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

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

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

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

GPU Parallel Computation of Morse-Smale Complexes

The Morse-Smale complex is a well studied topological structure that represents the gradient flow behavior of a scalar function. It supports multi-scale topological analysis and visualization of large scientific data. Its computation poses significant algorithmic challenges when considering large scale data and increased feature complexity. Several parallel algorithms have been proposed towards the fast computation of the 3D Morse-Smale complex. The non-trivial structure of the saddle-saddle connections are not amenable to parallel computation. This paper describes a fine grained parallel method for computing the Morse-Smale complex that is implemented on a GPU. The saddle-saddle reachability is first determined via a transformation into a sequence of vector operations followed by the path traversal, which is achieved via a sequence of matrix operations. Computational experiments show that the method achieves up to 7x speedup over current shared memory implementations

Conforming Morse-Smale complexes

pre-printMorse-Smale (MS) complexes have been gaining popularity as a tool for feature-driven data analysis and visualization. However, the quality of their geometric embedding and the sole dependence on the input scalar field data can limit their applicability when expressing application-dependent features. In this paper we introduce a new combinatorial technique to compute an MS complex that conforms to both an input scalar field and an additional, prior segmentation of the domain. The segmentation constrains the MS complex computation guaranteeing that boundaries in the segmentation are captured as separatrices of the MS complex. We demonstrate the utility and versatility of our approach with two applications. First, we use streamline integration to determine numerically computed basins/mountains and use the resulting segmentation as an input to our algorithm. This strategy enables the incorporation of prior flow path knowledge, effectively resulting in an MS complex that is as geometrically accurate as the employed numerical integration. Our second use case is motivated by the observation that often the data itself does not explicitly contain features known to be present by a domain expert. We introduce edit operations for MS complexes so that a user can directly modify their features while maintaining all the advantages of a robust topology-based representation

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

The persistent cosmic web and its filamentary structure II: Illustrations

The recently introduced discrete persistent structure extractor (DisPerSE, Soubie 2010, paper I) is implemented on realistic 3D cosmological simulations and observed redshift catalogues (SDSS); it is found that DisPerSE traces equally well the observed filaments, walls, and voids in both cases. In either setting, filaments are shown to connect onto halos, outskirt walls, which circumvent voids. Indeed this algorithm operates directly on the particles without assuming anything about the distribution, and yields a natural (topologically motivated) self-consistent criterion for selecting the significance level of the identified structures. It is shown that this extraction is possible even for very sparsely sampled point processes, as a function of the persistence ratio. Hence astrophysicists should be in a position to trace and measure precisely the filaments, walls and voids from such samples and assess the confidence of the post-processed sets as a function of this threshold, which can be expressed relative to the expected amplitude of shot noise. In a cosmic framework, this criterion is comparable to friend of friend for the identifications of peaks, while it also identifies the connected filaments and walls, and quantitatively recovers the full set of topological invariants (Betti numbers) {\sl directly from the particles} as a function of the persistence threshold. This criterion is found to be sufficient even if one particle out of two is noise, when the persistence ratio is set to 3-sigma or more. The algorithm is also implemented on the SDSS catalogue and used to locat interesting configurations of the filamentary structure. In this context we carried the identification of an ``optically faint'' cluster at the intersection of filaments through the recent observation of its X-ray counterpart by SUZAKU. The corresponding filament catalogue will be made available online.Comment: A higher resolution version is available at http://www.iap.fr/users/sousbie together with complementary material (movie and data). Submitted to MNRA