Ajith Mascarenhas

Six degree-of-freedom haptic display of polygonal models

We present an algorithm for haptic display of moderately complex polygonal models with a six degree of freedom (DOF) force feedback device. We make use of incremental algorithms for contact determination between convex primitives. The resulting contact information is used for calculating the restoring forces and torques and thereby used to generate a sense of virtual touch. To speed up the computation, our approach exploits a combination of geometric locality, temporal coherence, and predictive methods to compute object-object contacts at kHz rates. The algorithm has been implemented and interfaced with a 6-DOF PHANToM Premium 1.5. We demonstrate its performance on force display of the mechanical interaction between moderately complex geometric structures that can be decomposed into convex primitives.

Understanding the Structure of the Turbulent Mixing Layer in Hydrodynamic Instabilities

When a heavy fluid is placed above a light fluid, tiny vertical perturbations in the interface create a characteristic structure of rising bubbles and falling spikes known as Rayleigh-Taylor instability. Rayleigh-Taylor instabilities have received much attention over the past half-century because of their importance in understanding many natural and man-made phenomena, ranging from the rate of formation of heavy elements in supernovae to the design of capsules for Inertial Confinement Fusion. We present a new approach to analyze Rayleigh-Taylor instabilities in which we extract a hierarchical segmentation of the mixing envelope surface to identify bubbles and analyze analogous segmentations of fields on the original interface plane. We compute meaningful statistical information that reveals the evolution of topological features and corroborates the observations made by scientists. We also use geometric tracking to follow the evolution of single bubbles and highlight merge/split events leading to the formation of the large and complex structures characteristic of the later stages. In particular we (i) Provide a formal definition of a bubble; (ii) Segment the envelope surface to identify bubbles; (iii) Provide a multi-scale analysis technique to produce statistical measures of bubble growth; (iv) Correlate bubble measurements with analysis of fields on the interface plane; (v) Track the evolution of individual bubbles over time. Our approach is based on the rigorous mathematical foundations of Morse theory and can be applied to a more general class of applications

Time-varying reeb graphs for continuous space-time data

We study the evolution of the Reeb graph of a time-varying continuous function defined in three-dimensional space. While maintaining the Reeb graph, we compress the evolving sequence into a single, partially persistent data structure. We envision this data structure as a useful tool in visualizing real-valued space-time data obtained from computational simulations of physical processes.

Time-varying Reeb graphs for continuous space--time data

The Reeb graph is a useful tool in visualizing real-valued data obtained from computational simulations of physical processes. We characterize the evolution of the Reeb graph of a time-varying continuous function defined in three-dimensional space. We show how to maintain the Reeb graph over time and compress the entire sequence of Reeb graphs into a single, partially persistent data structure, and augment this data structure with Betti numbers to describe the topology of level sets and with path seeds to assist in the fast extraction of level sets for visualization.

Topological feature extraction and tracking

Scientific datasets obtained by measurement or produced by computational simulations must be analyzed to understand the phenomenon under study. The analysis typically requires a mathematically sound definition of the features of interest and robust algorithms to identify these features, compute statistics about them, and often track them over time. Because scientific datasets often capture phenomena with multi-scale behaviour, and almost always contain noise the definitions and algorithms must be designed with sufficient flexibility and care to allow multi-scale analysis and noise-removal. In this paper, we present some recent work on topological feature extraction and tracking with applications in molecular analysis, combustion simulation, and structural analysis of porous materials.

Encoding volumetric grids for streaming isosurface extraction

Gridded volumetric data sets representing simulation or tomography output are commonly visualized by displaying a triangulated isosurface for a particular isovalue. When the grid is stored in a standard format, the entire volume must be loaded from disk, even though only a fraction of the grid cells may intersect the isosurface. We propose a compressed ondisk representation for regular volume grids that allows streaming, I/O-efficient, out-of-core isosurface extraction. Unlike previous methods, we provide a guaranteed bound on the ratio between the number of cells loaded and number of cells intersecting the isosurface that applies for any isovalue. As grid cells are decompressed, we immediately extract vertices and triangles of the isosurface. Our output is a coherent streaming mesh, which facilitates subsequent processing, including on-the-fly simplification and compression.

Implementing time-varying contour trees

In this video, we describe our experiences in implementing an algo-rithm to compute time-varying contour trees and highlight the chal-lenges in applying this algorithm to real-world scientific datasets. For ease of illustration we restrict our explanations to contour trees of time-varying functions defined on the plane.

Multiscale Morse theory for science discovery

Computational scientists employ increasingly powerful parallel supercomputers to model and simulate fundamental physical phenomena. These simulations typically produce massive amounts of data easily running into terabytes and petabytes in the near future. The future ability of scientists to analyze such data, validate their models, and understand the physics depends on the development of new mathematical frameworks and software tools that can tackle this unprecedented complexity in feature characterization and extraction problems. We present recent advances in Morse theory and its use in the development of robust data analysis tools. We demonstrate its practical use in the analysis of two large scale scientific simulations: (i) a direct numerical simulation and a large eddy simulation of the mixing layer in a hydrodynamic instability and (ii) an atomistic simulation of a porous medium under impact. Our ability to perform these two fundamentally different analyses using the same mathematical framework of Morse theory demonstrates the flexibility of our approach and its robustness in managing massive models.

Time-varying reeb graphs: a topological framework supporting the analysis of continuous time-varying data

I present time-varying Reeb graphs as a topological framework to support the analysis of continuous time-varying data. Such data is captured in many studies, including computational fluid dynamics, oceanography, medical imaging, and climate modeling, by measuring physical processes over time, or by modeling and simulating them on a computer. Analysis tools are applied to these data sets by scientists and engineers who seek to understand the underlying physical processes. A popular tool for analyzing scientific datasets is level sets, which are the points in space with a fixed data value s. Displaying level sets allows the user to study their geometry, their topological features such as connected components, handles, and voids, and to study the evolution of these features for varying s. For static data, the Reeb graph encodes the evolution of topological features and compactly represents topological information of all level sets. The Reeb graph essentially contracts each level set component to a point. It can be computed efficiently, and it has several uses: as a succinct summary of the data, as an interface to select meaningful level sets, as a data structure to accelerate level set extraction, and as a guide to remove noise. I extend these uses of Reeb graphs to time-varying data. I characterize the changes to Reeb graphs over time, and develop an algorithm that can maintain a Reeb graph data structure by tracking these changes over time. I store this sequence of Reeb graphs compactly, and call it a time-varying Reeb graph. I augment the time-varying Reeb graph with information that records the topology of level sets of all level values at all times, that maintains the correspondence of level set components over time, and that accelerates the extraction of level sets for a chosen level value and time. Scientific data sampled in space-time must be extended everywhere in this domain using an interpolant. A poor choice of interpolant can create degeneracies that are difficult to resolve, making construction of time-varying Reeb graphs impractical. I investigate piecewise-linear, piecewise-trilinear, and piecewise-prismatic interpolants, and conclude that piecewise-prismatic is the best choice for computing time-varying Reeb graphs. Large Reeb graphs must be simplified for an effective presentation in a visualization system. I extend an algorithm for simplifying static Reeb graphs to compute simplifications of time-varying Reeb graphs as a first step towards building a visualization system to support the analysis of time-varying data.

Understanding the structure of the turbulent mixing layer in hydrodynamic instabilities

When a heavy fluid is placed above a light fluid, tiny vertical perturbations in the interface create a characteristic structure of rising bubbles and falling spikes known as Rayleigh-Taylor instability. Rayleigh-Taylor instabilities have received much attention over the past half-century because of their importance in understanding many natural and man-made phenomena, ranging from the rate of formation of heavy elements in supernovae to the design of capsules for Inertial Confinement Fusion. We present a new approach to analyze Rayleigh-Taylor instabilities in which we extract a hierarchical segmentation of the mixing envelope surface to identify bubbles and analyze analogous segmentations of fields on the original interface plane. We compute meaningful statistical information that reveals the evolution of topological features and corroborates the observations made by scientists.