Topology-Based Analysis of Time-Varying Data

Scalar data is often used in scientific data to represent the distribution of a particular value of interest, such as density of a material, electrostatic potential, or distribution of matter. We use the theoretical framework described by Morse theory to find topological features in scalar valued volumetric data. A topology-based analysis allows us to simplify components of the topology representation to extract the “important” features. In

this application, we use the pairing of critical points given by the Morse-Smale complex to identify a stable filament structure for a porous material. We use this stable extraction to compute quantitative and qualitative comparisons of how this structure changes over various timesteps when the simulated material undergoes a particle impact.

Attila Gyulassy, University of California, Davis,

CAR/CASC, Valerio Pascucci, Lawrence Livermore National Laboratory,

Morse-Smale ComplexMorse-Smale ComplexThe Morse-Smale complex is a segmentation of the domain into regions of uniform gradient flow behaviour. Integral lines are lines that agree with the gradient at every point. They trace paths from one critical point to another. Each cell of the complex groups integral lines that share a common origin and destination. The cells of dimension 0, 1, 2, and 3 are called nodes, arcs, quads, and crystals.

This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. UCRL-POST-234479

OverviewOverview- Compute the Morse-Smale complex for a complete description of the topology of the dataset.- Perform filtering on the complex and topology-based simplification to extract the components corresponding to features in the data.- Analyse the features through visualization and quantitative measurements.

Node Arc Quad Crystal

The cells of the Morse-Smale complex have regular combinatorial structure, and form a regular CW-complex that covers the domain.

Computing the Morse-Smale complex for volumetric data has traditionally been a difficult problem. Our construction uses simplification of an artificial complex, a complex with predictable structure that can be computed for any cell-based data.

Simplification/ Multi-scale AnalysisSimplification/ Multi-scale AnalysisThe Index lemma states that critical points connected by an arc of the Morse-Smale complex can be cancelled with only a local change to the function. This permits local control for removing low persistence critical points, and permits a multi-scalar critical point analysis.

In 3D, there are two kinds of cancellations. Each removes a pair of critical points and simulates an overall smoothing of the function.

Saddle-Extremum 1-Saddle-2-Saddle

Left: A volume rendering of the simulated impact at four timesteps. In the first step, the particle is visible as a sphere separate from the porous medium. As the particle hits the material, a crater is formed.

Right: A visual comparison between timesteps. The filament structure of the initial step is rendered as yellow tubes, and the subsequent timestep is green tubes. The red and blue “struts” connect the arcs in each filament structure with the corresponding closest arc in the other.

P

A Visual Comparison Filtering/Feature ExtractionFiltering/Feature ExtractionThe data is available as a signed distance field to an interface surface that marks the boundary between empty space and interior of the material. The location of this interface is a parameter used only in the construction of the distance field, and may contribute “extra” topology to the analysis of the data. Additionally, noise and artefacts from construction make a straightforward analysis difficult. We compute the Morse-Smale complex and analyse its structure to resolve these difficulties.

The complete Morse-Smale complex of a single timestep. Excess critical points makes initial analysis difficult.

A visual depiction of the 2-Saddle – Maximum arcs of the complex. Each arc is plotted (left) with coordinates of the function values of its 2-Saddle and Maximum, and the density of such points are displayed. Our features lie in the upper-left of this plot, and we wish to cancel all other arcs. Integrating along each axis shows where “stable” values are for simplification.

Using the values obtained by analysing the arcs of the Morse-Smale complex, we can filter the arc by cancelling those that do not pass certain requirements. The result is a “clean” structure, where the 2-Saddle – Maximum arcs that remain exactly correspond to the stable filaments of the material.

The filtered arcs reveal the filament structure of the material.Time Varying FeaturesTime Varying Features

Computing the filament structure of the material at each timestep in a controlled manner allows us to perform qualitative and quantitative comparisons of material properties throughout the particle impact.

Top: Isolating the portions of the material that are represented by filaments, we can compute the change in density of the material along the Z-axis in the impact crater (left), and over the entire material (right).

Bottom: Changes in the number of cycles and length of the filaments measures the loss in porosity of the material.