The Topology ToolKit

This system paper presents the Topology ToolKit (TTK), a software platform designed for topological data analysis in scientific visualization. TTK provides a unified, generic, efficient, and robust implementation of key algorithms for the topological analysis of scalar data, including: critical points, integral lines, persistence diagrams, persistence curves, merge trees, contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots, Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due to a tight integration with ParaView. It is also easily accessible to developers through a variety of bindings (Python, VTK/C++) for fast prototyping or through direct, dependence-free, C++, to ease integration into pre-existing complex systems. While developing TTK, we faced several algorithmic and software engineering challenges, which we document in this paper. In particular, we present an algorithm for the construction of a discrete gradient that complies to the critical points extracted in the piecewise-linear setting. This algorithm guarantees a combinatorial consistency across the topological abstractions supported by TTK, and importantly, a unified implementation of topological data simplification for multi-scale exploration and analysis. We also present a cached triangulation data structure, that supports time efficient and generic traversals, which self-adjusts its memory usage on demand for input simplicial meshes and which implicitly emulates a triangulation for regular grids with no memory overhead. Finally, we describe an original software architecture, which guarantees memory efficient and direct accesses to TTK features, while still allowing for researchers powerful and easy bindings and extensions. TTK is open source (BSD license) and its code, online documentation and video tutorials are available on TTK's website

The parallel computation of morse-smale complexes

pre-printTopology-based techniques are useful for multi-scale exploration of the feature space of scalar-valued functions, such as those derived from the output of large-scale simulations. The Morse-Smale (MS) complex, in particular, allows robust identification of gradient-based features, and therefore is suitable for analysis tasks in a wide range of application domains. In this paper, we develop a two-stage algorithm to construct the Morse-Smale complex in parallel, the first stage independently computing local features per block and the second stage merging to resolve global features. Our implementation is based on MPI and a distributed-memory architecture. Through a set of scalability studies on the IBM Blue Gene/P supercomputer, we characterize the performance of the algorithm as block sizes, process counts, merging strategy, and levels of topological simplification are varied, for datasets that vary in feature composition and size. We conclude with a strong scaling study using scientific datasets computed by combustion and hydrodynamics simulations

Singular Continuation: Generating Piece-wise Linear Approximations to Pareto Sets via Global Analysis

We propose a strategy for approximating Pareto optimal sets based on the global analysis framework proposed by Smale (Dynamical systems, New York, 1973, pp. 531-544). The method highlights and exploits the underlying manifold structure of the Pareto sets, approximating Pareto optima by means of simplicial complexes. The method distinguishes the hierarchy between singular set, Pareto critical set and stable Pareto critical set, and can handle the problem of superposition of local Pareto fronts, occurring in the general nonconvex case. Furthermore, a quadratic convergence result in a suitable set-wise sense is proven and tested in a number of numerical examples.Comment: 29 pages, 12 figure

Parallel Mapper

The construction of Mapper has emerged in the last decade as a powerful and effective topological data analysis tool that approximates and generalizes other topological summaries, such as the Reeb graph, the contour tree, split, and joint trees. In this paper, we study the parallel analysis of the construction of Mapper. We give a provably correct parallel algorithm to execute Mapper on multiple processors and discuss the performance results that compare our approach to a reference sequential Mapper implementation. We report the performance experiments that demonstrate the efficiency of our method

Combinatorial Gradient Fields for 2D Images with Empirically Convergent Separatrices

This paper proposes an efficient probabilistic method that computes combinatorial gradient fields for two dimensional image data. In contrast to existing algorithms, this approach yields a geometric Morse-Smale complex that converges almost surely to its continuous counterpart when the image resolution is increased. This approach is motivated using basic ideas from probability theory and builds upon an algorithm from discrete Morse theory with a strong mathematical foundation. While a formal proof is only hinted at, we do provide a thorough numerical evaluation of our method and compare it to established algorithms.Comment: 17 pages, 7 figure

Computing morse-smale complexes with accurate geometry

pre-printTopological techniques have proven highly successful in analyzing and visualizing scientific data. As a result, significant efforts have been made to compute structures like the Morse-Smale complex as robustly and efficiently as possible. However, the resulting algorithms, while topologically consistent, often produce incorrect connectivity as well as poor geometry. These problems may compromise or even invalidate any subsequent analysis. Moreover, such techniques may fail to improve even when the resolution of the domain mesh is increased, thus producing potentially incorrect results even for highly resolved functions. To address these problems we introduce two new algorithms: (i) a randomized algorithm to compute the discrete gradient of a scalar field that converges under refinement; and (ii) a deterministic variant which directly computes accurate geometry and thus correct connectivity of the MS complex. The first algorithm converges in the sense that on average it produces the correct result and its standard deviation approaches zero with increasing mesh resolution. The second algorithm uses two ordered traversals of the function to integrate the probabilities of the first to extract correct (near optimal) geometry and connectivity. We present an extensive empirical study using both synthetic and real-world data and demonstrates the advantages of our algorithms in comparison with several popular approaches

Parallel Computation of Piecewise Linear Morse-Smale Segmentations

This paper presents a well-scaling parallel algorithm for the computation of Morse-Smale (MS) segmentations, including the region separators and region boundaries. The segmentation of the domain into ascending and descending manifolds, solely defined on the vertices, improves the computational time using path compression and fully segments the border region. Region boundaries and region separators are generated using a multi-label marching tetrahedra algorithm. This enables a fast and simple solution to find optimal parameter settings in preliminary exploration steps by generating an MS complex preview. It also poses a rapid option to generate a fast visual representation of the region geometries for immediate utilization. Two experiments demonstrate the performance of our approach with speedups of over an order of magnitude in comparison to two publicly available implementations. The example section shows the similarity to the MS complex, the useability of the approach, and the benefits of this method with respect to the presented datasets. We provide our implementation with the paper.Comment: Journal: IEEE Transactions on Visualization and Computer Graphics / Submitted: 22-Jun-2022 / Accepted: 13-Mar-202

The Discrete Morse Complex of Images: Algorithms, Modeling and Applications

The Morse complex can be used for studying the topology of a function, e.g., an image or terrain height field when understood as bivariate functions. We present an algorithm for the computation of the discrete Morse complex of two-dimensional images using an edge-based data structure. By using this data structure, it is possible to perform local operations efficiently, which is important to construct the complex and make the structure useful for areas like visualization, persistent homology computation, or construction of a topological hierarchy. We present theoretical and applied results to demonstrate benefits and use of our method