Summer 2021

Tensors are a a much better way to store and analyze multidimensional data than traditional, matrix-based representation. However, generalizations of effective numerical techniques for extracting features from matrices such as the Singular Value Decomposition to tensors are neither straightforward nor uniquely defined. Those techniques are crucial to identify relationships in high-dimensional data. In this project, we study tensor SVDs, propose a projection-based classification algorithm, and experiment with the four-dimensional StarPlus fMRI dataset.

Our group explored two methods - a machine learning method and an atlas-based image registration method - to automatically identify the cerebellum and brain stem in magnetic resonance image (MRI) data. Identifying these two brain regions is needed to analyze DENSE MRI data and quantify the brain movement caused by the subject’s heart beat. This in turn can help determine whether or not a subject has a Chiari Malformation. We compared the strengths and weaknesses of both methods, and how they might work in the future to improve Chiari diagnosis.

Our team has devised a way to improve point-of-care tomographic imaging and thereby making medical diagnoses more effectively. The work behind this new algorithm relies heavily upon mathematical techniques that connect numerical linear algebra with optimization methods.

Our team developed efficient numerical linear algebra techniques to improve generative models based on potential flows. We derive a preconditioner from the stochastic Lanzcos quadratures and found that it reduced the number of conjugate gradient (CG) iterations that CG requires to converge.

The role of mathematical modeling in clinics is particularly evident in cardiology, as computational mechanics for many historical reasons is a mature field of applied mathematics; on the other hand, many important cardiovascular pathologies have a significant mechanical component, in terms of fluid, structure and their interactions. The clinical impact of mathematical models strongly relies on reconstructing patient geometries to customize and personalize numerical simulations. Advances in medical image processing made over the last two decades have enabled virtual patient-specific models. A key step of the processing pipeline in Cardiology is the extraction of complex vascular geometries like an aortic dissection from medical images (typically, Computed Tomographies, Magnetic Resonance, and Optical Coherence Tomography). Our team evaluated the relation between PDEs and image segmentation/reconstruction through the level set method, and compared this segmentation approach with deep learning ones based on Convolutional Neural Networks.