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).

Understanding the complex topography of polar sea ice is essential for assessing its impacts on sea ice processes. Recently, large volumes of high-resolution Light Detection and Ranging (LiDAR) altimeter data covering extensive polar regions became available. Current methods for extracting topographic features either rely on along-track elevation profiles limited to one dimension, as it is the case with ICESat-2 data, or convert point clouds into a raster grid. In our work, we model the ice surface as a Triangulated Irregular Network (TIN) connecting the original data points, and we focus on extracting pressure ridges from TINs. Pressure ridges are important factors, for instance, in sea ice navigability analysis and atmosphere-ice-momentum transfer models. We present a technique for extracting the structure of pressure ridges based on combinatorial topology, specifically discrete Morse theory. We show our results by using LiDAR point clouds collected from the OIB Airborne Topographic Mapper (ATM). The extracted ridges are compared with those obtained from one-dimensional profiles extracted from ICESat-2 data in the same region, showing that the proposed strategy provides information that cannot be derived from one-dimensional profiles alone.

The scale-space method is a well-established framework that constructs a hierarchical representation of an input signal and facilitates coarse-to-fine visual reasoning. Considering the terrain elevation function as the input signal, the scale-space method can identify and track significant topographic features across different scales. The number of scales a feature persists, called its life span, indicates the importance of that feature. In this way, important topographic features of a landscape can be selected, which are useful for many applications, including cartography, nautical charting, and land-use planning. The scale-space methods developed for terrain data use gridded Digital Elevation Models (DEMs) to represent the terrain. However, gridded DEMs lack the flexibility to adapt to the irregular distribution of input data and the varied topological complexity of different regions. Instead, Triangulated Irregular Networks (TINs) can be directly generated from irregularly distributed point clouds and accurately preserve important features. In this work, we introduce a novel scale-space analysis pipeline for TINs, addressing the multiple challenges in extending grid-based scale-space methods to TINs. Our pipeline can efficiently identify and track topologically important features on TINs. Moreover, it is capable of analyzing terrains with irregular boundaries, which poses challenges for grid-based methods. Comprehensive experiments show that, compared to grid-based methods, our TIN-based pipeline is more efficient, accurate, and has better resolution robustness.

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.