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Studying Brain Morphometry Using Conformal Equivalence Class Studying Brain Morphometry using Conformal Equivalence Class Yalin Wang Wei Dai Yi-Yu Chou Dept of Neurology/Math Dept of Math Dept of Neurology UCLA Zhejiang Univ UCLA [email protected] [email protected] [email protected] Xianfeng Gu Tony F. Chan Arthur W. Toga Paul M. Thompson Dept of Comp Sci Dept of Math Dept of Neurology Dept of Neurology Stony Brook Univ UCLA UCLA UCLA [email protected] [email protected] [email protected] [email protected] Abstract 1. Introduction 3D Surface shape analysis is a key research topic in face recognition [12], anatomical modeling, statistical com- parisons of anatomy and medical image registration. In Two surfaces are conformally equivalent if there exists research studies that analyze brain morphometry, many a bijective angle-preserving map between them. The Te- shape analysis methods have been proposed, such as spher- ichmuller¨ space for surfaces with the same topology is a ical harmonic analysis (SPHARM) [4], medial represen- finite-dimensional manifold, where each point represents a tations (M-reps) [14], and minimum description length ap- conformal equivalence class, and the conformal map is ho- proaches [5], etc.; these methods may be applied to analyze motopic to the identity map. In this paper, we propose a shape changes or abnormalities in subcortical brain struc- novel method to apply conformal equivalence based shape tures. Even so, a stable method to compute transformation- index to study brain morphometry. The shape index is de- invariant shape descriptors would be highly advantageous fined based on Teichmuller¨ space coordinates. It is intrinsic, in this research field. Here we propose a novel and in- and invariant under conformal transformations, rigid mo- trinsic method to compute a Teichm¨uller space coordinate tions and scaling. It is also simple to compute; no registra- (shape indices) and we apply it to study brain morphometry tion of surfaces is needed. Using the Yamabe flow method, in Alzheimers disease (AD), Williams syndrome (WS) and we can conformally map a genus-zero open boundary sur- HIV/AIDS. Our Teichm¨uller space coordinates are based on face to the Poincare´ disk. The shape indices that we com- the surface conformal structure and can be accurately com- pute are the lengths of a special set of geodesics under hy- puted using the Yamabe flow method. perbolic metric. By computing and studying this shape in- According to Klein’s Erlangen program, different ge- dex and its statistical behavior, we can analyze differences ometries study the invariants under different transforma- in anatomical morphometry due to disease or development. tion groups. Conformal geometry corresponds to the angle- Study on twin lateral ventricular surface data shows it may preserving transformations. If there exists a conformal map help detect generic influence on lateral ventricular shapes. between two surfaces, they are conformally equivalent. All In leave-one-out validation tests, we achieved 100% accu- surfaces can be classified by the conformal equivalence re- rate classification (versus only 68% accuracy for volume lation. For surfaces with the same topology, the Teichm¨uller measures) in distinguishing 11 HIV/AIDS individuals from space is a natural finite-dimensional manifold, where each 8 healthy control subjects, based on Teichmuller¨ coordi- point represents a conformal equivalence class and the dis- nates for lateral ventricular surfaces extracted from their tance between two shapes can be accurately measured. A 3D MRI scans.Our conformal invariants, the Teichmuller¨ shape index can be defined based on Teichm¨uller space co- coordinates, successfully classified all lateral ventricular ordinates. This shape index is intrinsic, and invariant under surfaces, showing their promise for analyzing anatomical conformal transformations, rigid motions and scaling. It is surface morphometry. simple to compute; no surface registration is needed. It is very general; it can handle all arbitrary topology surfaces with negative Euler numbers. By computing and studying tricular surfaces of HIV/AIDS individuals versus matched Teichm¨uller space coordinates and their statistical behav- healthy control subjects. Our work may inspire more re- ior, we can provide a promising approach to describe local searchers to adopt conformal invariant based shape analysis changes or abnormalities in anatomical morphometry due in their own research. to disease or development. In this work, we propose to study the Teichm¨uller space 1.1. Related Work coordinate based shape index with genus-zero surfaces with three boundaries. With the discrete version of the surface In the computational analysis of brain anatomy, volu- Ricci flow method (also called the discrete Yamabe flow), metric measures of structures identified on 3D MRI have we conformally projected the surfaces to the hyperbolic been used to study group differences in brain structure and plane and isometrically embedded them in the Poincar´e also to predict diagnosis [1]. Recent work has also used disk. The proposed Teichm¨uller space coordinates are the shape-based features [13], analyzing surface changes using lengths of a special set of geodesics under this special hy- pointwise displacements of surface meshes, local deforma- perbolic metric. For applications in brain morphometry re- tion tensors, or surface expansion factors, such as the Jaco- search, we first converted a closed 3D surface model of bian determinant of a surface-based mapping. For closed the cerebral cortex into a multiple-boundary surface by cut- surfaces homotopic to a sphere, spherical harmonics have ting it along selected anatomical landmark curves. Sec- commonlybeen used for shape analysis, as have their gener- ondly, we conformally parameterized each cortical surface alizations, e.g., eigenfunctions of the Laplace-Beltrami op- using the Yamabe flow method. Next, we computed the erator in a system of spherical coordinates. These shape in- Teichm¨uller space coordinates - the lengths of three bound- dices are also rotation invariant, i.e., their values do not de- aries (geodesics) on the hyperbolic space - as a 3×1 feature pend on the orientation of the surface in space. Shape analy- vector. This measure is invariant in the hyperbolicplane un- sis based on spherical harmonic basis functions (SPHARM) der conformal transformations of the original surface, and is is usually conductedin three steps, based on a pre-computed the same for surfaces that differ at most by a rigid motion. spherical parameterization of the surface: (1) estimating SH We tested our algorithm on cortical and lateral ventricu- coefficients for the x, y and z-components with a least- lar surfaces extracted from 3D anatomical brain MRI scans. squares procedure, (2) normalizing the orientation of the We applied our algorithm to analyze ventricular shapes in first-order ellipsoid, and (3) reconstructing the surface at 3D volumetric MRI scans from 76 identical and 56 same- regularly spaced points on the sphere [16]. Chung et al. [4] sex fraternal twins. The proposed Teichm¨uller space co- proposed a weighted spherical harmonicrepresentation. For ordinate features picked up stronger generic influence than a specific choice of weights, the weighted SPHARM is volume measures. The proposed algorithm can map the pro- shown to be the least squares approximation to the solution file of differences in surface morphometry between healthy of an anisotropic heat diffusion on the unit sphere. Davies controls and subjects with HIV/AIDS. Finally, we used a et al. studied anatomical shape abnormalities in schizophre- nearest-neighbor classifier together with our feature vector nia, using the minimal distance length approach to statisti- on the lateral ventricular surface data from a group of 11 cally align hippocampal parameterizations [5]. For classi- HIV/AIDS individuals and a group of 8 matched healthy fication, Linear Discriminant Analysis (LDA) or principal control subjects. Our classifier achieved a 100% accuracy geodesic analysis can be used to find the best discriminant rate and outperformed a nearest neighbor classifier based vector in the feature space for distinguishing diseased sub- on total brain volume, which achieved an overall 68.42% jects from controls. Gorczowski [7] presented a framework accuracy rate on the same dataset. for discriminant analysis of populations of 3D multi-object Our major contributions in this work include: a way sets. In addition to a sampled medial mesh representation, to compute a new conformal equivalence based shape in- m-rep [14], they also considered pose differences as an ad- dex, the Teichm¨uller space coordinate, on the Poincar´e ditional statistical feature to improve the shape classifica- disk in the parameter domain of a surface. Our proposed tion results. singularity-free Yamabe flow method preserves this invari- With the Ricci flow method, Wang et al. [19] solved the ant very well, so our method offers a stable way to calculate Yamabe equation and conformally mapped the cortical sur- it in 2D parametric coordinates. To the best of our knowl- face of the brain to a Euclidean multi-hole punctured disk. edge, it is the first work to apply the Teichm¨uller space Gu et al. [8] applied the surface Ricci flow method to study coordinate to brain morphometry research. We treated the general 3D shape matching and registration. The hyperbolic Teichm¨uller space coordinates as a random vector; by cal- Ricci flow has also been applied to study 3D face match- culating the Mahalanobis distance from any vector to in- ing [20]. Recently, Jin et al. [11] introduced the Teichm¨uller dividual members of two groups of subjects, our method shape space to index and compare general surfaces with var- achieved a 100% accuracy rate in classifying the lateral ven- ious topologies, geometries and resolutions. 2. Theoretical Background and Definitions Let φ be a Fuchsian transformation, let z ∈ H2, n the attractor and repulser of φ are limn→∞ φ (z) and This section briefly introduces the theoretic background −n lim →∞ φ (z) respectively. The axis of φ is the unique necessary for the current work.
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