Data Analysis and Visualization Using the Morse-Smale complex

Attila Gyulassy

Institute for Data Analysis and VisualizationComputer Science DepartmentUniversity of California, Davis

Center for Applied Scientific ComputingLLNS

Seminar Overview

• Introduce basic concepts from topology

• Intuitive definition of the Morse-Smale complex

• What is a feature?

• Examples from various application areas

• Algorithm

Topology Background - Critical Points

Regular Minimum 1-Saddle 2-Saddle Maximum Index 0 Index 1 Index 2 Index 3

Let ƒ be a scalar valued function whose critical points are not degenerate. We call ƒ a Morse function, and in the neighborhood of a critical point p, the function can be represented as

ƒ(p) = 0ƒ(x, y, z) = ƒ(p) ± x2 ± y2 ± z2

∆

Topology Background - Integral Lines

An integral line is a maximal path that agrees with the gradient of at every point

Topology Background - Manifolds

Ascending ManifoldsA(p) = {p} { x | x є l, orig(l) = p}

Descending ManifoldsD(p) = {p} { x | x є l, dest(l) = p}

∩ ∩

3-Manifold

2-Manifold

1-Manifold

0-Manifold

3-Manifold 1-Manifold

2-Manifold 0-Manifold

Maximum 2-Saddle Minimum1-Saddle

The Morse-Smale complex is a segmentation of the domain that clusters integral lines that share a common origin and destination.

What is the Morse-Smale Complex?

• The intersection of all descending and ascending manifolds

D(p) ∩ A(q), for all pairs p,q of f

• Any cell in the complex has the property that all integral lines in that cell share an origin and a destination

Morse-Smale Complex - 1D Example

minimummaximumminimummaximumminimummaximum

Morse-Smale Complex - 1D Example

minimummaximumminimummaximumminimummaximum

Morse-Smale Complex - 1D Example

Morse-Smale Complex - 2D Example

Morse-Smale Complex - 2D Example

Morse-Smale Complex - 2D Example

Morse-Smale Complex - 2D Example

Morse-Smale Complex - 3D Example

Cells of dimension i connect critical points with index that differ by i.

Crystal Quad Arc Node

Morse-Smale Complex - 3D Example

Index LemmaCritical points can be

created or destroyed in pairswith index that differs by one.

Topology based simplificationTopology based simplification

What is a feature?

What is a feature?

What is a feature?

What is a feature?

What is a feature?

What is a feature?

What is a feature?

What is a feature?

What is a feature?

What is a feature?

What is a feature?

What is a feature?

What is a feature?

What is a feature?

Critical Points

What is a feature?

Arcs

What is a feature?

Higher Degree Cells

Examples from applicationsPersistent extrema are the features

Finding atom locations in molecular simulations

Motivation

Tracking the formation of bubbles in turbulent mixing fluids (Laney et al.)

(Multi-Scale Analysis)Previous WorkMorse-Smale comple in 2D

Examples from applicationsPersistent extrema are the features

• Testing the “smoothness” of a generated function– How does the critical point count change as a

function of persistence?– Length of the persistent arcs?– Size of the persistent cells?

Examples from applicationsPersistent extrema are the features

Critical point count

Persistence

Terrain representation (Bremer et al.)

Examples from applicationsPersistent arcs are the features

Surface Quadrangulation (Dong et al.)

Examples from applicationsPersistent arcs are the features

Examples from applicationsPersistent arcs are the features

Analysis of porous media

Time comparison of the reconstructions

Examples from applicationsPersistent arcs are the features

Examples from applicationsPersistent arcs are the features

Analysis of the structure of galaxies

Examples from applicationsPersistent cells are the features

Analysis of a combustion simulation

A Simple Algorithm For Constructing the Morse-Smale Complex

• Construct the known complex for a similar function called the augmented function

• Simplify the artificial complex

Contributions

Constructing the Morse-Smale complex of anAugmented Morse Function

The augmented Morse function has a very regular structure. Every vertex of S is critical, with index = dimension of its cell in K. Arcs of the complex are the edges of S.

A Simple Algorithm For Constructing the Morse-Smale Complex

ContributionsTopology based simplification

A Simple Algorithm For Constructing the Morse-Smale Complex

●Remove extra critical points●Correct Morse-Smale complex within small error bound

ContributionsTopology based simplification

A Simple Algorithm For Constructing the Morse-Smale Complex

A Simple Algorithm For Constructing the Morse-Smale Complex

Original data points

A Simple Algorithm For Constructing the Morse-Smale Complex

Questions?

aggyulassy@ucdavis.edu