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Tomography is a widely used tool for analyzing microstructures in three dimensions (3D). The analysis, however, faces difficulty because the constituent materials produce similar grey-scale values. Sometimes, this prompts the image segmentation process to assign a pixel/voxel to the wrong phase (active material or pore). Consequently, errors are introduced in the microstructure characteristics calculation. In this work, we develop a filtering algorithm called PerSplat based on topological persistence (a technique used in topological data analysis) to improve segmentation quality. One problem faced when evaluating filtering algorithms is that real image data in general are not equipped with the `ground truth' for the microstructure characteristics. For this study, we construct synthetic images for which the ground-truth values are known. On the synthetic images, we compare the pore tortuosity and Minkowski functionals (volume and surface area) computed with our PerSplat filter and other methods such as total variation (TV) and non-local means (NL-means). Moreover, on a real 3D image, we visually compare the segmentation results provided by our filter against TV and NL-means. The experimental results indicate that PerSplat provides a significant improvement in segmentation quality. 

Graphs model real-world circumstances in many applications where they may constantly change to capture the dynamic behavior of the phenomena. Topological persistence which provides a set of birth and death pairs for the topological features is one instrument for analyzing such changing graph data. However, standard persistent homology defined over a growing space cannot always capture such a dynamic process unless shrinking with deletions is also allowed. Hence, zigzag persistence which incorporates both insertions and deletions of simplices is more appropriate in such a setting. Unlike standard persistence which admits nearly linear-time algorithms for graphs, such results for the zigzag version improving the general O(m^ω) time complexity are not known, where ω < 2.37286 is the matrix multiplication exponent. In this paper, we propose algorithms for zigzag persistence on graphs which run in near-linear time. Specifically, given a filtration with m additions and deletions on a graph with n vertices and edges, the algorithm for 0-dimension runs in O(mlog² n+mlog m) time and the algorithm for 1-dimension runs in O(mlog⁴ n) time. The algorithm for 0-dimension draws upon another algorithm designed originally for pairing critical points of Morse functions on 2-manifolds. The algorithm for 1-dimension pairs a negative edge with the earliest positive edge so that a 1-cycle containing both edges resides in all intermediate graphs. Both algorithms achieve the claimed time complexity via dynamic graph data structures proposed by Holm et al. In the end, using Alexander duality, we extend the algorithm for 0-dimension to compute the (p-1)-dimensional zigzag persistence for ℝ^p-embedded complexes in O(mlog² n+mlog m+nlog n) time. 

Persistent cycles, especially the minimal ones, are useful geometric features functioning as augmentations for the intervals in the purely topological persistence diagrams (also termed as barcodes). In our earlier work, we showed that computing minimal 1-dimensional persistent cycles (persistent 1-cycles) for finite intervals is NP-hard while the same for infinite intervals is polynomially tractable. In this paper, we address this problem for general dimensions with Z2 coefficients. In addition to proving that it is NP-hard to compute minimal persistent d-cycles (d>1) for both types of intervals given arbitrary simplicial complexes, we identify two interesting cases which are polynomially tractable. These two cases assume the complex to be a certain generalization of manifolds which we term as weak pseudomanifolds. For finite intervals from the d-th persistence diagram of a weak (d+1)-pseudomanifold, we utilize the fact that persistent cycles of such intervals are null-homologous and reduce the problem to a minimal cut problem. Since the same problem for infinite intervals is NP-hard, we further assume the weak (d+1)-pseudomanifold to be embedded in R^{d+1}Rd+1 so that the complex has a natural dual graph structure and the problem reduces to a minimal cut problem. Experiments with both algorithms on scientific data indicate that the minimal persistent cycles capture various significant features of the data.