Joint Spatial-Angular Sparse Coding, Compressed Sensing, and Dictionary Learning for Diffusion MRI

Neuroimaging provides a window into the inner workings of the human brain to diagnose and prevent neurological diseases and understand biological brain function, anatomy, and psychology. Diffusion Magnetic Resonance Imaging (dMRI) is an emerging medical imaging modality used to study the anatomical network of neurons in the brain, which form cohesive bundles, or fiber tracts, that connect various parts of the brain. Since about 73% of the brain is water, measuring the flow, or diffusion of water molecules in the presence of fiber bundles, allows researchers to estimate the orientation of fiber tracts and reconstruct the internal wiring of the brain, in vivo. Diffusion MRI signals can be modeled within two domains: the spatial domain consisting of voxels in a brain volume and the diffusion or angular domain, where fiber orientation is estimated in each voxel. Researchers aim to estimate the probability distribution of fiber orientation in every voxel of a brain volume in order to trace paths of fiber tracts from voxel to voxel over the entire brain. Therefore, the traditional framework for dMRI processing and analysis has been from a voxel-wise vantage point with added spatial regularization considered post-hoc. In contrast, we propose a new joint spatial-angular representation of dMRI data which pairs signals in each voxel with the global spatial environment, jointly. This has the ability to improve many aspects of dMRI processing and analysis and re-envision the core representation of dMRI data from a local perspective to a global one. In this thesis, we propose three main contributions which take advantage of such joint spatial-angular representations to improve major machine learning tasks applied to dMRI: sparse coding, compressed sensing, and dictionary learning. First, we will show that we can achieve sparser representations of dMRI by utilizing a global spatial-angular dictionary instead of a purely voxel-wise angular dictionary. As dMRI data is very large in size, we provide a number of novel extensions to popular spare coding algorithms that perform efficient optimization on a global-scale by exploiting the separability of our dictionaries over the spatial and angular domains. Next, compressed sensing is used to accelerate signal acquisition based on an underlying sparse representation of the data. We will show that our proposed representation has the potential to push the limits of the current state of scanner acceleration within a new compressed sensing model for dMRI. Finally, sparsity can be further increased by learning dictionaries directly from datasets of interest. Prior dictionary learning for dMRI learn angular dictionaries alone. Our third contribution is to learn spatial-angular dictionaries jointly from dMRI data directly to better represent the global structure. Traditionally, the problem of dictionary learning is non-convex with no guarantees of finding a globally optimal solution. We derive the first theoretical results of global optimality for this class of dictionary learning problems. We hope the core foundation of a joint spatial-angular representation will open a new perspective on dMRI with respect to many other processing tasks and analyses. In addition, our contributions are applicable to any general signal types that can benefit from separable dictionaries. We hope the contributions in this thesis may be adopted in the larger signal processing, computer vision, and machine learning communities. dMRI signals can be modeled within two domains: the spatial domain consisting of voxels in a brain volume and the diffusion or angular domain, where fiber orientation is estimated in each voxel. Computationally speaking, researchers aim to estimate the probability distribution of fiber orientation in every voxel of a brain volume in order to trace paths of fiber tracts from voxel to voxel over the entire brain. Therefore, the traditional framework for dMRI processing and analysis is from a voxel-wise, or angular, vantage point with post-hoc consideration of their local spatial neighborhoods. In contrast, we propose a new global spatial-angular representation of dMRI data which pairs signals in each voxel with the global spatial environment, jointly, to improve many aspects of dMRI processing and analysis, including the important need for accelerating the otherwise time-consuming acquisition of advanced dMRI protocols. In this thesis, we propose three main contributions which utilize our joint spatial-angular representation to improve major machine learning tasks applied to dMRI: sparse coding, compressed sensing, and dictionary learning. We will show that sparser codes are possible by utilizing a global dictionary instead of a voxel-wise angular dictionary. This allows for a reduction of the number of measurements needed to reconstruct a dMRI signal to increase acceleration using compressed sensing. Finally, instead of learning angular dictionaries alone, we learn spatial-angular dictionaries jointly from dMRI data directly to better represent the global structure. In addition, this problem is non-convex and so we derive the first theories to guarantee convergence to a global minimum. As dMRI data is very large in size, we provide a number of novel extensions to popular algorithms that perform efficient optimization on a global-scale by exploiting the separability of our global dictionaries over the spatial and angular domains. We hope the core foundation of a joint spatial-angular representation will open a new perspective on dMRI with respect to many other processing tasks and analyses. In addition, our contributions are applicable to any separable dictionary setting which we hope may be adopted in the larger image processing, computer vision, and machine learning communities

Super Resolution of HARDI images Using Compressed Sensing Techniques

Effective techniques of inferring the condition of neural tracts in the brain is invaluable for clinicians and researchers towards investigation of neurological disorders in patients. It was not until the advent of diffusion Magnetic Resonance Imaging (dMRI), a noninvasive imaging method used to detect the diffusion of water molecules, that scientists have been able to assess the characteristics of cerebral diffusion in vivo. Among different dMRI methods, High Angular Resolution Diffusion Imaging (HARDI) is well known for striking a balance between ability to distinguish crossing neural fibre tracts while requiring a modest number of diffusion measurements (which is directly related to acquisition time). HARDI data provides insight into the directional properties of water diffusion in cerebral matter as a function of spatial coordinates. Ideally, one would be interested in having this information available at fine spatial resolution while minimizing the probing along different spatial orientations (so as to minimize the acquisition time). Unfortunately, availability of such datasets in reasonable acquisition times are hindered by limitations in current hardware and scanner protocols. On the other hand, post processing techniques prove promising in increasing the effective spatial resolution, allowing more detailed depictions of cerebral matter, while keeping the number of diffusion measurements within a feasible range. In light of the preceding developments, the main purpose of this research is to look into super resolution of HARDI data, using the modern theory of compressed sensing. The method proposed in this thesis allows an accurate approximation of HARDI signals at a higher spatial resolution compared to data obtained with a typical scanner. At the same time, ideas for reducing the number of diffusion measurements in the angular domain to improve the acquisition time are explored. Accordingly, the novel method of applying two distinct compressed sensing approaches in both spatial and angular domain, and combining them into a single framework for performing super resolution forms the main contribution provided by this thesis

Fast high spatio-angular resolution estimation of the neuronal fiber orientations in the brain with diffusion MRI

The human brain is a very complex organ. The white matter tissue composing the brain is made of threadlike structures, called axons, which are responsible for the transmission of the impulses between different areas of the brain. Axons are extremely fine and cannot be visualized by any in vivo imaging technique, leaving a lot to explore about which brain regions are connected and how information is carried through these structures. Diffusion Magnetic Resonance Imaging (dMRI) is a unique technique that allows to investigate the inner structures of the brain in vivo and in a non-invasive way. High spatio-angular resolution dMRI techniques have been shown to provide accurate fiber reconstructions even in the presence of complex fiber configurations. However, high resolution techniques are characterized by long acquisition times, which hamper their application into the clinical practice. In this manuscript we present a novel method to recover the fiber orientation distribution (FOD) of the bundles of axons at high spatio-angular resolution via practical kq-space under-sampling that enables both acceleration and super-resolution. The quality of the recovered fibers is preserved by making use of advanced anatomical priors for the FOD reconstruction. Prior knowledge of the spatial distribution of the white matter, the gray matter and the cerebrospinal fluid is taken into account for the recovery of the FOD coefficients. In addition, the simultaneous voxelwise sparsity and spatial smoothness of fiber orientations is accounted for by means of a structured sparsity prior. A convex minimization problem is formulated and solved via an accelerated stochastic Forward-Backward algorithm. Simulations show that the proposed method outperforms state-of-the-art kq-space approaches in terms of reconstruction quality. Real data analysis suggests that accurate FOD mapping can be achieved from severe kq-space under-sampling regimes, potentially enabling the application of high spatio-angular resolution dMRI into the clinical practice