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/10243 node grid on a laptop computer with 2 Gb memory.

/pdf/a-practical-approach-to-morse-smale-complex-computation-1j6wmk9h2i.pdf

A Practical Approach to Morse-Smale Complex Computation: Scalability and Generality

In this paper, we present a topological approach for simplifying continuous functions defined on volumetric domains. We introduce two atomic operations that remove pairs of critical points of the function and design a combinatorial algorithm that simplifies the Morse-Smale complex by repeated application of these operations. The Morse-Smale complex is a topological data structure that provides a compact representation of gradient flow between critical points of a function. Critical points paired by the Morse-Smale complex identify topological features and their importance. The simplification procedure leaves important critical points untouched, and is therefore useful for extracting desirable features. We also present a visualization of the simplified topology.

/pdf/topology-based-simplification-for-feature-extraction-from-3d-2rsofwzf9y.pdf

Topology-based simplification for feature extraction from 3D scalar fields

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.

/pdf/efficient-computation-of-morse-smale-complexes-for-three-18ea4r3r99.pdf

Efficient Computation of Morse-Smale Complexes for Three-dimensional Scalar Functions

Almost all mesh processing procedures cause some more or less visible changes in the appearance of objects represented by polygonal meshes In many cases, such as mesh watermarking, simplification or lossy compression, the objective is to make the change in appearance negligible, or as small as possible, given some other constraints Measuring the amount of distortion requires taking into account the final purpose of the data In many applications, the final consumer of the data is a human observer, and therefore the perceptibility of the introduced appearance change by a human observer should be the criterion that is taken into account when designing and configuring the processing algorithms In this review, we discuss the existing comparison metrics for static and dynamic (animated) triangle meshes We describe the concepts used in perception-oriented metrics used for 2D image comparison, and we show how these concepts are employed in existing 3D mesh metrics We describe the character of subjective data used for evaluation of mesh metrics and provide comparison results identifying the advantages and drawbacks of each method Finally, we also discuss employing the perception-correlated metrics in perception-oriented mesh processing algorithms

/pdf/perceptual-metrics-for-static-and-dynamic-triangle-meshes-2p71ikqyee.pdf

Perceptual metrics for static and dynamic triangle meshes

We propose a robust 2D shape reconstruction and simplification algorithm which takes as input a defect-laden point set with noise and outliers. We introduce an optimal-transport driven approach where the input point set, considered as a sum of Dirac measures, is approximated by a simplicial complex considered as a sum of uniform measures on 0- and 1-simplices. A fine-to-coarse scheme is devised to construct the resulting simplicial complex through greedy decimation of a Delaunay triangulation of the input point set. Our method performs well on a variety of examples ranging from line drawings to grayscale images, with or without noise, features, and boundaries.

/pdf/an-optimal-transport-approach-to-robust-reconstruction-and-2sru3bhunc.pdf

An Optimal Transport Approach to Robust Reconstruction and Simplification of 2D Shapes

We present a novel generalization of the quadric error metric used in surface simplification that can be used for simplifying simplicial complexes of any type embedded in Euclidean spaces of any dimension. We demonstrate that our generalized simplification system can produce high quality approximations of plane and space curves, triangulated surfaces, tetrahedralized volume data, and simplicial complexes of mixed type. Our method is both efficient and easy to implement. It is capable of processing complexes of arbitrary topology, including nonmanifolds, and can preserve intricate boundaries.

/pdf/quadric-based-simplification-in-any-dimension-asu8pto6n0.pdf

Quadric-based simplification in any dimension

We propose a new algorithm for constructing finite-element meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Given a triangular mesh of some planar domain Ω and a target time value T, our method constructs a tetrahedral mesh of the spacetime domain Ω X [0,T] in constant running time per tetrahedron in R3 using an advancing front method. Elements are added to the evolving mesh in small patches by moving a vertex of the front forward in time. Spacetime discontinuous Galerkin methods allow the numerical solution within each patch to be computed as soon as the patch is created. Our algorithm employs new mechanisms for adaptively coarsening and refining the front in response to a posteriori error estimates returned by the numerical code. A change in the front induces a corresponding refinement or coarsening of future elements in the spacetime mesh. Our algorithm adapts the duration of each element to the local quality, feature size, and degree of refinement of the underlying space mesh. We directly exploit the ability of discontinuous Galerkin methods to accommodate discontinuities in the solution fields across element boundaries.

/pdf/spacetime-meshing-with-adaptive-refinement-and-coarsening-5ccvjkkjw5.pdf

Spacetime meshing with adaptive refinement and coarsening

Computational simulation of time-varying physical processes is of fundamental importance for many scientific and engineering applications. Most frequently, time-varying simulations are performed over multiple spatial grids at discrete points in time. In this paper, we investigate a new approach to time-varying simulation: spacetime discontinuous Galerkin finite element methods. The result of this simulation method is a simplicial tessellation of spacetime with per-element polynomial solutions for physical quantities such as strain, stress, and velocity. To provide accurate visualizations of the resulting solutions, we have developed a method for per-pixel evaluation of solution data on the GPU.We demonstrate the importance of per-pixel rendering versus simple linear interpolation for producing high quality visualizations. We also show that our system can accommodate reasonably large datasets - spacetime meshes containing up to 20 million tetrahedra are not uncommon in this domain.

http://vis.computer.org/vis2004/DVD/vis/papers/zhou.pdf

Pixel-Exact Rendering of Spacetime Finite Element Solutions

This paper addresses the computation of normalized solid angle measure of polyhedral cones. This is well understood in dimensions two and three. For higher dimensions, assuming that a positive-definite criterion is met, the measure can be computed via a multivariable hypergeometric series. We present two decompositions of full-dimensional simplicial cones into finite families of cones satisfying the positive-definite criterion, enabling the use of the hypergeometric series to compute the solid angle measure of any polyhedral cone. Additionally, our second decomposition method yields cones with a special tridiagonal structure, reducing the number of required coordinates for the hypergeometric series formula. Furthermore, we investigate the convergence of the hypergeometric series for this case. Our findings provide a powerful tool for computing solid angle measures in high-dimensional spaces.

Yuan Zhou

Papers

Quadric-based simplification in any dimension

Spacetime meshing with adaptive refinement and coarsening

Pixel-Exact Rendering of Spacetime Finite Element Solutions

Solid angle measure of polyhedral cones