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**11 - 15**of**15**### Persistent Homology in Sparse Regression and Its Application to Brain Morphometry

"... Abstract—Sparse systems are usually parameterized by a tuning parameter that determines the sparsity of the system. How to choose the right tuning parameter is a fundamental and difficult problem in learning the sparse system. In this paper, by treating the the tuning parameter as an additional dime ..."

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Abstract—Sparse systems are usually parameterized by a tuning parameter that determines the sparsity of the system. How to choose the right tuning parameter is a fundamental and difficult problem in learning the sparse system. In this paper, by treating the the tuning parameter as an additional dimension, persistent homological structures over the parameter space is in-troduced and explored. The structures are then further exploited in drastically speeding up the computation using the proposed soft-thresholding technique. The topological structures are further used as multivariate features in the tensor-based morphometry (TBM) in characterizing white matter alterations in children who have experienced severe early life stress and maltreatment. These analyses reveal that stress-exposed children exhibit more diffuse anatomical organization across the whole white matter region. Index Terms—GLASSO, maltreated children, persistent ho-mology, sparse brain networks, sparse correlations, tensor-based morphometry. I.

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"... rea Received 20 October 2013 Laplace–Beltrami eigenfunctions. Starting with the heat kernel constructed from the eigenfunctions, we formulate a new bivariate kernel regression framework as a weighted eigenfunction expansion with heat diffusion on surfaces has been introduced in brain imaging for sub ..."

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rea Received 20 October 2013 Laplace–Beltrami eigenfunctions. Starting with the heat kernel constructed from the eigenfunctions, we formulate a new bivariate kernel regression framework as a weighted eigenfunction expansion with heat diffusion on surfaces has been introduced in brain imaging for subsequent statistical analysis involving the random field the-ory (RFT) that assumes an isotropic covariance function as a noise model (Andrade et al., 2001; Chung and Taylor, 2004; Cachia et al., agler et al., 2006; its simplicity, it oothing in brain ights are sp a discrete anifold, th kernel with a small bandwidth can be approximated linearl the Gaussian kernel. The heat kernel with a large bandw then constructed by iteratively applying the Gaussian kernel with the small bandwidth. However, this process compounds the lin-earization error at each iteration. We propose a new kernel regression framework that constructs the heat kernel analytically using the eigenfunctions of the Laplace–Beltrami (LB) operator, avoiding the need for the linear approximation used by Chung et al. (2005) and Han et al. (2006).

### Improved Statistical Power with a Sparse Shape Model in Detecting an Aging Effect in the Hippocampus and Amygdala

"... The sparse regression framework has been widely used in medical image processing and analysis. However, it has been rarely used in anatomical studies. We present a sparse shape modeling framework using the Laplace-Beltrami (LB) eigenfunctions of the underlying shape and show its improvement of stati ..."

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The sparse regression framework has been widely used in medical image processing and analysis. However, it has been rarely used in anatomical studies. We present a sparse shape modeling framework using the Laplace-Beltrami (LB) eigenfunctions of the underlying shape and show its improvement of statistical power. Tradition-ally, the LB-eigenfunctions are used as a basis for intrinsically representing surface shapes as a form of Fourier descriptors. To reduce high frequency noise, only the first few terms are used in the expansion and higher frequency terms are simply thrown away. However, some lower frequency terms may not necessarily contribute significantly in reconstructing the surfaces. Motivated by this idea, we present a LB-based method to filter out only the significant eigenfunctions by imposing a sparse penalty. For dense anatomical data such as deformation fields on a surface mesh, the sparse regression behaves like a smoothing process, which will reduce the error of incorrectly detecting false negatives. Hence the statistical power improves. The sparse shape model is then applied in investigating the influence of age on amygdala and hippocampus shapes in the normal population. The advantage of the LB sparse framework is demonstrated by showing the increased statistical power. 1.

### Unified Heat Kernel Regression for Diffusion, Kernel Smoothing and Wavelets on Manifolds and Its Application to Mandible Growth Modeling in CT Images

, 2014

"... We present a novel kernel regression framework for smoothing scalar sur-face data using the Laplace-Beltrami eigenfunctions. Starting with the heat kernel constructed from the eigenfunctions, we formulate a new bivariate kernel regression framework as a weighted eigenfunction expansion with the heat ..."

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We present a novel kernel regression framework for smoothing scalar sur-face data using the Laplace-Beltrami eigenfunctions. Starting with the heat kernel constructed from the eigenfunctions, we formulate a new bivariate kernel regression framework as a weighted eigenfunction expansion with the heat kernel as the weights. The new kernel regression is mathematically equivalent to isotropic heat diffusion, kernel smoothing and recently pop-ular diffusion wavelets. Unlike many previous partial differential equation based approaches involving diffusion, our approach represents the solution of diffusion analytically, reducing numerical inaccuracy and slow convergence. The numerical implementation is validated on a unit sphere using spherical harmonics. As an illustration, we have applied the method in characterizing the localized growth pattern of mandible surfaces obtained in CT images from subjects between ages 0 and 20 years by regressing the length of dis-placement vectors with respect to the template surface.

### Shape Analysis of Corpus Callosum in Autism Subtype using Planar Conformal Mapping

"... A number of studies have documented that autism has a neurobiological basis, but the anatomical extent of these neurobiological abnormalities is largely unknown. In this study, we aimed at analyzing highly localized shape abnormalities of the corpus callosum in a homogeneous group of autism children ..."

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A number of studies have documented that autism has a neurobiological basis, but the anatomical extent of these neurobiological abnormalities is largely unknown. In this study, we aimed at analyzing highly localized shape abnormalities of the corpus callosum in a homogeneous group of autism children. Thirty patients with essential autism and twenty-four controls participated in this study. 2D contours of the corpus callosum were extracted from MR images by a semiautomatic segmentation method, and the 3D model was constructed by stacking the contours. The resulting 3D model had two openings at the ends, thus a new conformal parameterization for high genus surfaces was applied in our shape analysis work, which mapped each surface onto a planar domain. Surface matching among different individual meshes was achieved by re-triangulating each mesh according to a template surface. Statistical shape analysis was used to compare the 3D shapes point by point between patients with autism and their controls. The results revealed significant abnormalities in the anterior most and anterior body in essential autism group.