Lognormal Distributions and Geometric Averages of Positive Definite Matrices

This article gives a formal definition of a lognormal family of probability distributions on the set of symmetric positive definite (PD) matrices, seen as a matrix-variate extension of the univariate lognormal family of distributions. Two forms of this distribution are obtained as the large sample limiting distribution via the central limit theorem of two types of geometric averages of i.i.d. PD matrices: the log-Euclidean average and the canonical geometric average. These averages correspond to two different geometries imposed on the set of PD matrices. The limiting distributions of these averages are used to provide large-sample confidence regions for the corresponding population means. The methods are illustrated on a voxelwise analysis of diffusion tensor imaging data, permitting a comparison between the various average types from the point of view of their sampling variability.Comment: 28 pages, 8 figure

A Kernel-based Approach to Diffusion Tensor and Fiber Clustering in the Human Skeletal Muscle

In this report, we present a kernel-based approach to the clustering of diffusion tensors in images of the human skeletal muscle. Based on the physical intuition of tensors as a means to represent the uncertainty of the position of water protons in the tissues, we propose a Mercer (i.e. positive definite) kernel over the tensor space where both spatial and diffusion information are taken into account. This kernel highlights implicitly the connectivity along fiber tracts. We show that using this kernel in a kernel-PCA setting compounded with a landmark-Isomap embedding and k-means clustering provides a tractable framework for tensor clustering. We extend this kernel to deal with fiber tracts as input using the multi-instance kernel by considering the fiber as set of tensors centered in the sampled points of the tract. The obtained kernel reflects not only interactions between points along fiber tracts, but also the interactions between diffusion tensors. We give an interpretation of the obtained kernel as a comparison of soft fiber representations and show that it amounts to a generalization of the Gaussian kernel Correlation. As in the tensor case, we use the kernel-PCA setting and k-means for grouping of fiber tracts. This unsupervised method is further extended by way of an atlas-based registration of diffusion-free images, followed by a classification of fibers based on non-linear kernel Support Vector Machines (SVMs) and kernel diffusion. The experimental results on a dataset of diffusion tensor images of the calf muscle of 25 patients (of which 5 affected by myopathies, i.e. neuromuscular diseases) show the potential of our method in segmenting the calf in anatomically relevant regions both at the tensor and fiber level

Mining Brain Networks using Multiple Side Views for Neurological Disorder Identification

Mining discriminative subgraph patterns from graph data has attracted great interest in recent years. It has a wide variety of applications in disease diagnosis, neuroimaging, etc. Most research on subgraph mining focuses on the graph representation alone. However, in many real-world applications, the side information is available along with the graph data. For example, for neurological disorder identification, in addition to the brain networks derived from neuroimaging data, hundreds of clinical, immunologic, serologic and cognitive measures may also be documented for each subject. These measures compose multiple side views encoding a tremendous amount of supplemental information for diagnostic purposes, yet are often ignored. In this paper, we study the problem of discriminative subgraph selection using multiple side views and propose a novel solution to find an optimal set of subgraph features for graph classification by exploring a plurality of side views. We derive a feature evaluation criterion, named gSide, to estimate the usefulness of subgraph patterns based upon side views. Then we develop a branch-and-bound algorithm, called gMSV, to efficiently search for optimal subgraph features by integrating the subgraph mining process and the procedure of discriminative feature selection. Empirical studies on graph classification tasks for neurological disorders using brain networks demonstrate that subgraph patterns selected by the multi-side-view guided subgraph selection approach can effectively boost graph classification performances and are relevant to disease diagnosis.Comment: in Proceedings of IEEE International Conference on Data Mining (ICDM) 201

Systems modeling of white matter microstructural abnormalities in Alzheimer's disease

INTRODUCTION: Microstructural abnormalities in white matter (WM) are often reported in Alzheimer's disease (AD). However, it is unclear which brain regions have the strongest WM changes in presymptomatic AD and what biological processes underlie WM abnormality during disease progression. METHODS: We developed a systems biology framework to integrate matched diffusion tensor imaging (DTI), genetic and transcriptomic data to investigate regional vulnerability to AD and identify genetic risk factors and gene subnetworks underlying WM abnormality in AD. RESULTS: We quantified regional WM abnormality and identified most vulnerable brain regions. A SNP rs2203712 in CELF1 was most significantly associated with several DTI-derived features in the hippocampus, the top ranked brain region. An immune response gene subnetwork in the blood was most correlated with DTI features across all the brain regions. DISCUSSION: Incorporation of image analysis with gene network analysis enhances our understanding of disease progression and facilitates identification of novel therapeutic strategies for AD

Statistical Computing on Non-Linear Spaces for Computational Anatomy

International audienceComputational anatomy is an emerging discipline that aims at analyzing and modeling the individual anatomy of organs and their biological variability across a population. However, understanding and modeling the shape of organs is made difficult by the absence of physical models for comparing different subjects, the complexity of shapes, and the high number of degrees of freedom implied. Moreover, the geometric nature of the anatomical features usually extracted raises the need for statistics on objects like curves, surfaces and deformations that do not belong to standard Euclidean spaces. We explain in this chapter how the Riemannian structure can provide a powerful framework to build generic statistical computing tools. We show that few computational tools derive for each Riemannian metric can be used in practice as the basic atoms to build more complex generic algorithms such as interpolation, filtering and anisotropic diffusion on fields of geometric features. This computational framework is illustrated with the analysis of the shape of the scoliotic spine and the modeling of the brain variability from sulcal lines where the results suggest new anatomical findings

Surface-Based Analysis on Shape and Fractional Anisotropy of White Matter Tracts in Alzheimer's Disease

10.1371/journal.pone.0009811PLoS ONE53

Combination of Resting State fMRI, DTI, and sMRI Data to Discriminate Schizophrenia by N-way MCCA + jICA

Multimodal brain imaging data have shown increasing utility in answering both scientifically interesting and clinically relevant questions. Each brain imaging technique provides a different view of brain function or structure, while multimodal fusion capitalizes on the strength of each and may uncover hidden relationships that can merge findings from separate neuroimaging studies. However, most current approaches have focused on pair-wise fusion and there is still relatively little work on N-way data fusion and examination of the relationships among multiple data types. We recently developed an approach called “mCCA + jICA” as a novel multi-way fusion method which is able to investigate the disease risk factors that are either shared or distinct across multiple modalities as well as the full correspondence across modalities. In this paper, we applied this model to combine resting state fMRI (amplitude of low-frequency fluctuation, ALFF), gray matter (GM) density, and DTI (fractional anisotropy, FA) data, in order to elucidate the abnormalities underlying schizophrenia patients (SZs, n = 35) relative to healthy controls (HCs, n = 28). Both modality-common and modality-unique abnormal regions were identified in SZs, which were then used for successful classification for seven modality-combinations, showing the potential for a broad applicability of the mCCA + jICA model and its results. In addition, a pair of GM-DTI components showed significant correlation with the positive symptom subscale of Positive and Negative Syndrome Scale (PANSS), suggesting that GM density changes in default model network along with white-matter disruption in anterior thalamic radiation are associated with increased positive PANSS. Findings suggest the DTI anisotropy changes in frontal lobe may relate to the corresponding functional/structural changes in prefrontal cortex and superior temporal gyrus that are thought to play a role in the clinical expression of SZ

Visual Exploration And Information Analytics Of High-Dimensional Medical Images

Data visualization has transformed how we analyze increasingly large and complex data sets. Advanced visual tools logically represent data in a way that communicates the most important information inherent within it and culminate the analysis with an insightful conclusion. Automated analysis disciplines - such as data mining, machine learning, and statistics - have traditionally been the most dominant fields for data analysis. It has been complemented with a near-ubiquitous adoption of specialized hardware and software environments that handle the storage, retrieval, and pre- and postprocessing of digital data. The addition of interactive visualization tools allows an active human participant in the model creation process. The advantage is a data-driven approach where the constraints and assumptions of the model can be explored and chosen based on human insight and confirmed on demand by the analytic system. This translates to a better understanding of data and a more effective knowledge discovery. This trend has become very popular across various domains, not limited to machine learning, simulation, computer vision, genetics, stock market, data mining, and geography. In this dissertation, we highlight the role of visualization within the context of medical image analysis in the field of neuroimaging. The analysis of brain images has uncovered amazing traits about its underlying dynamics. Multiple image modalities capture qualitatively different internal brain mechanisms and abstract it within the information space of that modality. Computational studies based on these modalities help correlate the high-level brain function measurements with abnormal human behavior. These functional maps are easily projected in the physical space through accurate 3-D brain reconstructions and visualized in excellent detail from different anatomical vantage points. Statistical models built for comparative analysis across subject groups test for significant variance within the features and localize abnormal behaviors contextualizing the high-level brain activity. Currently, the task of identifying the features is based on empirical evidence, and preparing data for testing is time-consuming. Correlations among features are usually ignored due to lack of insight. With a multitude of features available and with new emerging modalities appearing, the process of identifying the salient features and their interdependencies becomes more difficult to perceive. This limits the analysis only to certain discernible features, thus limiting human judgments regarding the most important process that governs the symptom and hinders prediction. These shortcomings can be addressed using an analytical system that leverages data-driven techniques for guiding the user toward discovering relevant hypotheses. The research contributions within this dissertation encompass multidisciplinary fields of study not limited to geometry processing, computer vision, and 3-D visualization. However, the principal achievement of this research is the design and development of an interactive system for multimodality integration of medical images. The research proceeds in various stages, which are important to reach the desired goal. The different stages are briefly described as follows: First, we develop a rigorous geometry computation framework for brain surface matching. The brain is a highly convoluted structure of closed topology. Surface parameterization explicitly captures the non-Euclidean geometry of the cortical surface and helps derive a more accurate registration of brain surfaces. We describe a technique based on conformal parameterization that creates a bijective mapping to the canonical domain, where surface operations can be performed with improved efficiency and feasibility. Subdividing the brain into a finite set of anatomical elements provides the structural basis for a categorical division of anatomical view points and a spatial context for statistical analysis. We present statistically significant results of our analysis into functional and morphological features for a variety of brain disorders. Second, we design and develop an intelligent and interactive system for visual analysis of brain disorders by utilizing the complete feature space across all modalities. Each subdivided anatomical unit is specialized by a vector of features that overlap within that element. The analytical framework provides the necessary interactivity for exploration of salient features and discovering relevant hypotheses. It provides visualization tools for confirming model results and an easy-to-use interface for manipulating parameters for feature selection and filtering. It provides coordinated display views for visualizing multiple features across multiple subject groups, visual representations for highlighting interdependencies and correlations between features, and an efficient data-management solution for maintaining provenance and issuing formal data queries to the back end

Multimodal Imaging Evidence for Axonal and Myelin Deterioration in Amnestic Mild Cognitive Impairment

White matter (WM) microstructural declines have been demonstrated in Alzheimer\u27s disease and amnestic mild cognitive impairment (aMCI). However, the pattern of WM microstructural changes in aMCI after controlling for WM atrophy is unknown. Here, we address this issue through joint consideration of aMCI alterations in fractional anisotropy, mean diffusivity, axial diffusivity, and radial diffusivity, as well as macrostructural volume in WM and gray matter compartments. Participants were 18 individuals with aMCI and 24 healthy seniors. Voxelwise analyses of diffusion tensor imaging data was carried out using tract-based spatial statistics (TBSS) and voxelwise analyses of high-resolution structural data was conducted using voxel based morphometry. After controlling for WM atrophy, the main pattern of TBSS findings indicated reduced fractional anisotropy with only small alterations in mean diffusivity/radial diffusivity/axial diffusivity. These WM microstructural declines bordered and/or were connected to gray matter structures showing volumetric declines. However, none of the potential relationships between WM integrity and volume in connected gray matter structures was significant, and adding fractional anisotropy information improved the classificatory accuracy of aMCI compared to the use of hippocampal atrophy alone. These results suggest that WM microstructural declines provide unique information not captured by atrophy measures that may aid the magnetic resonance imaging contribution to aMCI detection

Non Rigid Registration of Diffusion Tensor Images

We propose a novel variational framework for the dense non-rigid registration of Diffusion Tensor Images (DTI). Our approach relies on the differential geometrical properties of the Riemannian manifold of multivariate normal distributions endowed with the metric derived from the Fisher information matrix. The availability of closed form expressions for the geodesics and the Christoffel symbols allows us to define statistical quantities and to perform the parallel transport of tangent vectors in this space. We propose a matching energy that aims to minimize the difference in the local statistical content (means and covariance matrices) of two DT images through a gradient descent procedure. The result of the algorithm is a dense vector field that can be used to wrap the source image into the target image. This article is essentially a mathematical study of the registration problem. Some numerical experiments are provided as a proof of concept