Shape analysis is important in fields like computational geometry, biology, and machine learning, where understanding differences in structure and tracking changes over time is useful. Topological Data Analysis (TDA) provides tools to study shape in a way that is resistant to noise and captures both fine and large-scale features. This dissertation focuses on directional transforms, a method that encodes shape by looking at its structure from different directions.First, we introduce the Labeled Merge Tree Transform (LMTT), a new way to compare embedded graphs using merge trees and directional transforms. We test this method on real-world datasets and show that it works better than existing distance measures in some cases. Next, we develop a kinetic data structure (KDS) for bottleneck distance, which allows us to update shape comparisons efficiently when the data changes over time. We apply this method to the Persistent Homology Transform (PHT) and show that it reduces computation time while keeping accurate results. Finally, we explore a future direction that extends the kinetic data structure to the Wasserstein distance. These contributions improve the use of topology in studying dynamic shapes and open new research possibilities in both theory and practical applications.
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