Research Description

Deconstructing human brain organization is a staggeringly complex endeavor. My research is geared towards development of mathematical and computational methods for brain signal and image analysis. Recent advances in Neuroimaging have allowed researchers to study the brain in-vivo by imaging its structures and functions. Various imaging modalities such as Magnetic resonance imaging (MRI), functional Magnetic Resonance Imaging (fMRI), Electroencephalography (EEG), Diffusion Tensor Imaging (DTI) collect rich imaging data of structural and functional aspects of the brain. The functional data is often reconstructed on the cortical surface (cortex) of the brain, which represents the boundary between gray and white matter. Specifically, the human cortex can be modeled as a highly convoluted 2D surface, and therefore the data is modeled as non-flat (non-euclidean) images. The mathematical and computational challenges in the analysis of this data has led to the development of novel and interesting image processing theory and algorithms, that use partial differential equations (PDEs) as well as differential and Riemannian geometry. My PhD work was focused on development of novel geometric techniques for image analysis that accounts for the non-euclidean geometry of the cortex while performing registration and subsequent signal processing of anatomical and functional signals.

Morphometric studies of anatomical changes in the brain over time or of differences between populations are often performed to study changes in the brain in disease and development. Such studies require that the imaging data first be transformed to a common coordinate system in which anatomical structures are aligned. Similarly, inter-subject longitudinal studies or group analysis of functional data also require that the images first be anatomically aligned. I developed a method based on p-harmonic mapping for performing surface parameterization that generates a 2D coordinate system on the cortical surface. This coordinate system is then used to compute the surface metric and discretize derivatives in the surface geometry. The surface alignment problem then translates to the problem of alignment of the two coordinate systems. For performing inter-subject cortical registration, we present a Finite Element Method (FEM) for simultaneous parameterization and registration of landmarks based on elastic energy minimization. These can be used to bring surface signals from individual brains to a common atlas surface.

I then reformulated the isotropic and anisotropic diffusion filters as well as classification methods using covariant PDEs for processing of the non-flat cortical data. When the surface data is a point-set on the cortex, we propose a method to quantify its mean and variance with respect to the surface geometry by using a heat-kernel to model the probability distributions of point-set on the surface.

The registration techniques presented for surface alignment are then extended to volumes to perform full surface and volume registration. This is done by using volumetric harmonic mappings that extend the surface point correspondence to the cortical brain volume. Finally, the volumetric registration is refined by using inverse-consistent linear elastic intensity registration. This set of methods presents a unified framework for registration and analysis of the brain signals for inter-subject neuroanatomical studies. Morphometric studies performed on twins show improved statistical power using our registration algorithm.

Another aspect of my research is focused on developing methods for finding structural and functional connections in the human brain. The full understanding of brain connections: the ‘human connectome’ is critical in elucidating the neural pathways that underlie brain function. Diffusion weighted imaging (DWI) produces in vivo images that are weighted by the directional characteristics of water molecule diffusion in the white matter brain tissue. This imaging modality is particularly useful for inferring white-matter fiber connectivity in the brain. The tensor data produced by DTI images can be used to reconstruct the neuronal fiber tracts in white matter (tractography). In order to perform intersubject comparison and analysis of diffusion imaging data, accurate alignment of white matter is important. Because folding pattern on the cortex is intimately related to the development of white matter connectivity, any such comparison needs accurate alignment of the sulcal folds on the cortical surface. The surface and volumetric combined registration technique we presented in makes such an alignment possible. We will extend the volumetric registration approaches that we have developed for structural images to the alignment of diffusion data. In developing these methods we will combine our cortically-constrained approach to volume alignment with the fluid-based information theoretic approach to diffusion registration.

Another aspect of the human connectome, is inferring functional connectivity during resting state and tasks. Functional magnetic resonance imaging (fMRI), and Electroencephalography (EEG) and Magnetoencephalography (MEG) generate multivariate time series of the electrical and physiological brain signals. Inferring undirected and causal connectivity graphs from this data is challenging due to confounding effects of experimental design, noise, unrelated electrophysiological signals in the brain as well as low sample size availability. Using sparse models, such as Gaussian graphical models, Bayesian networks and LASSO can help in this task. I am currently working on development of a undirected partial correlation algorithm, that uses a Markov property of the Gaussian concentration graphs to infer the brain connectivity.

My research will help in development of techniques for brain signal analysis and will help in reaching the scientific objective of development of the human connectome. It will also lead to development of novel signal processing algorithms for multivariate and geometrical signals.