topology based selection and curation of level sets
DESCRIPTION
Topology Based Selection and Curation of Level Sets. Andrew Gillette Joint work with Chandrajit Bajaj and Samrat Goswami. Problem Statement. Given a trivariate function we want to select a level set L(r) = with the following properties: - PowerPoint PPT PresentationTRANSCRIPT
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer SciencesUniversity of Texas at Austin March 2007
Topology Based Selection and Curation of Level
SetsAndrew Gillette
Joint work with
Chandrajit Bajaj and Samrat Goswami
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Problem Statement
Given a trivariate function we want to select a level set L(r) = with the following properties:
1) L(r) is a single, smooth component.
2) L(r) does not have any topological or geometrical features of size less than where the size of a feature is measured in the complementary space. The value of is determined by the application domain.
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Application: Molecular Surface Selection
• We need a molecular surface model to study molecular function (charge, binding affinity, hydrophobicity, etc).
• We can create an implicit solvation surface as the level set of an electron density function.
• Our selected level set should be a single component and have no small features (tunnels, pockets, or voids).
“The World of the Cell” 1996
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Computational Pipeline
Physical Observation
Volumetric Data (e.g. cryo-EM for
viruses)
Atomic Data (e.g. pdb files for
proteins)
Gaussian Decay Model
Trivariate Electron Density
Function
Level Set (isosurface)
Selection
Level Set (isosurface)
Curation
Our algorithm:
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Example 1: Gramicidin A
• Three topologically distinct isosurfaces for the molecule are shown
• We need information on the topology of the complementary space to select a correct isosurface
Images created from Protein Data Bank file 1MAG
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Example 2: mouse Acetylcholinesterase
• Two isosurfaces for the molecule are shown, with an important pocket magnified
• We need information on the geometry of the complementary space to select a correct isosurface and ensure correct energetics calculations
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Example 3: Nodavirus
• A rendering of the cryo-EM map and two isosurfaces of the virus capsid are shown
• We need to locate symmetrical topological features to select a correct isosurface
Data from Tim Baker, UCSD; Images generated at CVC, UT Austin
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Mathematical Preliminaries
A. Contour Tree
B. Voronoi / Delaunay Triangulation
C.Distance Function and Stable Manifolds
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Prior Related WorkIsosurface Selection via Contour TreeModern application of contour trees:
“Trekking in the alps without freezing or getting tired” (de Berg, van Kreveld: 1997)
“Contour trees and small seed sets for isosurface traversal” (van Kreveld, van Oostrum, Bajaj, Pascucci, Schikore: 1997)
Computation via split and join trees:
“Computing contour trees in all dimensions” (Carr, Snoeyink, Axen: 2001)
Betti numbers and augmented contour trees:
“Parallel computation of the topology of level sets” (Pascucci, Cole-McLaughlin: 2003)
Distance Function and Stable Manifold Computation“Shape segmentation and matching with flow discretization” (Dey, Giesen, Goswami:
2003)
“Surface reconstruction by wrapping finite point sets in space” (Edelsbrunner: 2002)
“The flow complex: a data structure for geometric modeling.” (Giesen, John: 2003)
“Identifying flat and tubular regions of a shape by unstable manifolds” (Goswami, Dey, Bajaj: 2006)
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Level Sets and Contours
• Each component of an isosurface is called a contour
• We select an isosurface with a single component via the contour tree
• In this talk, f(x,y,z) will denote the electron density at the point (x,y,z)
• An isosurface in this context is a level set of the function f, that is, a set of the type
Isosurface with three contours
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Contour Tree
• Recall • A critical isovalue of f is a value r
such that f -1(r) is not a 2-manifold• Examples: r is a value where contours
emerge, merge, split, or vanish.
r = 1 r = 2 r = 3
non-critical critical non-critical
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Contour Tree
• The contour tree is a tool used to aid in the selection of an isosurface
• Vertices: subset of critical values of f
• Edges: connect vertices along which a contour smoothly deforms
Increasing isovalues Isovalue selector
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Isosurface
(from 1AOR pdb: Hyperthormophilic
Tungstopterin Enzyme, Aldehyde
Ferredoxin Oxidoreductase)
Bar below green square indicates
isovalue selection
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Isosurface
(from 1AOR pdb: Hyperthormophilic
Tungstopterin Enzyme, Aldehyde
Ferredoxin Oxidoreductase)
Bar below green square indicates
isovalue selection
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Isosurface
(from 1AOR pdb: Hyperthormophilic
Tungstopterin Enzyme, Aldehyde
Ferredoxin Oxidoreductase)
Bar below green square indicates
isovalue selection
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Isosurface
(from 1AOR pdb: Hyperthormophilic
Tungstopterin Enzyme, Aldehyde
Ferredoxin Oxidoreductase)
Bar below green square indicates
isovalue selection
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Isosurface
(from 1AOR pdb: Hyperthormophilic
Tungstopterin Enzyme, Aldehyde
Ferredoxin Oxidoreductase)
Bar below green square indicates
isovalue selection
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Voronoi Diagram
• Let P be a finite set of points in
• The set of Vp partition and “meet nicely” along faces and edges.
• A 2-D example is shown
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Delaunay Diagram
• Voronoi diagram = Vor P
• Delaunay diagram = Del P
• Del P is defined to be the dual of Vor P– Vertices = P
– Edges = dual to Vp facets
– Facets = dual to Vp edges
– Tetrahedra = centered at Vor P vertices
Vor P
Del P
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The distance function
• Let S be a surface smoothly embedded in
• Let P be a finite sampling of points on S. Then we approximate:
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Critical points of hP by analogy
hS hP
Smooth Not smooth
Gradient Flow
Gradient = 0 Intersection of Vor P and Del P
Minimum Point of P
Index 1 saddle Intersection of Vor P facet and Del P edge
Index 2 saddle Intersection of Del P facet and Vor P edge
Maximum Vertex of Vor P
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Flow
MinimumSaddleMaximum
Sample Point
Orbit
• Flow describes how a point x moves if it is allowed to move in the direction of steepest ascent, that is, the direction that most rapidly increases the distance of x from all points in P.
• The corresponding path is called an orbit of x.
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Stable Manifolds
Given a critical value c of hP, the stable manifold of c is the set of points whose orbits end at c.
Stable manifold of a… …has boundary S.M. of a…
Max Index 2 saddle
Index 2 saddle Index 1 saddle
Index 1 saddle Min
Min (no boundary)
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Algorithm and Results
A. Description of Algorithm
B. Results
C.Future Work
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Algorithm in words
1.Find critical points of distance function hP
2.Classify critical points exterior to S as max, saddle, or saddle incident on infinity
3.Cluster points based on stable manifolds
4.Classify clusters based on number of mouths
5.Rank clusters based on geometric significance
Given an isosurface S sampled by pointset P:
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Algorithm in pictures
1 2 3 4 5
Void:
Pocket:
Tunnel:
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Results
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Results
From 1RIE pdb(Rieske Iron-Sulfur
Protein of the bovine heart mitochondrial cytochrome BC1-
complex)
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Results
• The chaperon GroEL; generated from cryo-EM density map.
• The large tunnel is used for forming and folding proteins.
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Future Work
What makes a point set P sufficient for applying our algorithm?
How can we provide a “quick update” to the distance function for a range of isovalues?
Compare energy calculations on our pre- and post-curation surfaces.
Center for Computational VisualizationUniversity of Texas at Austin March 2007
Thank you!
(Danke)