Terrain topology plays an important role in simulations and segmentation. A widely used terrain representation is the Triangulated Irregular Network (TIN). However, topological analysis on TINs is challenging due to high time and memory requirements, which limit the size of the terrain that can be analyzed. We address this problem by proposing a novel framework for efficient and scalable analysis of large TINs based on Morse theory using Apache Spark. The proposed framework, named Morse–Spark, is based on a data structure for encoding the minimal information of a triangle mesh. Morse–Spark provides optimized methods for the local extraction of many connectivity relations, beginning with the global retrieval of the Vertex–Triangle relation. These relations serve as the foundation for computing terrain morphology through integrated, scalable algorithms. To evaluate the effectiveness and scalability of such a framework, we compare Morse–Spark against a vanilla Spark implementation, a MPI-supported Topology Toolkit (MPI-TTK) implementation, and three well-established software libraries for the topological analysis of TINs. Our experimental evaluation with real-world TINs shows that Morse–Spark can effectively handle datasets around 13 times larger than those processed by state-of-the-art tools for distributed computing (e.g. MPI-TTK).

Unstructured meshes present challenges in scientific data analysis due to irregular distribution and complex connectivity. Computing and storing connectivity information is a major bottleneck for visualization algorithms, affecting both time and memory performance. Recent task-parallel data structures address this by precomputing connectivity information at runtime while the analysis algorithm executes, effectively hiding computation costs and improving performance. However, existing approaches are CPU-bound, forcing the data structure and analysis algorithm to compete for the same computational resources, limiting potential speedups. To overcome this limitation, we introduce a novel task-parallel approach optimized for heterogeneous CPU-GPU systems. Specifically, we offload the computation of mesh connectivity information to GPU threads, enabling CPU threads to focus on executing the visualization algorithm. Following this paradigm, we propose GPU-Aided Localized data structurE (GALE), the first open-source CUDA-based data structure designed for heterogeneous task parallelism. Experiments on two 20-core CPUs and an NVIDIA V100 GPU show that GALE achieves up to $2.7\times$ speedup over state-of-the-art localized data structures while maintaining memory efficiency.

We consider 2D scalar fields sampled on a regular grid. When the gradient is low relative to the resolution of the dataset’s range, the signal may contain flat spots : connected areas where all points share the same value. Flat spots hinder certain analyses, such as topological characterization or drainage network computations. We present an algorithm to determine a symbolic slope inside flat spots and consistently place a minimal set of critical points, in a way that is less biased than state-of-the-art methods. We present experimental results on both synthetic and real data, demonstrating how our method provides a more plausible positioning of critical points and a better recovery of the Morse–Smale complex.

Unstructured meshes are characterized by data points irregularly distributed in the Euclidian space. Due to the irregular nature of these data, computing connectivity information between the mesh elements requires much more time and memory than on uniformly distributed data. To lower storage costs, dynamic data structures have been proposed. These data structures compute connectivity information on the fly and discard them when no longer needed. However, on-the-fly computation slows down algorithms and results in a negative impact on the time performance. To address this issue, we propose a new task-parallel approach to proactively compute mesh connectivity. Unlike previous approaches implementing data-parallel models, where all threads run the same type of instructions, our task-parallel approach allows threads to run different functions. Specifically, some threads run the algorithm of choice while other threads compute connectivity information before they are actually needed. The approach was implemented in the new Accelerated Clustered TOPOlogical (ACTOPO) data structure, which can support any processing algorithm requiring mesh connectivity information. Our experiments show that ACTOPO combines the benefits of state-of-the-art memory-efficient (TTK CompactTriangulation) and time-efficient (TTK ExplicitTriangulation) topological data structures. It occupies a similar amount of memory as TTK CompactTriangulation while providing up to 5x speedup. Moreover, it achieves comparable time performance as TTK ExplicitTriangulation while using only half of the memory space.

Unstructured data are collections of points with irregular topology, often represented through simplicial meshes, such as triangle and tetrahedral meshes. Whenever possible such representations are avoided in visualization since they are computationally demanding if compared with regular grids. In this work, we aim at simplifying the encoding and processing of simplicial meshes. The article proposes TopoCluster, a new localized data structure for tetrahedral meshes. TopoCluster provides efficient computation of the connectivity of the mesh elements with a low memory footprint. The key idea of TopoCluster is to subdivide the simplicial mesh into clusters. Then, the connectivity information is computed locally for each cluster and discarded when it is no longer needed. We define two instances of TopoCluster. The first instance prioritizes time efficiency and provides only a modest savings in memory, while the second instance drastically reduces memory consumption up to an order of magnitude with respect to comparable data structures. Thanks to the simple interface provided by TopoCluster, we have been able to integrate both data structures into the existing Topological Toolkit (TTK) framework. As a result, users can run any plugin of TTK using TopoCluster without changing a single line of code.

Persistent homology is a fundamental tool in topological data analysis used for the most diverse applications. Information captured by persistent homology is commonly visualized using scatter plots representations. Despite being widely adopted, such a visualization technique limits user understanding and is prone to misinterpretation. This article proposes a new approach for the efficient computation of persistence cycles, a geometric representation of the features captured by persistent homology. We illustrate the importance of rendering persistence cycles when analyzing scalar fields, and we discuss the advantages that our approach provides compared to other techniques in topology-based visualization. We provide an efficient implementation of our approach based on discrete Morse theory, as a new module for the Topology Toolkit. We show that our implementation has comparable performance with respect to state-of-the-art toolboxes while providing a better framework for visually analyzing persistent homology information.

A common first step in the terrain processing pipeline of large Triangulated Irregular Networks (TINs) is simplifying the TIN to make it manageable for further processing. The major problem with TIN simplification algorithms is that they create or remove critical points in an uncontrolled way. Topology-aware operators have been defined to solve this issue by coarsening a TIN without affecting the topology of its underlying terrain, i.e., without modifying critical simplices describing pits, saddles, peaks, and their connectivity. While effective, existing algorithms are sequential in nature and are not scalable enough to perform well with large terrains on multicore systems. Here, we consider the problem of topology-aware simplification of very large meshes. We define a topology-aware simplification algorithm on a compact and distributed data structure for triangle meshes, namely the Terrain trees. Terrain trees reduce both the memory and time requirements of the simplification procedure by adopting a batched processing strategy of the mesh elements. Furthermore, we define a new parallel topology-aware simplification algorithm that takes advantage of the spatial domain decomposition at the basis of Terrain trees. Scalability and efficiency are experimentally demonstrated on real-world TINs originated from topographic and bathymetric LiDAR data. Our experiments show that topology-aware simplification on Terrain trees uses 40% less memory and half the time than the same approach implemented on the most compact and efficient connectivity-based data structure for TINs. Beyond that, our parallel algorithm on the Terrain trees reaches a 12x speedup when using 20 threads.

Light Detection and Ranging (LiDAR) sensors generate dense point clouds that can be used to map forest structures at a high spatial resolution level. In this work, we consider the problem of identifying individual trees in a LiDAR point cloud. Existing techniques generally require intense parameter tuning and user interactions. Our goal is defining an automatic approach capable of providing robust results with minimal user interactions.

Abstract Simplicial complexes are widely used to discretize shapes. In low dimensions, a 3D shape is represented by discretizing its boundary surface, encoded as a triangle mesh, or by discretizing the enclosed volume, encoded as a tetrahedral mesh. High‐dimensional simplicial complexes have recently found their application in topological data analysis. Topological data analysis aims at studying a point cloud P, possibly embedded in a high‐dimensional metric space, by investigating the topological characteristics of the simplicial complexes built on P. Analysing such complexes is not feasible due to their size and dimensions. To this aim, the idea of simplifying a complex while preserving its topological features has been proposed in the literature. Here, we consider the problem of efficiently simplifying simplicial complexes in arbitrary dimensions. We provide a new definition for the edge contraction operator, based on a top‐based data structure, with the objective of preserving structural aspects of a simplicial shape (i.e., its homology), and a new algorithm for verifying the link condition on a top‐based representation. We implement the simplification algorithm obtained by coupling the new edge contraction and the link condition on a specific top‐based data structure, that we use to demonstrate the scalability of our approach.

We consider the problem of efficiently computing a discrete Morse complex on simplicial complexes of arbitrary dimension and very large size. Based on a common graph-based formalism, we analyze existing data structures for simplicial complexes, and we define an efficient encoding for the discrete Morse gradient on the most compact of such representations. We theoretically compare methods based on reductions and coreductions for computing a discrete Morse gradient, proving that the combination of reductions and coreductions produces new mutually equivalent approaches. We design and implement a new algorithm for computing a discrete Morse complex on simplicial complexes. We show that our approach scales very well with the size and the dimension of the simplicial complex also through comparisons with the only existing public-domain algorithm for discrete Morse complex computation. We discuss applications to the computation of multi-parameter persistent homology and of extremum graphs for visualization of time-varying 3D scalar fields.

Terrain trees are a new in-core family of spatial indexes for the representation and analysis of Triangulated Irregular Networks (TINs). Terrain trees combine a minimal encoding of the connectivity of the underlying triangle mesh with a hierarchical spatial index, implicitly representing the topological relations among vertices, edges and triangles. Topological relations are extracted locally within each leaf block of the hierarchal index at runtime, based on specific application needs. We have developed a tool based on Terrain trees for terrain analysis, which includes state-of-the-art estimators for slope and curvature, and for the extraction of critical points, as well as algorithms for topology-based terrain segmentation and multifield terrain analysis. By working on TINs generated from very large LiDAR (Light, Detection and Ranging) data sets, we demonstrate the effectiveness and scalability of the Terrain trees against a state-of-the-art compact data structures.

Abstract This paper presents the state of the art in the area of topology‐based visualization. It describes the process and results of an extensive annotation for generating a definition and terminology for the field. The terminology enabled a typology for topological models which is used to organize research results and the state of the art. Our report discusses relations among topological models and for each model describes research results for the computation, simplification, visualization, and application. The paper identifies themes common to subfields, current frontiers, and unexplored territory in this research area.

Abstract Morse theory offers a natural and mathematically‐sound tool for shape analysis and understanding. It allows studying the behavior of a scalar function defined on a manifold. Starting from a Morse function, we can decompose the domain of the function into meaningful regions associated with the critical points of the function. Such decompositions, called Morse complexes, provide a segmentation of a shape and are extensively used in terrain modeling and in scientific visualization. Discrete Morse theory, a combinatorial counterpart of smooth Morse theory defined over cell complexes, provides an excellent basis for computing Morse complexes in a robust and efficient way. Moreover, since a discrete Morse complex computed over a given complex has the same homology as the original one, but fewer cells, discrete Morse theory is a fundamental tool for efficiently detecting holes in shapes through homology and persistent homology. In this survey, we review, classify and analyze algorithms for computing and simplifying Morse complexes in the context of such applications with an emphasis on discrete Morse theory and on algorithms based on it.

We consider the problem of efficient computing and simplifying Morse complexes on a Triangulated Irregular Network (TIN) based on discrete Morse theory. We develop a compact encoding for the discrete Morse gradient field, defined by the terrain elevation, by attaching it to the triangles of the TIN. This encoding is suitable to be combined with any TIN data structure storing just its vertices and triangles. We show how to compute such gradient field from the elevation values given at the TIN vertices, and how to simplify it effectively in order to reduce the number of critical elements. We demonstrate the effectiveness and scalability of our approach over large terrains by developing algorithms for extracting the cells of the Morse complexes as well as the graph joining the critical elements from the discrete gradient field. We compare implementations of our approach on a widely-used and compact adjacency-based topological data structure for a TIN and on a compact spatio-topological data structure that we have recently developed, the PR-star quadtree.

We consider the problem of modeling a terrain from both a geometric and a morphological point of view for efficient and effective terrain analysis on large data sets. We devise and implement a simplification hierarchy for a triangulated terrain, where the terrain is represented as a triangle mesh and its morphology is described by a discrete Morse gradient field defined on the basis on the elevation values given at the vertices of the mesh. The discrete Morse gradient is attached to the triangles, edges and vertices of the mesh. We define a new edge-contraction operator for the edges of the triangle mesh, which does not change the behavior of the gradient flow and does not create new critical points, and we apply it to the original full-resolution mesh in combination with a topological simplification operator which eliminates critical simplices in pair. We build the simplification hierarchy based on suitably combining such operators and we evaluate it experimentally.

We propose a new technique for eliminating flat edges from a Triangulated Irregular Network (TIN) in a morphologically consistent way. The algorithm is meant to be a preprocessing step for performing morphological computations on a terrain. Terrain morphology is rooted in Morse theory for smooth functions. Segmentation algorithms have been defined for TINs, mostly based on discrete versions of Morse theory, and under the assumption that the terrain model does not include flat edges. On the other hand, flat edges often occur in real data, and thus either they are eliminated through data perturbation, or the segmentation algorithms must be able to deal with them. In both cases, the resulting Morse segmentations are highly affected by the presence of flat edges. The new technique we propose provides a better solution, as it preserves the set of maxima and minima of the original terrain, and improves consistency among the terrain decompositions produced by different segmentation algorithms.

Abstract We consider the problem of computing discrete Morse and Morse‐Smale complexes on an unstructured tetrahedral mesh discretizing the domain of a 3D scalar field. We use a duality argument to define the cells of the descending Morse complex in terms of the supplied (primal) tetrahedral mesh and those of the ascending complex in terms of its dual mesh. The Morse‐Smale complex is then described combinatorially as collections of cells from the intersection of the primal and dual meshes. We introduce a simple compact encoding for discrete vector fields attached to the mesh tetrahedra that is suitable for combination with any topological data structure encoding just the vertices and tetrahedra of the mesh. We demonstrate the effectiveness and scalability of our approach over large unstructured tetrahedral meshes by developing algorithms for computing the discrete gradient field and for extracting the cells of the Morse and Morse‐Smale complexes. We compare implementations of our approach on an adjacency‐based topological data structure and on the PR‐star octree, a compact spatio‐topological data structure.

We describe a dual graph-based representation for the ascending and descending Morse complexes of a scalar field, and a compact and dimension-independent data structure based on it, which assumes a discrete representation of the field as a simplicial mesh. We present atomic dimension-independent simplification operators on the graph-based representation. Based on such operators, we have developed a simplification algorithm, which allows generalization of the ascending and descending Morse complexes at different levels of resolution. We show here the results of our implementation, discussing the computation times and the size of the resulting simplified graphs, also in comparison with the size of the original full-resolution graph.