The Morse‐Smale complex is a topological structure that captures the behavior of the gradient of a scalar function on a manifold. This paper discusses scalable techniques to compute the Morse‐Smale complex of scalar functions defined on large three‐dimensional structured grids. Computing the Morse‐Smale complex of three‐dimensional domains is challenging as compared to two‐dimensional domains because of the non‐trivial structure introduced by the two types of saddle criticalities. We present a parallel shared‐memory algorithm to compute the Morse‐Smale complex based on Forman's discrete Morse theory. The algorithm achieves scalability via synergistic use of the CPU and the GPU. We first prove that the discrete gradient on the domain can be computed independently for each cell and hence can be implemented on the GPU. Second, we describe a two‐step graph traversal algorithm to compute the 1‐saddle‐2‐saddle connections efficiently and in parallel on the CPU. Simultaneously, the extremasaddle connections are computed using a tree traversal algorithm on the GPU.

Abstract-The large-scale structure of the universe is comprised of virialized blob-like clusters, linear filaments, sheet-like walls and huge near empty three-dimensional voids. Characterizing the large scale universe is essential to our understanding of the formation and evolution of galaxies. The density range of clusters, walls and voids are relatively well separated, when compared to filaments, which span a relatively larger range. The large scale filamentary network thus forms an intricate part of the cosmic web. In this paper, we describe Felix, a topology based framework for visual exploration of filaments in the cosmic web. The filamentary structure is represented by the ascending manifold geometry of the 2-saddles in the Morse-Smale complex of the density field. We generate a hierarchy of Morse-Smale complexes and query for filaments based on the density ranges at the end points of the filaments. The query is processed efficiently over the entire hierarchical Morse-Smale complex, allowing for interactive visualization. We apply Felix to computer simulations based on the heuristic Voronoi kinematic model and the standard ΛCDM cosmology, and demonstrate its usefulness through two case studies. First, we extract cosmic filaments within and across cluster like regions in Voronoi kinematic simulation datasets. We demonstrate that we produce similar results to existing structure finders. Second, we extract different classes of filaments based on their density characteristics from the ΛCDM simulation datasets. Filaments that form the spine of the cosmic web, which exist in high density regions in the current epoch, are isolated using Felix. Also, filaments present in void-like regions are isolated and visualized. These filamentary structures are often over shadowed by higher density range filaments and are not easily characterizable and extractable using other filament extraction methodologies.Index Terms-Morse-Smale complexes, tessellations, cosmology theory, cosmic web, large-scale structure of the universe.

Abstract-The Morse-Smale complex is a useful topological data structure for the analysis and visualization of scalar data. This paper describes an algorithm that processes all mesh elements of the domain in parallel to compute the Morse-Smale complex of large twodimensional data sets at interactive speeds. We employ a reformulation of the Morse-Smale complex using Forman's Discrete Morse Theory and achieve scalability by computing the discrete gradient using local accesses only. We also introduce a novel approach to merge gradient paths that ensures accurate geometry of the computed complex. We demonstrate that our algorithm performs well on both multicore environments and on massively parallel architectures such as the GPU.

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Background: Aligning similar molecular structures is an important step in the process of bio-molecular structure and function analysis. Molecular surfaces are simple representations of molecular structure that are easily constructed from various forms of molecular data such as 3D atomic coordinates (PDB) and Electron Microscopy (EM) data. Methods: We present a Multi-Scale Morse-Smale Molecular-Surface Alignment tool, MS3ALIGN, which aligns molecular surfaces based on significant protrusions on the molecular surface. The input is a pair of molecular surfaces represented as triangle meshes. A key advantage of MS3ALIGN is computational efficiency that is achieved because it processes only a few carefully chosen protrusions on the molecular surface. Furthermore, the alignments are partial in nature and therefore allows for inexact surfaces to be aligned. Results:The method is evaluated in four settings. First, we establish performance using known alignments with varying overlap and noise values. Second, we compare the method with SurfComp, an existing surface alignment method. We show that we are able to determine alignments reported by SurfComp, as well as report relevant alignments not found by SurfComp. Third, we validate the ability of MS3ALIGN to determine alignments in the case of structurally dissimilar binding sites. Fourth, we demonstrate the ability of MS3ALIGN to align iso-surfaces derived from cryo-electron microscopy scans. Conclusions:We have presented an algorithm that aligns Molecular Surfaces based on the topology of surface curvature. A webserver and standalone software implementation of the algorithm available at http://vgl.serc.iisc.ernet.in/ms3align.

The shape of biomolecules, such as proteins, may be represented using different representations like space-fill, ballstick, backbone, or secondary structures. The secondary structure of proteins, comprising of sheet-like, helix-like and loops structures, represent a higher level abstraction of its structure. With ever increasing sizes of protein structure data produced by high resolution x-ray crystallography and cryo-electron microscopy, biologists often rely on visualizations to better understand the overall structure of proteins. In this paper, we present a unified framework for accelerating the rendering of various representations of these structure using GPUs. The framework first produces "impostor primitives", which are simple linear element approximations of quadric objects, such as spheres, cylinders, and helices. Next, the rasterizations of the impostors are corrected to produce pixel-precise renderings of the quadric objects. We incorporate this framework into a bio-molecular visualization tool proteinvis to demonstrate quantitative and qualitative performance gains over earlier approaches for rendering various representations of proteins.