Topological Data Analysis in Sub-cellular Motion Reconstruction and Filament Networks Classification

Topological Data Analysis is a powerful tool in the image data analysis. In this dissertation, we focus on studying cell physiology by the sub-cellular motions of organelles and generation process of filament networks, relying on topology of the cellular image data. We first develop a novel, automated algorithm, which tracks organelle movements and reconstructs their trajectories on stacks of microscopy image data. Our tracking method proceeds with three steps: (i) identification, (ii) localization, and (iii) linking, and does not assume a specific motion model. This method combines topological data analysis principles with Ensemble Kalman Filtering in the computation of associated nerve during the linking step. Moreover, we show a great success of our method with several applications. We then study filament networks as a classification problem, and propose a distancebased classifier. This algorithm combines topological data analysis with a supervised machine learning framework, and is built based on the foundation of persistence diagrams on the data.We adopt a new metric, the dcp distance, on the space of persistence diagrams, and show it is useful in catching the geometric difference of filament networks. Furthermore, our classifier succeeds in classifying filament networks with high accuracy rate

Advanced Statistical Methods for Atomic-Level Quantification of Multi-Component Alloys

This thesis comprises a collection of papers whose common theme is data analysis of high entropy alloys. The experimental technique used to view these alloys at the nano-scale produces a dataset that, while comprised of approximately 10^7 atoms, is corrupted by observational noise and sparsity. Our goal is to developstatistical methods to quantify the atomic structure of these materials. Understanding the atomic structure of these materials involves three parts: 1. Determining the crystal structure of the material 2. Finding the optimal transformation onto a reference structure 3. Finding the optimal matching between structures and the lattice constantFrom identifying these elements, we may map a noisy and sparse representation of an HEA onto its reference structure and determine the probabilities of different elemental types that are immediately adjacent, i.e., first neighbors, or are one-level removed and are second neighbors. Having these elemental descriptors of a material, researchers may then develop interaction potentials for molecular dynamics simulations, and make accurate predictions about these novel metallic alloys