TopoMS: Comprehensive topological exploration for molecular and condensed-matter systems.

We introduce TopoMS, a computational tool enabling detailed topological analysis of molecular and condensed-matter systems, including the computation of atomic volumes and charges through the quantum theory of atoms in molecules, as well as the complete molecular graph. With roots in techniques from computational topology, and using a shared-memory parallel approach, TopoMS provides scalable, numerically robust, and topologically consistent analysis. TopoMS can be used as a command-line tool or with a GUI (graphical user interface), where the latter also enables an interactive exploration of the molecular graph. This paper presents algorithmic details of TopoMS and compares it with state-of-the-art tools: Bader charge analysis v1.0 (Arnaldsson et al., 01/11/17) and molecular graph extraction using Critic2 (Otero-de-la-Roza et al., Comput. Phys. Commun. 2014, 185, 1007). TopoMS not only combines the functionality of these individual codes but also demonstrates up to 4× performance gain on a standard laptop, faster convergence to fine-grid solution, robustness against lattice bias, and topological consistency. TopoMS is released publicly under BSD License. © 2018 Wiley Periodicals, Inc.

Exploring Classification of Topological Priors with Machine Learning for Feature Extraction.

In many scientific endeavors, increasingly abstract representations of data allow for new interpretive methodologies and conceptualization of phenomena. For example, moving from raw imaged pixels to segmented and reconstructed objects allows researchers new insights and means to direct their studies toward relevant areas. Thus, the development of new and improved methods for segmentation remains an active area of research. With advances in machine learning and neural networks, scientists have been focused on employing deep neural networks such as U-Net to obtain pixel-level segmentations, namely, defining associations between pixels and corresponding/referent objects and gathering those objects afterward. Topological analysis, such as the use of the Morse-Smale complex to encode regions of uniform gradient flow behavior, offers an alternative approach: first, create geometric priors, and then apply machine learning to classify. This approach is empirically motivated since phenomena of interest often appear as subsets of topological priors in many applications. Using topological elements not only reduces the learning space but also introduces the ability to use learnable geometries and connectivity to aid the classification of the segmentation target. In this paper, we describe an approach to creating learnable topological elements, explore the application of ML techniques to classification tasks in a number of areas, and demonstrate this approach as a viable alternative to pixel-level classification, with similar accuracy, improved execution time, and requiring marginal training data.

Towards replacing physical testing of granular materials with a Topology-based Model.

In the study of packed granular materials, the performance of a sample (e.g., the detonation of a high-energy explosive) often correlates to measurements of a fluid flowing through it. The "effective surface area," the surface area accessible to the airflow, is typically measured using a permeametry apparatus that relates the flow conductance to the permeable surface area via the Carman-Kozeny equation. This equation allows calculating the flow rate of a fluid flowing through the granules packed in the sample for a given pressure drop. However, Carman-Kozeny makes inherent assumptions about tunnel shapes and flow paths that may not accurately hold in situations where the particles possess a wide distribution in shapes, sizes, and aspect ratios, as is true with many powdered systems of technological and commercial interest. To address this challenge, we replicate these measurements virtually on micro-CT images of the powdered material, introducing a new Pore Network Model based on the skeleton of the Morse-Smale complex. Pores are identified as basins of the complex, their incidence encodes adjacency, and the conductivity of the capillary between them is computed from the cross-section at their interface. We build and solve a resistive network to compute an approximate laminar fluid flow through the pore structure. We provide two means of estimating flow-permeable surface area: (i) by direct computation of conductivity, and (ii) by identifying dead-ends in the flow coupled with isosurface extraction and the application of the Carman-Kozeny equation, with the aim of establishing consistency over a range of particle shapes, sizes, porosity levels, and void distribution patterns.

Improving the Usability of Virtual Reality Neuron Tracing with Topological Elements.

Researchers in the field of connectomics are working to reconstruct a map of neural connections in the brain in order to understand at a fundamental level how the brain processes information. Constructing this wiring diagram is done by tracing neurons through high-resolution image stacks acquired with fluorescence microscopy imaging techniques. While a large number of automatic tracing algorithms have been proposed, these frequently rely on local features in the data and fail on noisy data or ambiguous cases, requiring time-consuming manual correction. As a result, manual and semi-automatic tracing methods remain the state-of-the-art for creating accurate neuron reconstructions. We propose a new semi-automatic method that uses topological features to guide users in tracing neurons and integrate this method within a virtual reality (VR) framework previously used for manual tracing. Our approach augments both visualization and interaction with topological elements, allowing rapid understanding and tracing of complex morphologies. In our pilot study, neuroscientists demonstrated a strong preference for using our tool over prior approaches, reported less fatigue during tracing, and commended the ability to better understand possible paths and alternatives. Quantitative evaluation of the traces reveals that users' tracing speed increased, while retaining similar accuracy compared to a fully manual approach.

Toward Localized Topological Data Structures: Querying the Forest for the Tree.

Topological approaches to data analysis can answer complex questions about the number, connectivity, and scale of intrinsic features in scalar data. However, the global nature of many topological structures makes their computation challenging at scale, and thus often limits the size of data that can be processed. One key quality to achieving scalability and performance on modern architectures is data locality, i.e., a process operates on data that resides in a nearby memory system, avoiding frequent jumps in data access patterns. From this perspective, topological computations are particularly challenging because the implied data structures represent features that can span the entire data set, often requiring a global traversal phase that limits their scalability. Traditionally, expensive preprocessing is considered an acceptable trade-off as it accelerates all subsequent queries. Most published use cases, however, explore only a fraction of all possible queries, most often those returning small, local features. In these cases, much of the global information is not utilized, yet computing it dominates the overall response time. We address this challenge for merge trees, one of the most commonly used topological structures. In particular, we propose an alternative representation, the merge forest, a collection of local trees corresponding to regions in a domain decomposition. Local trees are connected by a bridge set that allows us to recover any necessary global information at query time. The resulting system couples (i) a preprocessing that scales linearly in practice with (ii) fast runtime queries that provide the same functionality as traditional queries of a global merge tree. We test the scalability of our approach on a shared-memory parallel computer and demonstrate how data structure locality enables the analysis of large data with an order of magnitude performance improvement over the status quo. Furthermore, a merge forest reduces the memory overhead compared to a global merge tree and enables the processing of data sets that are an order of magnitude larger than possible with previous algorithms.

High-throughput feature extraction for measuring attributes of deforming open-cell foams.

Metallic open-cell foams are promising structural materials with applications in multifunctional systems such as biomedical implants, energy absorbers in impact, noise mitigation, and batteries. There is a high demand for means to understand and correlate the design space of material performance metrics to the material structure in terms of attributes such as density, ligament and node properties, void sizes, and alignments. Currently, X-ray Computed Tomography (CT) scans of these materials are segmented either manually or with skeletonization approaches that may not accurately model the variety of shapes present in nodes and ligaments, especially irregularities that arise from manufacturing, image artifacts, or deterioration due to compression. In this paper, we present a new workflow for analysis of open-cell foams that combines a new density measurement to identify nodal structures, and topological approaches to identify ligament structures between them. Additionally, we provide automated measurement of foam properties. We demonstrate stable extraction of features and time-tracking in an image sequence of a foam being compressed. Our approach allows researchers to study larger and more complex foams than could previously be segmented only manually, and enables the high-throughput analysis needed to predict future foam performance.

Loop surgery for volumetric meshes: Reeb graphs reduced to contour trees.

This paper introduces an efficient algorithm for computing the Reeb graph of a scalar function f defined on a volumetric mesh M in R3. We introduce a procedure called "loop surgery" that transforms M into a mesh M' by a sequence of cuts and guarantees the Reeb graph of f(M') to be loop free. Therefore, loop surgery reduces Reeb graph computation to the simpler problem of computing a contour tree, for which well-known algorithms exist that are theoretically efficient (O(n log n)) and fast in practice. Inverse cuts reconstruct the loops removed at the beginning. The time complexity of our algorithm is that of a contour tree computation plus a loop surgery overhead, which depends on the number of handles of the mesh. Our systematic experiments confirm that for real-life data, this overhead is comparable to the computation of the contour tree, demonstrating virtually linear scalability on meshes ranging from 70 thousand to 3.5 million tetrahedra. Performance numbers show that our algorithm, although restricted to volumetric data, has an average speedup factor of 6,500 over the previous fastest techniques, handling larger and more complex data-sets.We demonstrate the versatility of our approach by extending fast topologically clean iso surface extraction to non simply-connected domains. We apply this technique in the context of pressure analysis for mechanical design. In this case, our technique produces results in matter of seconds even for the largest meshes. For the same models, previous Reeb graph techniques do not produce a result.

Shared-Memory Parallel Computation of Morse-Smale Complexes with Improved Accuracy.

Topological techniques have proven to be a powerful tool in the analysis and visualization of large-scale scientific data. In particular, the Morse-Smale complex and its various components provide a rich framework for robust feature definition and computation. Consequently, there now exist a number of approaches to compute Morse-Smale complexes for large-scale data in parallel. However, existing techniques are based on discrete concepts which produce the correct topological structure but are known to introduce grid artifacts in the resulting geometry. Here, we present a new approach that combines parallel streamline computation with combinatorial methods to construct a high-quality discrete Morse-Smale complex. In addition to being invariant to the orientation of the underlying grid, this algorithm allows users to selectively build a subset of features using high-quality geometry. In particular, a user may specifically select which ascending/descending manifolds are reconstructed with improved accuracy, focusing computational effort where it matters for subsequent analysis. This approach computes Morse-Smale complexes for larger data than previously feasible with significant speedups. We demonstrate and validate our approach using several examples from a variety of different scientific domains, and evaluate the performance of our method.

A practical approach to Morse-Smale complex computation: scalability and generality.

The Morse-Smale (MS) complex has proven to be a useful tool in extracting and visualizing features from scalar-valued data. However, efficient computation of the MS complex for large scale data remains a challenging problem. We describe a new algorithm and easily extensible framework for computing MS complexes for large scale data of any dimension where scalar values are given at the vertices of a closure-finite and weak topology (CW) complex, therefore enabling computation on a wide variety of meshes such as regular grids, simplicial meshes, and adaptive multiresolution (AMR) meshes. A new divide-and-conquer strategy allows for memory-efficient computation of the MS complex and simplification on-the-fly to control the size of the output. In addition to being able to handle various data formats, the framework supports implementation-specific optimizations, for example, for regular data. We present the complete characterization of critical point cancellations in all dimensions. This technique enables the topology based analysis of large data on off-the-shelf computers. In particular we demonstrate the first full computation of the MS complex for a 1 billion/1024(3) node grid on a laptop computer with 2Gb memory.

Efficient computation of Morse-Smale complexes for three-dimensional scalar functions.

The Morse-Smale complex is an efficient representation of the gradient behavior of a scalar function, and critical points paired by the complex identify topological features and their importance. We present an algorithm that constructs the Morse-Smale complex in a series of sweeps through the data, identifying various components of the complex in a consistent manner. All components of the complex, both geometric and topological, are computed, providing a complete decomposition of the domain. Efficiency is maintained by representing the geometry of the complex in terms of point sets.

Topologically clean distance fields.

Analysis of the results obtained from material simulations is important in the physical sciences. Our research was motivated by the need to investigate the properties of a simulated porous solid as it is hit by a projectile. This paper describes two techniques for the generation of distance fields containing a minimal number of topological features, and we use them to identify features of the material. We focus on distance fields defined on a volumetric domain considering the distance to a given surface embedded within the domain. Topological features of the field are characterized by its critical points. Our first method begins with a distance field that is computed using a standard approach, and simplifies this field using ideas from Morse theory. We present a procedure for identifying and extracting a feature set through analysis of the MS complex, and apply it to find the invariants in the clean distance field. Our second method proceeds by advancing a front, beginning at the surface, and locally controlling the creation of new critical points. We demonstrate the value of topologically clean distance fields for the analysis of filament structures in porous solids. Our methods produce a curved skeleton representation of the filaments that helps material scientists to perform a detailed qualitative and quantitative analysis of pores, and hence infer important material properties. Furthermore, we provide a set of criteria for finding the "difference" between two skeletal structures, and use this to examine how the structure of the porous solid changes over several timesteps in the simulation of the particle impact.

ISAVS: Interactive Scalable Analysis and Visualization System.

Modern science is inundated with ever increasing data sizes as computational capabilities and image acquisition techniques continue to improve. For example, simulations are tackling ever larger domains with higher fidelity, and high-throughput microscopy techniques generate larger data that are fundamental to gather biologically and medically relevant insights. As the image sizes exceed memory, and even sometimes local disk space, each step in a scientific workflow is impacted. Current software solutions enable data exploration with limited interactivity for visualization and analytic tasks. Furthermore analysis on HPC systems often require complex hand-written parallel implementations of algorithms that suffer from poor portability and maintainability. We present a software infrastructure that simplifies end-to-end visualization and analysis of massive data. First, a hierarchical streaming data access layer enables interactive exploration of remote data, with fast data fetching to test analytics on subsets of the data. Second, a library simplifies the process of developing new analytics algorithms, allowing users to rapidly prototype new approaches and deploy them in an HPC setting. Third, a scalable runtime system automates mapping analysis algorithms to whatever computational hardware is available, reducing the complexity of developing scaling algorithms. We demonstrate the usability and performance of our system using a use case from neuroscience: filtering, registration, and visualization of tera-scale microscopy data. We evaluate the performance of our system using a leadership-class supercomputer, Shaheen II.

A topological approach to simplification of three-dimensional scalar functions.

This paper describes an efficient combinatorial method for simplification of topological features in a 3D scalar function. The Morse-Smale complex, which provides a succinct representation of a function's associated gradient flow field, is used to identify topological features and their significance. The simplification process, guided by the Morse-Smale complex, proceeds by repeatedly applying two atomic operations that each remove a pair of critical points from the complex. Efficient storage of the complex results in execution of these atomic operations at interactive rates. Visualization of the simplified complex shows that the simplification preserves significant topological features while removing small features and noise.

Interstitial and Interlayer Ion Diffusion Geometry Extraction in Graphitic Nanosphere Battery Materials.

Large-scale molecular dynamics (MD) simulations are commonly used for simulating the synthesis and ion diffusion of battery materials. A good battery anode material is determined by its capacity to store ion or other diffusers. However, modeling of ion diffusion dynamics and transport properties at large length and long time scales would be impossible with current MD codes. To analyze the fundamental properties of these materials, therefore, we turn to geometric and topological analysis of their structure. In this paper, we apply a novel technique inspired by discrete Morse theory to the Delaunay triangulation of the simulated geometry of a thermally annealed carbon nanosphere. We utilize our computed structures to drive further geometric analysis to extract the interstitial diffusion structure as a single mesh. Our results provide a new approach to analyze the geometry of the simulated carbon nanosphere, and new insights into the role of carbon defect size and distribution in determining the charge capacity and charge dynamics of these carbon based battery materials.

Conforming Morse-Smale Complexes.

Morse-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.

Direct Feature Visualization Using Morse-Smale Complexes.

In this paper, we characterize the range of features that can be extracted from an Morse-Smale complex and describe a unified query language to extract them. We provide a visual dictionary to guide users when defining features in terms of these queries. We demonstrate our topology-rich visualization pipeline in a tool that interactively queries the MS complex to extract features at multiple resolutions, assigns rendering attributes, and combines traditional volume visualization with the extracted features. The flexibility and power of this approach is illustrated with examples showing novel features.