data analysis and visualization using the morse-smale complex attila gyulassy institute for data...
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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?