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To this day, the origins and prognosis of Alzheimer's disease (AD) pose significant challenges. This thesis presents a methodology to understand AD by systematically applying state-of-the-art techniques from graph neural networks (GNNs) to medical images, producing a set of individualized graphs suitable for analysis. We accomplish this task by defining a latent graph variable as an intermediate latent variable inferred in a machine learning task. As one of the first researchers to explore the use of GNNs with medical images, this thesis begins with simple machine learning tasks that gradually increase in complexity. In particular, the thesis is organized into the following four main parts: (1) establishing a baseline with standard attention-based models applied to gray matter and PET probability regions of interest (ROIs), (2) developing a 3D-CNN model for processing voxel images directly to infer a latent graph more interpretable and closely aligned to known forms of connectivity within the brain, (3) learning latent graphs for a challenging network diffusion task with synthetic and real longitudinal datasets, (4) and developing a pipeline based on statistical analysis and metrics for comparing the learned latent graphs to baseline graphs derived from standard methods to assess the performance of the latent graphs. Our experiments show that the careful application of GNNs to medical images can provide a suitable inductive bias to enhance performance compared to standard baselines such as support vector machines and multilayer perceptrons. The latent graph connectivity structure learned from our new 3D-CNN-GNN model alone is significant enough to achieve state-of-the-art precision when differentiating between patients with mild cognitive impairment (MCI) who will progress to AD and those who will maintain an MCI diagnosis, providing a set of interpretable features to uncover the mysteries surrounding AD.

Due to the rapid development of imaging acquisition techniques, a large amount of large-scale biomedical studies on neuro-related diseases have been established, such as Alzheimer's Disease Neuroimaging Initiative (ADNI) study, the Human Connectome Project (HCP) study and the UK biobank (UKB) study. These databases have myriad data types, including genomic data, phenotypic data, imaging data, clinical outcomes, and many others. Among the studies on biomedical data, functional regression has received the most attention in functional data analysis (FDA). However, existing statistical methods in this field still face challenges while considering unknown confounders caused by imaging heterogeneity, non-Euclidean imaging data, or imaging data with jump discontinuities. To tackle these challenges, we propose several functional regression models for biomedical data with different structures. First, the function-on-scalar regression model is a powerful tool for building the relationship between the functional responses and covariates of interest. Although it has been widely applied in many large-scale neuroimaging studies, most existing approaches fail to take into account the nonlinear relationship and the unknown confounders caused by imaging heterogeneity. In particular, the potential heterogeneity may be caused by the differences in the study environment, population, design, protocols, and other unknown factors. To address this challenge, we develop a single index function-on-scalar regression model to investigate the nonlinear relationship between functional responses and covariates of interest while adjusting for unknown confounding factors caused by potential imaging heterogeneity. Both estimation and inference procedures are established for unknown quantities in our proposed model. In addition, the asymptotic properties of estimated functions and detected confounding factors are also systematically investigated. The finite-sample performance of our proposed method is assessed by using both Monte Carlo simulations and a real data example on the diffusion tensor images from the ADNI study. Second, causal mediation analysis is widely utilized in neuroscience to investigate the role of brain image phenotypes in the neurological pathways from genetic exposures to clinical outcomes. However, it is still difficult to conduct a genome-wide mediation analysis with the shapes of brain regions as mediators due to several challenges including (i) large-scale genetic exposures, i.e., millions of single-nucleotide polymorphisms (SNPs); (ii) nonlinear Hilbert space for shape mediators; and (iii) statistical inference on the direct and indirect effects. To tackle these challenges, this paper proposes a mediation analysis framework with high dimensional genetic exposures and shape mediators. First, to address the issue caused by the high dimensionality in genetic exposures, a fast genome-wide association analysis is conducted to discover potential genetic variants with significant genetic effects on the clinical outcome. Second, the square-root velocity function representations are extracted from the shapes of the subcortical brain structure, which fall in an unconstrained linear Hilbert subspace. Third, to identify the underlying causal pathways from the detected SNPs to the clinical outcome implicitly through the shape mediators, we proposed a framework consisting of a shape-on-scalar model and a scalar-on-shape model. Furthermore, the bootstrap resampling approach is adopted to investigate both global and spatial significant mediation effects. Finally, our framework is applied to the ADNI corpus callosum shape data, and we successfully identify the mediation effect of a subset of candidate SNPs on Alzheimer's Disease through some specific subregions of the corpus callosum. Finally, most existing methods in functional data analysis assume the smoothness of the functional responses and the coefficient functions or require some smoothing techniques before fitting the functional responses into the statistical models. However, this is often violated in biomedical imaging data, which usually contains spatially contiguous regions or effect regions with relatively sharp edges due to the physical or biological reasons. Thus they tend to blur the image data near the edges or jumps. While the pixel or voxel based methods ignore the spatial information of imaging data, they fail to characterize the spatial correlations. Thus, it is essential to propose an appropriate framework to reveal the relationship between imaging responses and the covariates of interest, while taking the discontinuous property and spatial correlation into account simultaneously. To address these challenges, we proposed a Bayesian hypothesis testing framework on image-on-scalar regression model with mixed effects, and consider wavelet basis decomposition to account for the sudden transitions in the imaging responses. To depict the spatial correlation between grids for each observation, a conditional auto-regressive (CAR) model is utilized. Due to the advantage of Bayesian analysis in hypothesis testing, we propose a Bayesian framework for an image-on-scalar CAR model using a set of Normal - Inverse-Gamma - Uniform priors, introducing the local indicators to conduct block-wise hypothesis tests efficiently and flexibly.

Neurodegenerative diseases affect large portions of the population worldwide and account for a significant portion of medical spending in associated costs. This dissertation will attempt to utilize graph theory and Magnetic Resonance Imaging (MRI) techniques to monitor disease progression and treatment efficacy. The two forms of neurodegeneration presented here are Alzheimer's Disease (AD) and ischemic stroke. The two MRI methods utilized for this work are diffusion tensor imaging (DTI) and resting-state functional MRI (rs-fMRI). DTI provides a map of diffusional anisotropy based on the Brownian motion of water within the restricted boundaries of tissues. In the absence of direct stimulation, rs-fMRI displays cognitive activity based on the change in signal between oxygenated and deoxygenated hemoglobin to establish correlations between different neuroanatomical regions. In this work, a high angular resolution DTI acquired with 18 diffusion directions was acquired using an 11.75-T, 500-MHz MRI scanner located at the FAMU-FSU College of Engineering and a 21.1-T, 900-MHz MRI scanner located at the National High Magnetic Field Laboratory to perform tractography in 5xFAD and 3xTgAD mouse models of familial Alzheimer's Disease. The 21.1-T scanner was used to perform the same DTI acquisition and rs-fMRI acquisitions on ischemic stroke models in rats, some of which received a human mesenchymal stem cell treatment to evaluate the structural and functional changes that occur over time, post-ischemia. The main neural areas of focus for the DTI studies were the piriform area of the cortex, temporal cortex, parietal cortex and hippocampus. In the rs-fMRI study, regions included the amygdala, hippocampus, midbrain, thalamus and striatum. Changes in the network metrics of weighted degree, eccentricity, clustering coefficient, betweenness centrality, closeness centrality, harmonic closeness centrality, and eigenvector centrality were measured as a function of age/time post-ischemia, phenotype (AD) and treatment regimen.

As the push for autonomous systems gains momentum, there has been a shift from automated to semi-autonomous systems, and current and future applications of artificial intelligence will revolve around the development of autonomous systems. At present, research is being performed to enable autonomous systems in trading, transportation, weapons, aircrafts, homes, drones, robots, and diagnostic imaging. Several companies such as Apple, Google, Microsoft and Tesla as well as NASA, the US Navy and the US military are investing in research on autonomous systems. Autonomous systems may involve the use of a large number of hardware and software components. While sensors are responsible for collecting data about the surroundings, machine learning and algorithms are used to generate useful information from available data and determine the action that should be taken. For most automatic and autonomous systems, it is important for the system to have the capability to detect hazardous situations and provide precise details, enabling either a human to take over or allowing the system itself to handle the situation. While an autonomous system must have sufficient knowledge or training to be able to achieve its ultimate goal, it must be capable of dynamically modifying the path to the goal based on the information obtained by fusing data from various sources or sensors. Besides, several autonomous systems, such as autonomous vehicles or autonomous trading, need fast real-time data processing in order to act accurately in real-time. Finally, the goal of all autonomous systems is to minimize human intervention. This dissertation contributes to the development of autonomous systems through applications that target different aspects of complex autonomous systems and belong to three distinct domains. It includes automatic segmentation of brain tumors, which is a step towards autonomous diagnostic imaging as well as an example of detection and exact identification of a hazardous condition. The automatic generation of high-resolution undistorted MRIs from distorted low-resolution MRIs reduces the amount of human intervention and develops a system that generates usable information from available data. On the other hand, drone self-localization, road segmentation, and extraction of road network graphs involve fusion of data from various types of sensors. Road graph extraction and drone self-localization play an import role in routing, and is crucial in the ability of an unmanned vehicle to reach its final goal. Besides, in the field of autonomous trading, we explore the use of time-series data to predict stock prices and make decisions about real-time stock transactions.

Chan-Vese is a level set method that simultaneously evolves a level set surface and fits locally constant intensity models for the interior and exterior regions to minimize a Mumford-Shah integral. However, the length-based contour regularization in the Chan-Vese formulation is quite simple and too weak for many applications. In this dissertation we introduce a generalization of the Chan-Vese method to evolve a curve where the regularization is based on a Fully Convolutional Neural Network. We also show how to learn the curve model as a Recurrent Neural Network (RNN) using training examples. Our RNN differs from the standard ones because it has the Chan-Vese locally constant intensity model, which gives it better interpretability and flexibility.

Diffusion-weighted image(DWI) and tractography have been developed for decades and are key elements in recent, large-scale efforts for mapping the human brain. The two techniques together provide us a unique possibility to access the macroscopic structure and connectivity of the human brain non-invasively and in vivo. The information obtained not only can help visualize brain connectivity and help segment the brain into different functional areas but also provides tools for understanding some major cognitive diseases such as multiple sclerosis, schizophrenia, epilepsy, etc. There are lots of efforts have been put into this area. On the one hand, a vast spectrum of tractography algorithms have been developed in recent years, ranging from deterministic approaches through probabilistic methods to global tractography; On the other hand, various mathematical models, such as diffusion tensor, multi-tensor model, spherical deconvolution, Q-ball modeling, have been developed to better exploit the acquisition dependent signal of Diffusion-weighted image(DWI). Despite considerable progress in this area, current methods still face many challenges, such as sensitive to noise, lots of false positive/negative fibers, incapable of handling complex fiber geometry and expensive computation cost. More importantly, recent researches have shown that, even with high-quality data, the results using current tractography methods may not be improved, suggesting that it is unlikely to obtain an anatomically accurate map of the human brain solely based on the diffusion profile. Motivated by these issues, this dissertation develops a global approach that incorporates anatomical validated geometric shape prior when reconstructing neuron fibers. The fiber tracts between regions of interest are initialized and updated via deformations based on gradients of the posterior energy defined in this paper. This energy has contributions from diffusion data, shape prior information, and roughness penalty. The dissertation first describes and demonstrates the proposed method on the 2D dataset and then extends it to 3D Phantom data and the real brain data. The results show that the proposed method is relatively immune to issues such as noise, complicated fiber structure like fiber crossings and kissing, false positive fibers, and achieve more explainable tractography results.

Lung cancer has the highest mortality rate of all cancers in both men and women in the United States. The algorithmic detection, characterization, and diagnosis of abnormalities found in chest CT scan images can aid radiologists by providing additional medically-relevant information to consider in their assessment of medical images. Such algorithms, if robustly validated in clinical settings, carry the potential to improve the health of the general population. In this thesis, we first give an analysis of publicly available chest CT scan annotation data, in which we determine upper bounds on expected classification accuracy when certain radiological features are used as inputs to statistical learning algorithms for the purpose of inferring the likelihood of a lung nodule as being either malignant or benign. Second, a statistical extension of the level set method for image segmentation is introduced and applied to both synthetically-generated and real three-dimensional image volumes of lung nodules in chest CT scans, obtaining results comparable to the current state-of-the-art on the latter.

Diffusion tensor imaging (DTI) provides a map of diffusional anisotropy based on the Brownian motion of water within the restricted boundaries of tissues. In this work, a high angular resolution DTI acquired with 18 diffusion directions and four unweighted images was acquired using a 11.75-T, 500-MHz MRI scanner located at the FAMU-FSU College of Engineering in order to perform tractography in amyloid precursor protein/presenilin 1 (APP/PS1) mouse models of familial Alzheimer’s Disease. Evaluating phenotype (APP/PS1 versus wild type) and age (1, 2, 4 and 6 months), a structural network analysis was employed to assess DTI datasets acquired at an in-plane resolution was 100 x 100 microns with a matrix size of 256 x 256, repetition time of 2 s, echo time of 30 ms, diffusion gradient separation of 21 ms and diffusion gradient time of 3 ms. With 15 averages, high signal-to-noise ratios were achieved over an approximate acquisition time of 47 h per sample. This study used all female brains fixed with 4% paraformaldehyde. The five main neural areas of focus were the piriform area of the cortex, temporal cortex, parietal cortex, and left and right hippocampus. A significant decrease in FA of the temporal cortex was identified. Changes in the network metrics of weighted degree, eccentricity, clustering, betweenness centrality, and closeness centrality were observed as a function of age and phenotype.

Machine learning algorithms along with magnetic resonance imaging (MRI) provides promising techniques to overcome the drawbacks of the current clinical screening techniques. In this study the resting-state functional magnetic resonance imaging (fMRI) to see the level of activity in a patient's brain and dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) to explore the level of improvement of neo-adjuvant chemotherapy in patients with locally advanced breast cancer were considered. As the first project, we considered fMRI of patients before and after they underwent a double-blind smoking cessation treatment. For the first time, this study aims at developing new theory-driven biomarkers by implementing and evaluating novel techniques from resting-state scans that can be used in relapse prediction in nicotine-dependent patients and future treatment efficacy. In this regards, two classes of patients have been studied, one took the drug N-acetylcysteine and the other took a placebo. Our goal was to classify the patients as treatment or non-treatment, based on their fMRI scans. The image slices of brain are used as the variable. We have applied different voxel selection schemes and data reduction algorithms on all images. Then, we compared several multivariate classifiers and deep learning algorithms and also investigated how the different data reductions affect classification performance. For the second part, we have employed multi-parametric magnetic resonance imaging (mpMRI) using different morphological and functional MRI parameters such as T2-weighted, dynamic contrast-enhanced (DCE) MRI, and diffusion weighted imaging (DWI) has emerged as the method of choice for the early response assessments to NAC. Although, mpMRI is superior to conventional mammography for predicting treatment response, and evaluating residual disease, yet there is still room for improvement. In the past decade, the field of medical imaging analysis has grown exponentially, with an increased numbers of pattern recognition tools, and an increase in data sizes. These advances have heralded the field of radiomics. Radiomics allows the high-throughput extraction of the quantitative features that result in the conversion of images into mineable data, and the subsequent analysis of the data for an improved decision support with response monitoring during NAC being no exception. In this study. we determined the importance and ranking of the extracted parameters from mpMRI using T2-weighted, DCE, and DWI for prediction of pCR and patient outcomes with respect to metastases and disease-specific death employing different machine learning algorithms.

Alzheimer's disease (AD) is one of the top 10 leading causes of death in the US; it debilitates memory and impairs cognition. The current core clinical criteria for diagnosis of AD are based on functional deficits and cognitive impairments that do not include the advanced imaging techniques or cerebrospinal fluid analysis; the final confirmation of the disease is only possible at the time of autopsy when neurofibrillary tangles and beta-amyloids are present in a brain tissue examination. The distributions of these particular pathogens (neurofibrillary tangles and beta-amyloids) follow a pattern that is useful to identify different stages of AD at the time of the autopsy by looking at the presence of pathogens in the areas of the brain. The pathogens are first seen in entorhinal/perirhinal cortex, and then spread to hippocampus cornu ammonis subfields, followed by association cortex and finally the rest of the brain. This disease progression is standard and described in NIA-RI guidelines. In the last decades, with the introduction of advanced imaging techniques in research settings, many in vivo based research methods have been focusing on the volumetric measurements of the hippocampus and its subfields in MRI images and using them as additional information for early diagnosis of AD. While the hippocampal volume provides excellent diagnostic aid, it doesn't address both the pathogens associated with AD and the progression of the pathogens within the different subregions of the hippocampus. The hippocampus formation is a complex circuit that spans the temporal lobes and found to have distinctive subregions. These subregions are subject to different influence by AD at different stages. Since the disease progression as seen in pathogen distributions follows a pattern, studying the pattern of the regional changes will allow us to predict which stage the disease is at. These pathological shifts in regions of the brain are studied extensively in ex vivo MRI as well as during autopsy but not in in vivo MRI. Considering that the brain areas with neurofibrillary tangles and beta-amyloids show hypointensity (PD-weighted) and hyperintensity voxels (T2-weighted) in the MRI images, we suggest an in vivo study using normalized MRI images taken from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. We analyze the pattern of changes in the hippocampal region using a volume-based method in normalized T1-weighted MRI and an intensity-based analysis in normalized PD-weighted MRI. We use the volume-based method to calculate the changes in the volume of each of the hippocampus subfields. For the intensity-based method, we count the number of hypointensity (intensity value < 125) voxel combinations from the Gray-Level Co-occurrence Matrix (GLCM); a normalized MRI images are used inorder to minimize intensity variation. We then use the data (volume and intensity changes) to construct decision trees which classify the MRI images into three categories: normal control (NC), mild cognitive impairment (MCI) and AD. We have found that the volume-based decision trees detect AD MRI images with an accuracy of 75 % but failed to detect NC and MCI MRI images with the same level of accuracy. Whereas, with the intensity-based decision trees, we were able to classify MRI images into NC, MCI and AD categories each with an equally high level of accuracy (above 86 %). To find out how reliable the intensity-based method is in classifying MRI images, we introduced noises to our images. The addition of noises forced some adjustments in our decision trees. The accuracy of decision tree classification decreased in the presence of noises. However, even in the presence of the additional noises, we noticed that the intensity-based method outperforms volume-based method. The classification of MRI images improves when both measures (intensity-based and volume-based) are used in constructing our decision trees. This study has demonstrated that the inclusion of the intensity measurements of PD-weighted MRI images in AD studies may provide a more accurate way to model the natural progression of AD in vivo and contribute to the early diagnosis of AD.

A new method for the segmentation of 3D breast lesions in dynamic contrast enhanced magnetic resonance imaging (DCE-MRI) images, using parallel programming with general purpose computing on graphics processing units (GPGPUs), is proposed. The method has two main parts: a pre-processing step and a segmentation algorithm. In the pre-processing step, DCE-MRI images are registered using an intensity-based rigid transformation algorithm based on gradient descent. After the registration, voxels that correspond to breast lesions are enhanced using the Naïve Bayes machine learning classifier. This classifier is trained to identify four different classes inside breast images: lesion, normal tissue, chest and background. Training is performed by manually selecting 150 voxels for each of the four classes from images in which breast lesions have been confirmed by an expert in the field. Thirteen attributes obtained from the kinetic curves of the selected voxels are later used to train the classifier. Finally, the classifier is used to increase the intensity values of voxels labeled as lesions and to decrease the intensities of all other voxels. The post-processed images are used for volume segmentation of the breast lesions using a level set method based on the active contours without edges (ACWE) algorithm. The segmentation algorithm is implemented in OpenCL for the GPGPUs to accelerate the original model by parallelizing two main steps of the segmentation process: the computation of the signed distance function (SDF) and the evolution of the segmented curve. The proposed framework uses OpenGL to display the segmented volume in real time, allowing the physician to obtain immediate feedback on the current segmentation progress. The proposed implementation of the SDF is compared with an optimal implementation developed in Matlab and achieves speedups of 25 and 12 for 2D and 3D images, respectively. Moreover, the OpenCL implementation of the segmentation algorithm is compared with an optimal implementation of the narrow-band ACWE algorithm. Peak speedups of 55 and 40 are obtained for 2D and 3D images, respectively. The segmentation algorithm has been developed as open source software, with different versions for 2D and 3D images, and can be used in different areas of medical imaging as well as in areas within computer vision, such like tracking, robotics and navigation.

With the increasing popularity of information technology, especially electronic imaging techniques, large amount of high dimensional data such as 3D shapes become pervasive in science, engineering and even people's daily life, in the recent years. Though the data quantity is huge, the extraction of relevant knowledge on those data is still limited. How to understand data in a meaningful way is generally an open problem. The specific challenges include finding adequate mathematical representations of data and designing proper algorithms to process them. The existing tools for analyzing high-dimensional data, including 3D shape data, are found to be insufficient as they usually suffer from many factors, such as misalignments, noise, and clutter. This thesis attempts to develop a framework for processing, analyzing and understanding high-dimensional data, especially 3D shapes, by proposing a set of statistical tools including theory, algorithms and optimization applied to practical problems. In particular, the following aspects of shape analysis are considered: 1. A framework adopting the SRNF representation, based on parallel transport of deformations across surfaces in the shape space, leads to statistical analysis on shape data. Three main analyses are conducted under this framework: (1) computing geodesics when either two end surfaces or the starting surface and an initial deformation are given; (2) parallel transporting deformation across surfaces; and (3) sampling random surfaces. 2. Computational efficiency plays an important role in performing statistical shape analysis on large datasets of 3D objects. To speed up the previous method, a framework with numerical solution is introduced by approximating the inverse mapping, and it reduces the computational cost by an order of magnitude. 3. The geometrical and morphological information, or their shapes, of 3D objects can be analyzed explicitly using boundaries extracted from original image scans. An alternative idea is to consider variability in shapes directly from their embedding images. A novel framework is proposed to unify three important tasks, registering, comparing and modeling images. 4. Finally, the spatial deformations learned from registering images are modeled using the GRID based decomposition. This specific model provides a way to decompose a large deformation into local and fundamental ones so that shape differences between images are easily interpretable. We conclude this thesis with conclusions drawn in this research and discuss potential future directions of statistical shape analysis in the last chapter, both from methodological and application aspects.