We tested the algorithm on MRI-derived cortical surfaces from 42 subjects with genetically confirmed Williams syndrome and 40 age-matched controls, multivariate statistics on the local tensors gave greater effect sizes for detecting group differences relative to other TBM-based statistics including analysis of the Jacobian determinant and the largest eigenvalue of the surface metric. Our method also gave reasonable classification results relative to the Jacobian determinant, the pair of eigenvalues of the Jacobian matrix and volume features. This analysis pipeline may boost the power of morphometry studies, and may assist with image-based classification.

Hemispherically-unified surface maps of human cerebral cortex: reliability and hemispheric asymmetries

Understanding the anatomical and structural organization of the cerebral cortex is facilitated by surface-based analysis enabled by FreeSurfer, Caret, and related tools. Here, we examine the precision of FreeSurfer parcellation of the cortex and introduce a method to align FreeSurfer-registered left and right hemispheres onto a common template in order to characterize hemispheric asymmetries. The results are visualized using Mollweide projections, an area-preserving map. The regional distribution, inter-hemispheric asymmetries and intersubject variability in cortical curvature, sulcal depth, cortical thickness, and cortical surface area of 138 young, right handed subjects were analyzed on the Mollweide projection map of the common spherical space. The results show that gyral and sulcal structures are aligned with high but variable accuracy in different cortical regions and show consistent hemispheric asymmetries that are maximal in posterior temporal regions.

Brain surface conformal parameterization with the Ricci flow

In brain mapping research, parameterized 3-D surface models are of great interest for statistical comparisons of anatomy, surface-based registration, and signal processing. Here, we introduce the theories of continuous and discrete surface Ricci flow, which can create Riemannian metrics on surfaces with arbitrary topologies with user-defined Gaussian curvatures. The resulting conformal parameterizations have no singularities and they are intrinsic and stable. First, we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves. Secondly, we conformally parameterize each cortical surface to a parameter domain with a user-designed Gaussian curvature arrangement. In the parameter domain, a shape index based on conformal invariants is computed, and inter-subject cortical surface matching is performed by solving a constrained harmonic map. We illustrate various target curvature arrangements and demonstrate the stability of the method using longitudinal data. To map statistical differences in cortical morphometry, we studied brain asymmetry in 14 healthy control subjects. We used a manifold version of Hotelling's T(2) test, applied to the Jacobian matrices of the surface parameterizations. A permutation test, along with the cumulative distribution of p-values, were used to estimate the overall statistical significance of differences. The results show our algorithm's power to detect subtle group differences in cortical surfaces.

Algorithms to improve the reparameterization of spherical mappings of brain surface meshes

A spherical map of a cortical surface is often used for improved brain registration, for advanced morphometric analysis (eg, of brain shape), and for surface-based analysis of functional signals recorded from the cortex. Furthermore, for intersubject analysis, it is usually necessary to reparameterize the surface mesh into a common coordinate system. An isometric map conserves all angle and area information in the original cortical mesh; however, in practice, spherical maps contain some distortion. Here, we propose fast new algorithms to reduce the distortion of initial spherical mappings generated using one of three common spherical mapping methods. The algorithms iteratively solve a nonlinear optimization problem to reduce distortion. Our results demonstrate that our correction process is computationally inexpensive and the resulting spherical maps have improved distortion metrics. We show that our corrected spherical maps improve reparameterization of the cortical surface mesh, such that the distance error measures between the original and reparameterized surface are significantly decreased.

Multimodality data integration in epilepsy

An important goal of software development in the medical field is the design of methods which are able to integrate information obtained from various imaging and nonimaging modalities into a cohesive framework in order to understand the results of qualitatively different measurements in a larger context. Moreover, it is essential to assess the various features of the data quantitatively so that relationships in anatomical and functional domains between complementing modalities can be expressed mathematically. This paper presents a clinically feasible software environment for the quantitative assessment of the relationship among biochemical functions as assessed by PET imaging and electrophysiological parameters derived from intracranial EEG. Based on the developed software tools, quantitative results obtained from individual modalities can be merged into a data structure allowing a consistent framework for advanced data mining techniques and 3D visualization. Moreover, an effort was made to derive quantitative variables (such as the spatial proximity index, SPI) characterizing the relationship between complementing modalities on a more generic level as a prerequisite for efficient data mining strategies. We describe the implementation of this software environment in twelve children (mean age 5.2 +/- 4.3 years) with medically intractable partial epilepsy who underwent both high-resolution structural MR and functional PET imaging. Our experiments demonstrate that our approach will lead to a better understanding of the mechanisms of epileptogenesis and might ultimately have an impact on treatment. Moreover, our software environment holds promise to be useful in many other neurological disorders, where integration of multimodality data is crucial for a better understanding of the underlying disease mechanisms.

General multivariate linear modeling of surface shapes using SurfStat

Although there are many imaging studies on traditional ROI-based amygdala volumetry, there are very few studies on modeling amygdala shape variations. This paper presents a unified computational and statistical framework for modeling amygdala shape variations in a clinical population. The weighted spherical harmonic representation is used to parameterize, smooth out, and normalize amygdala surfaces. The representation is subsequently used as an input for multivariate linear models accounting for nuisance covariates such as age and brain size difference using the SurfStat package that completely avoids the complexity of specifying design matrices. The methodology has been applied for quantifying abnormal local amygdala shape variations in 22 high functioning autistic subjects.

Teichmöller shape space theory and its application to brain morphometry

Here we propose a novel method to compute Teichmiiller shape space based shape index to study brain morphometry. Such a shape index is intrinsic, and invariant under conformal transformations, rigid motions and scaling. We conformally map a genus-zero open boundary surface to the Poincaré disk with the Yamabe flow method. The shape indices that we compute are the lengths of a special set of geodesics under hyperbolic metric. Tests on longitudinal brain imaging data were used to demonstrate the stability of the derived feature vectors. In leave-one-out validation tests, we achieved 100% accurate classification (versus only 68% accuracy for volume measures) in distinguishing 11 HIV/AIDS individuals from 8 healthy control subjects, based on Teichmüller coordinates for lateral ventricular surfaces extracted from their 3D MRI scans.

Near-isometric flattening of brain surfaces

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

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.

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.

Quantitative brain surface mapping of an electrophysiologic/metabolic mismatch in human neocortical epilepsy

The spatial relationship between an intracranial EEG-defined epileptic focus and cortical hypometabolism on glucose PET has not been precisely described. In order to quantitatively evaluate the hypothesis that ictal seizure onset and/or rapid seizure propagation, detected by subdural EEG monitoring, commonly involves normometabolic cortex adjacent to hypometabolic cortical regions, we applied a novel, landmark-constrained conformal mapping approach in 14 children with refractory neocortical epilepsy. The 3D brain surface was parcellated into finite cortical elements (FCEs), and hypometabolism was defined using lobe- and side-specific asymmetry indices derived from normal adult controls. The severity and location of hypometabolic areas vs. ictal intracranial EEG abnormalities were compared on the 3D brain surface. Hypometabolism was more severe in the seizure onset zone than in cortical areas covered by non-onset electrodes. However, similar proportions of the onset electrodes were located over and adjacent to (within 2 cm) hypometabolic regions (46% vs. 41%, respectively), whereas rapid seizure spread electrodes preferred these "adjacent areas" rather than the hypometabolic area itself (51% vs. 22%). On average, 58% of the hypometabolic regions had no early seizure involvement. These findings strongly support that the seizure onset zone often extends from hypometabolic to adjacent normometabolic cortex, while large portions of hypometabolic cortex are not involved in seizure onset or early propagation. The clinical utility of FDG PET in guiding subdural electrode placement in neocortical epilepsy could be greatly enhanced by extending grid coverage to at least 2 cm beyond hypometabolic cortex, when feasible.

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.

Persistence diagrams of cortical surface data

We present a novel framework for characterizing signals in images using techniques from computational algebraic topology. This technique is general enough for dealing with noisy multivariate data including geometric noise. The main tool is persistent homology which can be encoded in persistence diagrams. These diagrams visually show how the number of connected components of the sublevel sets of the signal changes. The use of local critical values of a function differs from the usual statistical parametric mapping framework, which mainly uses the mean signal in quantifying imaging data. Our proposed method uses all the local critical values in characterizing the signal and by doing so offers a completely new data reduction and analysis framework for quantifying the signal. As an illustration, we apply this method to a 1D simulated signal and 2D cortical thickness data. In case of the latter, extra homological structures are evident in an control group over the autistic group.

Conformal slit mapping and its applications to brain surface parameterization

We propose a method that computes a conformal mapping from a multiply connected mesh to the so-called slit domain, which consists of a canonical rectangle or disk in which 3D curved landmarks on the original surfaces are mapped to concentric or parallel lines in the slit domain. In this paper, we studied its application to brain surface parameterization. After cutting along some landmark curve features on surface models of the cerebral cortex, we obtain multiple connected domains. By computing exact harmonic one-forms, closed harmonic one-forms, and holomorphic one-forms, we are able to build a circular slit mapping that conformally maps the surface to an annulus with some concentric arcs and a rectangle with some slits. The whole algorithm is based on solving linear systems so it is very stable. In the slit domain parameterization results, the feature curves are either mapped to straight lines or concentric arcs. This representation is convenient for anatomical visualization, and may assist statistical comparisons of anatomy, surface-based registration and signal processing. Preliminary experimental results parameterizing various brain anatomical surfaces are presented.

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.

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 healthy comparison subjects.

Brain surface conformal parameterization using Riemann surface structure

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

Cortical mapping of genotype-phenotype relationships in schizophrenia

Although schizophrenia is highly heritable, the search for susceptibility genes has been challenging. The "endophenotype" approach is an alternative method for measuring phenotypic variation that may make it easier to identify susceptibility genes in the context of complexly inherited traits. Neuroimaging methods in particular offer a powerful way to bridge the neurobiology of genes and behavior. Such investigations may be further empowered by complementary strategies involving chromosomal abnormalities associated with schizophrenia, which can help to localize causative genes and better understand the genetic complexity of the illness. Here, we illustrate our use of these convergent approaches, with a focus on neuroimaging studies using novel computational brain mapping algorithms, to investigate genetic influences on brain structure in the development of psychosis. These studies provide compelling evidence that specific genetic loci suspected to predispose to schizophrenia may affect quantitative variation in neural indicators underlying the neurobehavioral phenotype, and illustrate how genetic-neuroimaging paradigms can improve our understanding of the pathogenesis of this highly disabling mental illness.

Direct cortical mapping via solving partial differential equations on implicit surfaces

In this paper, we propose a novel approach for cortical mapping that computes a direct map between two cortical surfaces while satisfying constraints on sulcal landmark curves. By computing the map directly, we can avoid conventional intermediate parameterizations and help simplify the cortical mapping process. The direct map in our method is formulated as the minimizer of a flexible variational energy under landmark constraints. The energy can include both a harmonic term to ensure smoothness of the map and general data terms for the matching of geometric features. Starting from a properly designed initial map, we compute the map iteratively by solving a partial differential equation (PDE) defined on the source cortical surface. For numerical implementation, a set of adaptive numerical schemes are developed to extend the technique of solving PDEs on implicit surfaces such that landmark constraints are enforced. In our experiments, we show the flexibility of the direct mapping approach by computing smooth maps following landmark constraints from two different energies. We also quantitatively compare the metric preserving property of the direct mapping method with a parametric mapping method on a group of 30 subjects. Finally, we demonstrate the direct mapping method in the brain mapping applications of atlas construction and variability analysis.

Brain surface conformal parameterization with algebraic functions

In medical imaging, parameterized 3D surface models are of great interest for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal processing. Here we introduce a parameterization method based on algebraic functions. By solving the Yamabe equation with the Ricci flow method, we can conformally map a brain surface to a multi-hole disk. The resulting parameterizations do not have any singularities and are intrinsic and stable. To illustrate the technique, we computed parameterizations of several types of anatomical surfaces in MRI scans of the brain, including the hippocampi and the cerebral cortices with various landmark curves labeled. For the cerebral cortical surfaces, we show the parameterization results are consistent with selected landmark curves and can be matched to each other using constrained harmonic maps. Unlike previous planar conformal parameterization methods, our algorithm does not introduce any singularity points. It also offers a method to explicitly match landmark curves between anatomical surfaces such as the cortex, and to compute conformal invariants for statistical comparisons of anatomy.

Multiscale 3-D shape representation and segmentation using spherical wavelets

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

Brain surface parameterization using Riemann surface structure

We develop a general approach that uses holomorphic 1-forms to parameterize anatomical surfaces with complex (possibly branching) topology. Rather than evolve the surface geometry to a plane or sphere, we instead use the fact that all orientable surfaces are Riemann surfaces and admit conformal structures, which induce special curvilinear coordinate systems on the surfaces. Based on Riemann surface structure, we can then canonically partition the surface into patches. Each of these patches can be conformally mapped to a parallelogram. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable. To illustrate the technique, we computed conformal structures for several types of anatomical surfaces in MRI scans of the brain, including the cortex, hippocampus, and lateral ventricles. We found that the resulting parameterizations were consistent across subjects, even for branching structures such as the ventricles, which are otherwise difficult to parameterize. Compared with other variational approaches based on surface inflation, our technique works on surfaces with arbitrary complexity while guaranteeing minimal distortion in the parameterization. It also offers a way to explicitly match landmark curves in anatomical surfaces such as the cortex, providing a surface-based framework to compare anatomy statistically and to generate grids on surfaces for PDE-based signal processing.

Optimization of brain conformal mapping with landmarks

To compare and integrate brain data, data from multiple subjects are typically mapped into a canonical space. One method to do this is to conformally map cortical surfaces to the sphere. It is well known that any genus zero Riemann surface can be conformally mapped to a sphere. Therefore, conformal mapping offers a convenient method to parameterize cortical surfaces without angular distortion, generating an orthogonal grid on the cortex that locally preserves the metric. To compare cortical surfaces more effectively, it is advantageous to adjust the conformal parameterizations to match consistent anatomical features across subjects. This matching of cortical patterns improves the alignment of data across subjects, although it is more challenging to create a consistent conformal (orthogonal) parameterization of anatomy across subjects when landmarks are constrained to lie at specific locations in the spherical parameter space. Here we propose a new method, based on a new energy functional, to optimize the conformal parameterization of cortical surfaces by using landmarks. Experimental results on a dataset of 40 brain hemispheres showed that the landmark mismatch energy can be greatly reduced while effectively preserving conformality. The key advantage of this conformal parameterization approach is that any local adjustments of the mapping to match landmarks do not affect the conformality of the mapping significantly. We also examined how the parameterization changes with different weighting factors. As expected, the landmark matching error can be reduced if it is more heavily penalized, but conformality is progressively reduced.

Mapping heritability and molecular genetic associations with cortical features using probabilistic brain atlases: methods and applications to schizophrenia

There is an urgent need to decipher the complex nature of genotype-phenotype relationships within the multiple dimensions of brain structure and function that are compromised in neuropsychiatric syndromes such as schizophrenia. Doing so requires sophisticated methodologies to represent population variability in neural traits and to probe their heritable and molecular genetic bases. We have recently developed and applied computational algorithms to map the heritability of, as well as genetic linkage and association to, neural features encoded using brain imaging in the context of three-dimensional (3D), populationbased, statistical brain atlases. One set of algorithms builds on our prior work using classical twin study methods to estimate heritability by fitting biometrical models for additive genetic, unique, and common environmental influences. Another set of algorithms performs regression-based (Haseman-Elston) identical-bydescent linkage analysis and genetic association analysis of DNA polymorphisms in relation to neural traits of interest in the same 3D population-based brain atlas format. We demonstrate these approaches using samples of healthy monozygotic (MZ) and dizygotic (DZ) twin pairs, as well as MZ and DZ twin pairs discordant for schizophrenia, but the methods can be generalized to other classes of relatives and to other diseases. The results confirm prior evidence of genetic influences on gray matter density in frontal brain regions. They also provide converging evidence that the chromosome 1q42 region is relevant to schizophrenia by demonstrating linkage and association of markers of the Transelin-Associated-Factor-X and Disrupted-In- Schizophrenia-1 genes with prefrontal cortical gray matter deficits in twins discordant for schizophrenia.

LANDMARK MATCHING ON THE SPHERE USING DISTANCE FUNCTIONS

Nonlinear registration of 3D surfaces is important in many medical imaging applications, including the mapping of longitudinal changes in anatomy, or of multi-subject functional MRI data to a canonical surface for comparison and integration. To register 3D surfaces, such as the cortical surface of the brain, one approach is to transform them first to planar or spherical objects. Internal landmarks can then be matched on these simpler parameter domains. Here we study the diffeomorphic matching of landmarks on the sphere. Our method builds on the level set technique of Leow et al. [1] for the plane. Both forward and backward matching terms are included, thus ensuring the invertibility of the representation. We demonstrate our technique on a pair of lines on the sphere. The overall approach improves on earlier work in cortical matching by allowing the matching energy to be relaxed along sulcal landmarks, minimizing distortion, and also enables point and curve landmarks to be aligned in the same general framework as densely-defined scalar fields, such as curvature or cortical thickness maps.

Quantitative evaluation of three cortical surface flattening methods

During the past decade, several computational approaches have been proposed for the task of mapping highly convoluted surfaces of the human brain to simpler geometric objects such as a sphere or a topological disc. We report the results of a quantitative comparison of FreeSurfer, CirclePack, and LSCM with respect to measurements of geometric distortion and computational speed. Our results indicate that FreeSurfer performs best with respect to a global measurement of metric distortion, whereas LSCM performs best with respect to angular distortion and best in all but one case with a local measurement of metric distortion. FreeSurfer provides more homogeneous distribution of metric distortion across the whole cortex than CirclePack and LSCM. LSCM is the most computationally efficient algorithm for generating spherical maps, while CirclePack is extremely fast for generating planar maps from patches.

Computational anatomy: shape, growth, and atrophy comparison via diffeomorphisms

Computational anatomy (CA) is the mathematical study of anatomy I in I = I(alpha) o G, an orbit under groups of diffeomorphisms (i.e., smooth invertible mappings) g in G of anatomical exemplars I(alpha) in I. The observable images are the output of medical imaging devices. There are three components that CA examines: (i) constructions of the anatomical submanifolds, (ii) comparison of the anatomical manifolds via estimation of the underlying diffeomorphisms g in G defining the shape or geometry of the anatomical manifolds, and (iii) generation of probability laws of anatomical variation P(.) on the images I for inference and disease testing within anatomical models. This paper reviews recent advances in these three areas applied to shape, growth, and atrophy.

Validation of semiautomated methods for quantifying cingulate cortical metrics in schizophrenia

This paper validates semiautomated methods for reconstructing cortical surfaces of the cingulate gyrus from high-resolution magnetic resonance (MR) images. Bayesian segmentation was used to delineate the image voxels into five tissue types: cerebrospinal fluid (CSF), gray matter (GM), white matter (WM), and partial volumes of CSF/GM and GM/WM; the tissues were then recalibrated as CSF, GM, and WM via the Neyman-Pearson Likelihood Ratio Test. To generate cortical surfaces at the interface of GM and WM, the thresholds between the tissue types were first used to reassign partial volume voxels to CSF, GM, and WM with minimum error (that varied from 0.06 to 0.15 for the 10 subjects). Next, topology-correct cortical surfaces were generated and validated with almost all surface vertices lying within one voxel (0.5 mm) of hand contours. Dynamic programming was used to delineate and extract the cingulate gyrus from the cortical surfaces based on its gyral and sulcal boundaries. The intraclass correlation coefficient for surface area obtained by two raters for all 10 surfaces was 0.82. In addition, by repeating the entire procedure three times in one subject, we obtained a coefficient of variation of 0.0438 for surface area.