Topological Data Analysis (TDA), a fast-growing research topic in applied topology, uses techniques in algebraic topology to capture features from data. Its importance has been discovered in many areas, such as medical image processing, molecular biology, machine learning, and pattern recognition. Persistent homology (PH) is vital in topological data analysis that detects local changes in filtered topological spaces. It measures the robustness and significance of homological objects in spaces' deformation, such as connected components, loops, or higher dimensional voids. In Morse theory, filtered spaces for persistent homology usually rely on a single parameter, such as the sublevel set filtration of height functions. Recently, as a generalization of persistent homology, computational topologists began to be interested in multi-parameter persistent homology. Multi-parameter persistent homology (or multi-parameter persistence) is an algebraic structure established on a multi-parametrized network of topological spaces and has more fruitful geometric information than persistent homology. So far, finding methods to extract features in multi-parameter persistence is still an open and concentrating topic in TDA. Also, examples of multi-parameter filtration are still rare and limited. The three principal contributions of this dissertation are as follows. First, we combined persistent homology features (persistence statistics and persistence curves) and machine learning models for analyzing medical images. We found that adding topological information into machine learning models can improve recognition accuracy and stability. Second, unlike traditional construction for multi-parameter filtrations in Euclidean spaces, we propose a framework for constructing multi-parameter filtrations from digital images through mathematical morphology and discrete geometry. Multi-parameter persistence derived from mathematical morphology is more efficient for computing and contains intuitive geometric attributes of objects, such as the sizes or robustness of local objects in digital images. We involve these features to remove the salt and pepper noise in digital images as an application. Compared with current denoise algorithms, the proposed approach has a more stable accuracy and keeps the topological structures of original data. The third part of this dissertation focuses on using sheaf theory to analyze the lifespans of objects in multi-parameter persistence. The multi-parameter persistence has a natural sheaf structure by equipping the Alexandrov topology on the based partially ordered set. This sheaf structure uncovers the gluing properties of local image regions in the multi-parameter filtration. We referred to these properties as a fingerprint of the filtration and applied them for the character recognition task. Finally, we propose using sheaf operators to define ultrametric norms on local spaces in multi-parameter persistence. Like persistence barcodes, this metric provides finer geometric and topological quantities.

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