Contributions to MCMC Methods in Constrained Domains with Applications to Neuroimaging

Markov chain Monte Carlo (MCMC) methods form a rich class of computational techniques that help its user ascertain samples from target distributions when direct sampling is not possible or when their closed forms are intractable. Over the years, MCMC methods have been used in innumerable situations due to their flexibility and generalizability, even in situations involving nonlinear and/or highly parametrized models. In this dissertation, two major works relating to MCMC methods are presented. The first involves the development of a method to identify the number and directions of nerve fibers using diffusion-weighted MRI measurements. For this, the biological problem is first formulated as a model selection and estimation problem. Using the framework of reversible jump MCMC, a novel Bayesian scheme that performs both the above tasks simultaneously using customizable priors and proposal distributions is proposed. The proposed method allows users to set a prior level of spatial separation between the nerve fibers, allowing more crossing paths to be detected when desired or a lower number to potentially only detect robust nerve tracts. Hence, estimation that is specific to a given region of interest within the brain can be performed. In simulated examples, the method has been shown to resolve up to four fibers even in instances of highly noisy data. Comparative analysis with other state-of-the-art methods on in-vivo data showed the method\u27s ability to detect more crossing nerve fibers. The second work involves the construction of an MCMC algorithm that efficiently performs (Bayesian) sampling of parameters with support constraints. The method works by embedding a transformation called inversion in a sphere within the Metropolis-Hastings sampler. This creates an image of the constrained support that is amenable to sampling using standard proposals such as Gaussian. The proposed strategy is tested on three domains: the standard simplex, a sector of an n-sphere, and hypercubes. In each domain, a comparison is made with existing sampling techniques

A local wave tracking strategy for efficiently solving mid-and high-frequency Helmholtz problems.

International audienceWe propose a procedure for selecting basis function orientation to improve the efficiency of solution methodologies that employ local plane-wave approximations. The proposed adaptive approach consists of a local wave tracking strategy. Each plane-wave basis set within considered elements of the mesh partition is individually or collectively rotated to best align one function of the set with the local propagation direction of the field. Systematic determination of the direction of the field inside the computational domain is formulated as a minimization problem. As the resultant system is nonlinear with respect to the directions of propagation, the Newton method is employed with exact characterization of the Jacobian and Hessian. To illustrate the salient features and evaluate the performance of the proposed wave tracking approach, we present error estimates as well as numerical results obtained by incorporating the procedure into a prototypical plane-wave based approach, the least-squares method (LSM) developed by Monk and Wang (1999). The numerical results obtained for the case of a two-dimensional rigid scattering problem indicate that (a) convergence was achievable to a prescribed level of accuracy, even upon initial application of the tracking wave strategy outside the pre-asymptotic convergence region, and (b) the proposed approach reduced the size of the resulting system by up to two orders of magnitude, depending on the frequency range, with respect to the size of the standard LSM system

Resolving Intravoxel White Matter Structures in the Human Brain Using Regularized Regression and Clustering

The human brain is a complex system of neural tissue that varies significantly between individuals. Although the technology that delineates these neural pathways does not currently exist, medical imaging modalities, such as diffusion magnetic resonance imaging (dMRI), can be leveraged for mathematical identification. The purpose of this work is to develop a novel method employing machine learning techniques to determine intravoxel nerve number and direction from dMRI data. The method was tested on multiple synthetic datasets and showed promising estimation accuracy and robustness for multi-nerve systems under a variety of conditions, including highly noisy data and imprecision in parameter assumptions

Efficient solution methodology based on a local wave tracking strategy for high-frequency Helmholtz problems

International audienc