Conforming Morse-Smale complexes

pre-printMorse-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

The parallel computation of morse-smale complexes

pre-printTopology-based techniques are useful for multi-scale exploration of the feature space of scalar-valued functions, such as those derived from the output of large-scale simulations. The Morse-Smale (MS) complex, in particular, allows robust identification of gradient-based features, and therefore is suitable for analysis tasks in a wide range of application domains. In this paper, we develop a two-stage algorithm to construct the Morse-Smale complex in parallel, the first stage independently computing local features per block and the second stage merging to resolve global features. Our implementation is based on MPI and a distributed-memory architecture. Through a set of scalability studies on the IBM Blue Gene/P supercomputer, we characterize the performance of the algorithm as block sizes, process counts, merging strategy, and levels of topological simplification are varied, for datasets that vary in feature composition and size. We conclude with a strong scaling study using scientific datasets computed by combustion and hydrodynamics simulations

Computing morse-smale complexes with accurate geometry

pre-printTopological techniques have proven highly successful in analyzing and visualizing scientific data. As a result, significant efforts have been made to compute structures like the Morse-Smale complex as robustly and efficiently as possible. However, the resulting algorithms, while topologically consistent, often produce incorrect connectivity as well as poor geometry. These problems may compromise or even invalidate any subsequent analysis. Moreover, such techniques may fail to improve even when the resolution of the domain mesh is increased, thus producing potentially incorrect results even for highly resolved functions. To address these problems we introduce two new algorithms: (i) a randomized algorithm to compute the discrete gradient of a scalar field that converges under refinement; and (ii) a deterministic variant which directly computes accurate geometry and thus correct connectivity of the MS complex. The first algorithm converges in the sense that on average it produces the correct result and its standard deviation approaches zero with increasing mesh resolution. The second algorithm uses two ordered traversals of the function to integrate the probabilities of the first to extract correct (near optimal) geometry and connectivity. We present an extensive empirical study using both synthetic and real-world data and demonstrates the advantages of our algorithms in comparison with several popular approaches

Hierarchical Morse-Smale Complex in 3D

This paper investigates the topology of piecewise linear scalar functions, such as the interpolation function over a tetrahedralized volumetric grid. An algorithm is presented that creates a hierarchical representation of a Morse-Smale complex based on persistence. We show the correctness of our approach and discuss the applications

Time- and Space-efficient Error Calculation for Multiresolution Direct Volume Rendering

Summary. Multiresolution data representations are crucial for viewing large volumetric datasets interactively. When data is too large to fit into texture memory, or into main memory, a “cut ” must be made through the multiresolution data hierarchy to attain a subset of the data that satisfies the memory requirements. Ideally, a subset is chosen such that the error made when visualizing the subset (compared to a visualization of the full data set) is smaller than that of any other subset of the same size. For real-time applications it is computationally too expensive to calculate the exact error during runtime. Futher, computing error in a pre-processing step is usually not practical due to a large number of possible different configurations each requiring its own error computation. For example, when coupling a multiresolution representation with a direct volume rendering technique, screen-space error depends on the transfer function and viewing direction, making impossible its pre-computation. We present an algorithm that stores an intermediate form of the error, which allows us to approximate screen-space error efficiently. The input for our algorithm is any spatially subdivided multiresolution representation of grid-aligned scalar or multi-variate volume data. We focus on octree- and wavelet-based multiresolution techniques. For each level in the multiresolution hierarchy, the algorithm estimates screen-space error “on the fly, ” with respect to the current transfer function and viewing direction. The error is approximated by means of a two-dimensional histogram of error pairs. We have extended previous methods by presenting an approach that balances computational and memory costs with approximation quality of the error estimate.

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