Geometry‐Aware Alignment and Comparison of Hierarchical Morse Complexes with Applications

Scalar fields derived from 3D X‐ray CT scans of samples undergoing ex situ processes, such as thermal aging, chemical etching, or mechanical stress, pose unique challenges for characterizing similarities and differences across acquisitions. Typically, a sample A (source) is imaged, removed, and subjected to experimental conditions that alter its microstructure, and then re‐imaged as sample B (target) to study the resulting changes. Direct comparison between A and B is rendered impractical if not impossible for current techniques because the challenges of physical and morphological changes are compounded by the effects of geometric misalignment, differences in reconstruction parameters, discretization artifacts, and changes in acquisition settings such as position, beam intensity, or exposure time (the acquisition for sample B often happens at a much later time and the device may have been upgraded or changed). To overcome these challenges, we introduce a geometry‐rich topological representation that uses the hierarchical Morse complex to capture the structural relationships among regions segmented within each sample and shape descriptors to characterize their metric properties. With this data structure, we cast the similarity problem as a sequence of optimizations, each minimizing differences in structure and geometry at a given resolution. The sequence of optimizations begins by aligning the fine‐scale segmentations of the two samples. Following optimization minimizes differences across incrementally coarser levels, producing a fully synchronized hierarchical representation of the two samples. In addition, we introduce a visualization framework that enables interactive exploration and manual editing of the matched hierarchies, thereby allowing an expert user to further improve the quality of the comparison. We apply our workflow to characterize changes in grain structure for energetic materials undergoing aging, match segmentations for materials under different stress conditions, and perform image registration that outperforms state‐of‐the‐art techniques.