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

We present a novel kernel regression framework for smoothing scalar surface 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 popular 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 displacement vectors with respect to the template surface.Comment: Accepted in Medical Image Analysi

CLASSIFICATION OF NEUROANATOMICAL STRUCTURES BASED ON NON-EUCLIDEAN GEOMETRIC OBJECT PROPERTIES

Studying the observed morphological differences in neuroanatomical structures between individuals with neurodevelopmental disorders and a control group of typically developing individuals has been an important objective. Researchers study the differences with two goals: to assist an accurate diagnosis of the disease and to gain insights into underlying mechanisms of the disease that cause such changes. Shape classification is commonly utilized in such studies. An effective classification is difficult because it requires 1) a choice of an object model that can provide rich geometric object properties (GOPs) relevant for a given classification task, and 2) a choice of a statistical classification method that accounts for the non-Euclidean nature of GOPs. I lay out my methodological contributions to address the aforementioned challenges in the context of early diagnosis and detection of Autism Spectrum Disorder (ASD) in infants based on shapes of hippocampi and caudate nuclei; morphological deviations in these structures between individuals with ASD and typically developing individuals have been reported in the literature. These contributions respectively lead to 1) an effective modeling of shapes of objects of interest and 2) an effective classification. As the first contribution for modeling shapes of objects, I propose a method to obtain a set of skeletal models called s-reps from a set of 3D objects. First, the method iteratively deforms the object surface via Mean Curvature Flow (MCF) until the deformed surface is approximately ellipsoidal. Then, an s-rep of the approximate ellipsoid is obtained analytically. Finally, the ellipsoid s-rep is deformed via a series of inverse MCF transformations. The method has two important properties: 1) it is fully automatic, and 2) it yields a set of s-reps with good correspondence across the set. The method is shown effective in generating a set of s-reps for a few neuroanatomical structures. As the second contribution with respect to modeling shapes of objects, I introduce an extension to the current s-rep for representing an object with a narrowing sharp tail. This includes a spoke interpolation method for interpolating a discrete s-rep of an object with a narrowing sharp tail into a continuous object. This extension is necessary for representing surface geometry of objects whose boundary has a singular point. I demonstrate that this extension allows appropriate surface modeling of a narrowing sharp tail region of the caudate nucleus. In addition, I show that the extension is beneficial in classifying autistic and non-autistic infants at high risk of ASD based on shapes of caudate nuclei. As the first contribution with respect to statistical methods, I propose a novel shape classification framework that uses the s-rep to capture rich localized geometric descriptions of an object, a statistical method called Principal Nested Spheres (PNS) analysis to handle the non-Euclidean s-rep GOPs, and a classification method called Distance Weighted Discrimination (DWD). I evaluate the effectiveness of the proposed method in classifying autistic and non-autistic infants based on either hippocampal shapes or caudate shapes in terms of the Area Under the ROC curve (AUC). In addition, I show that the proposed method is superior to commonly used shape classification methods in the literature. As my final methodological contribution, I extend the proposed shape classification method to perform the classifcation task based on temporal shape differences. DWD learns a class separation direction based on the temporal shape differences that are obtained by taking differences of the temporal pair of Euclideanized s-reps. In the context of early diagnosis and detection of ASD in young infants, the proposed temporal shape difference classification produces some interesting results; the temporal differences in shapes of hippocampi and caudate nuclei do not seem to be as predictive as the cross-sectional shape of these structures alone.Doctor of Philosoph

Shape analysis of the human brain.

Autism is a complex developmental disability that has dramatically increased in prevalence, having a decisive impact on the health and behavior of children. Methods used to detect and recommend therapies have been much debated in the medical community because of the subjective nature of diagnosing autism. In order to provide an alternative method for understanding autism, the current work has developed a 3-dimensional state-of-the-art shape based analysis of the human brain to aid in creating more accurate diagnostic assessments and guided risk analyses for individuals with neurological conditions, such as autism. Methods: The aim of this work was to assess whether the shape of the human brain can be used as a reliable source of information for determining whether an individual will be diagnosed with autism. The study was conducted using multi-center databases of magnetic resonance images of the human brain. The subjects in the databases were analyzed using a series of algorithms consisting of bias correction, skull stripping, multi-label brain segmentation, 3-dimensional mesh construction, spherical harmonic decomposition, registration, and classification. The software algorithms were developed as an original contribution of this dissertation in collaboration with the BioImaging Laboratory at the University of Louisville Speed School of Engineering. The classification of each subject was used to construct diagnoses and therapeutic risk assessments for each patient. Results: A reliable metric for making neurological diagnoses and constructing therapeutic risk assessment for individuals has been identified. The metric was explored in populations of individuals having autism spectrum disorders, dyslexia, Alzheimers disease, and lung cancer. Conclusion: Currently, the clinical applicability and benefits of the proposed software approach are being discussed by the broader community of doctors, therapists, and parents for use in improving current methods by which autism spectrum disorders are diagnosed and understood

Estimation of probability distribution on multiple anatomical objects and evaluation of statistical shape models

The estimation of shape probability distributions of anatomic structures is a major research area in medical image analysis. The statistical shape descriptions estimated from training samples provide means and the geometric shape variations of such structures. These are key components in many applications. This dissertation presents two approaches to the estimation of a shape probability distribution of a multi-object complex. Both approaches are applied to objects in the male pelvis, and show improvement in the estimated shape distributions of the objects. The first approach is to estimate the shape variation of each object in the complex in terms of two components: the object's variation independent of the effect of its neighboring objects; and the neighbors' effect on the object. The neighbors' effect on the target object is interpreted using the idea on which linear mixed models are based. The second approach is to estimate a conditional shape probability distribution of a target object given its neighboring objects. The estimation of the conditional probability is based on principal component regression. This dissertation also presents a measure to evaluate the estimated shape probability distribution regarding its predictive power, that is, the ability of a statistical shape model to describe unseen members of the population. This aspect of statistical shape models is of key importance to any application that uses shape models. The measure can be applied to PCA-based shape models and can be interpreted as a ratio of the variation of new data explained by the retained principal directions estimated from training data. This measure was applied to shape models of synthetic warped ellipsoids and right hippocampi. According to two surface distance measures and a volume overlap measure it was empirically verified that the predictive measure reflects what happens in the ambient space where the model lies

Complexity Analysis of Cortical Surface Detects Changes in Future Alzheimer’s Disease Converters

Alzheimer’s disease (AD) is a neurological disorder that creates neurodegenerative changes at several structural and functional levels in human brain tissue. The fractal dimension (FD) is a quantitative parameter that characterizes the morphometric variability of the human brain. In this study we investigate spherical harmonic-based FD (SHFD), thickness and local gyrification index (LGI) to assess whether they identify cortical surface abnormalities toward the conversion to AD. We study 33 AD patients, 122 mild cognitive impairment (MCI) patients (50 MCI-converters and 29 MCI-non converters) and 32 healthy controls (HC). SHFD, thickness and LGI methodology allowed us to perform not only global but also local level assessments in each cortical surface vertex. First, we found that global SHFD decreased in AD and future MCI-converters compared to HC, and in MCI-converters compared to MCI-non-converters. Second, we found that local white matter SHFD was reduced in AD compared to HC and MCI mainly in medial temporal lobe. Third, local white matter SHFD was significantly reduced in MCI-converters compared to MCI-non-converters in distributed areas, including the medial frontal lobe. Thickness and LGI metrics presented a reduction in AD compared to HC. Thickness was significantly reduced in MCI-converters compared to healthy controls in entorhinal cortex and lateral temporal. In summary, SHFD was the only surface measure showing differences between MCI individuals that will convert or remain stable in the next four years. We suggest that SHFD may be an optimal complement to thickness loss analysis in monitoring longitudinal changes in preclinical and clinical stages of AD

BrainPrint: A discriminative characterization of brain morphology

We introduce BrainPrint, a compact and discriminative representation of brain morphology. BrainPrint captures shape information of an ensemble of cortical and subcortical structures by solving the eigenvalue problem of the 2D and 3D Laplace–Beltrami operator on triangular (boundary) and tetrahedral (volumetric) meshes. This discriminative characterization enables new ways to study the similarity between brains; the focus can either be on a specific brain structure of interest or on the overall brain similarity. We highlight four applications for BrainPrint in this article: (i) subject identification, (ii) age and sex prediction, (iii) brain asymmetry analysis, and (iv) potential genetic influences on brain morphology. The properties of BrainPrint require the derivation of new algorithms to account for the heterogeneous mix of brain structures with varying discriminative power. We conduct experiments on three datasets, including over 3000 MRI scans from the ADNI database, 436 MRI scans from the OASIS dataset, and 236 MRI scans from the VETSA twin study. All processing steps for obtaining the compact representation are fully automated, making this processing framework particularly attractive for handling large datasets.National Cancer Institute (U.S.) (1K25-CA181632-01)Athinoula A. Martinos Center for Biomedical Imaging (P41-RR014075)Athinoula A. Martinos Center for Biomedical Imaging (P41-EB015896)National Alliance for Medical Image Computing (U.S.) (U54-EB005149)Neuroimaging Analysis Center (U.S.) (P41-EB015902)National Center for Research Resources (U.S.) (U24 RR021382)National Institute of Biomedical Imaging and Bioengineering (U.S.) (5P41EB015896-15)National Institute of Biomedical Imaging and Bioengineering (U.S.) (R01EB006758)National Institute on Aging (AG022381)National Institute on Aging (5R01AG008122-22)National Institute on Aging (AG018344)National Institute on Aging (AG018386)National Center for Complementary and Alternative Medicine (U.S.) (RC1 AT005728-01)National Institute of Neurological Diseases and Stroke (U.S.) (R01 NS052585-01)National Institute of Neurological Diseases and Stroke (U.S.) (1R21NS072652-01)National Institute of Neurological Diseases and Stroke (U.S.) (1R01NS070963)National Institute of Neurological Diseases and Stroke (U.S.) (R01NS083534)National Institutes of Health (U.S.) ((5U01-MH093765

BrainPrint: A discriminative characterization of brain morphology

We introduce BrainPrint, a compact and discriminative representation of brain morphology. BrainPrint captures shape information of an ensemble of cortical and subcortical structures by solving the eigenvalue problem of the 2D and 3D Laplace–Beltrami operator on triangular (boundary) and tetrahedral (volumetric) meshes. This discriminative characterization enables new ways to study the similarity between brains; the focus can either be on a specific brain structure of interest or on the overall brain similarity. We highlight four applications for BrainPrint in this article: (i) subject identification, (ii) age and sex prediction, (iii) brain asymmetry analysis, and (iv) potential genetic influences on brain morphology. The properties of BrainPrint require the derivation of new algorithms to account for the heterogeneous mix of brain structures with varying discriminative power. We conduct experiments on three datasets, including over 3000 MRI scans from the ADNI database, 436 MRI scans from the OASIS dataset, and 236 MRI scans from the VETSA twin study. All processing steps for obtaining the compact representation are fully automated, making this processing framework particularly attractive for handling large datasets.National Cancer Institute (U.S.) (1K25-CA181632-01)Athinoula A. Martinos Center for Biomedical Imaging (P41-RR014075)Athinoula A. Martinos Center for Biomedical Imaging (P41-EB015896)National Alliance for Medical Image Computing (U.S.) (U54-EB005149)Neuroimaging Analysis Center (U.S.) (P41-EB015902)National Center for Research Resources (U.S.) (U24 RR021382)National Institute of Biomedical Imaging and Bioengineering (U.S.) (5P41EB015896-15)National Institute of Biomedical Imaging and Bioengineering (U.S.) (R01EB006758)National Institute on Aging (AG022381)National Institute on Aging (5R01AG008122-22)National Institute on Aging (AG018344)National Institute on Aging (AG018386)National Center for Complementary and Alternative Medicine (U.S.) (RC1 AT005728-01)National Institute of Neurological Diseases and Stroke (U.S.) (R01 NS052585-01)National Institute of Neurological Diseases and Stroke (U.S.) (1R21NS072652-01)National Institute of Neurological Diseases and Stroke (U.S.) (1R01NS070963)National Institute of Neurological Diseases and Stroke (U.S.) (R01NS083534)National Institutes of Health (U.S.) ((5U01-MH093765

Proceedings of the Second International Workshop on Mathematical Foundations of Computational Anatomy (MFCA'08) - Geometrical and Statistical Methods for Modelling Biological Shape Variability

International audienceThe goal of computational anatomy is to analyze and to statistically model the anatomy of organs in different subjects. Computational anatomic methods are generally based on the extraction of anatomical features or manifolds which are then statistically analyzed, often through a non-linear registration. There are nowadays a growing number of methods that can faithfully deal with the underlying biomechanical behavior of intra-subject deformations. However, it is more difficult to relate the anatomies of different subjects. In the absence of any justified physical model, diffeomorphisms provide a general mathematical framework that enforce topological consistency. Working with such infinite dimensional space raises some deep computational and mathematical problems, in particular for doing statistics. Likewise, modeling the variability of surfaces leads to rely on shape spaces that are much more complex than for curves. To cope with these, different methodological and computational frameworks have been proposed (e.g. smooth left-invariant metrics, focus on well-behaved subspaces of diffeomorphisms, modeling surfaces using courants, etc.) The goal of the Mathematical Foundations of Computational Anatomy (MFCA) workshop is to foster the interactions between the mathematical community around shapes and the MICCAI community around computational anatomy applications. It targets more particularly researchers investigating the combination of statistical and geometrical aspects in the modeling of the variability of biological shapes. The workshop aims at being a forum for the exchange of the theoretical ideas and a source of inspiration for new methodological developments in computational anatomy. A special emphasis is put on theoretical developments, applications and results being welcomed as illustrations. Following the very successful first edition of this workshop in 2006 (see http://www.inria.fr/sophia/asclepios/events/MFCA06/), the second edition was held in New-York on September 6, in conjunction with MICCAI 2008. Contributions were solicited in Riemannian and group theoretical methods, Geometric measurements of the anatomy, Advanced statistics on deformations and shapes, Metrics for computational anatomy, Statistics of surfaces. 34 submissions were received, among which 9 were accepted to MICCAI and had to be withdrawn from the workshop. Each of the remaining 25 paper was reviewed by three members of the program committee. To guaranty a high level program, 16 papers only were selected