Topology Based Selection and
Curation of Level Sets
Andrew Gillette
Joint work with
Chandrajit Bajaj and Samrat Goswami
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
March 2007
University of Texas at Austin
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 Visualization
University 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 Visualization
University of Texas at Austin
March 2007
Computational Pipeline
Physical
Observation
Atomic Data
(e.g. pdb files for
proteins)
Gaussian Decay Model
Volumetric Data
(e.g. cryo-EM for
viruses)
Trivariate
Electron Density
Function
Level Set
(isosurface)
Selection
Center for Computational Visualization
University of Texas at Austin
Our algorithm:
Level Set
(isosurface)
Curation
March 2007
Example 1: Gramicidin A
Images created from Protein Data Bank file 1MAG
• Three topologically distinct isosurfaces for the
molecule are shown
• We need information on the topology of the
complementary space to select a correct isosurface
Center for Computational Visualization
University 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 Visualization
University of Texas at Austin
March 2007
Example 3: Nodavirus
Data from Tim Baker, UCSD; Images generated at CVC, UT Austin
• 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
Center for Computational Visualization
University of Texas at Austin
March 2007
Mathematical Preliminaries
A. Contour Tree
B. Voronoi / Delaunay
Triangulation
C. Distance Function and
Stable Manifolds
Center for Computational Visualization
University of Texas at Austin
March 2007
Prior Related Work
Isosurface Selection via Contour Tree
Modern 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 Visualization
University of Texas at Austin
March 2007
Level Sets and Contours
• 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
• Each component of
an isosurface is
called a contour
• We select an
isosurface with a
single component via
the contour tree
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University of Texas at Austin
Isosurface
with three
contours
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
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University 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 Visualization
University 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 Visualization
University 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 Visualization
University 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 Visualization
University 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 Visualization
University 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 Visualization
University 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 
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University of Texas at Austin
March 2007
Delaunay Diagram
Vor P
• 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
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University of Texas at Austin
Del P
March 2007
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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University of Texas at Austin
March 2007
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 Visualization
University of Texas at Austin
March 2007
Flow
Sample Point
Minimum
Saddle
Maximum
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 Visualization
University 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 Visualization
University of Texas at Austin
March 2007
Algorithm and Results
A. Description of Algorithm
B. Results
C.Future Work
Center for Computational Visualization
University of Texas at Austin
March 2007
Algorithm in words
Given an isosurface S sampled by pointset P:
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
Center for Computational Visualization
University of Texas at Austin
March 2007
Algorithm in pictures
1
2
3
4
5
Void:
Pocket:
Tunnel:
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University of Texas at Austin
March 2007
Results
Center for Computational Visualization
University of Texas at Austin
March 2007
Results
From 1RIE pdb
(Rieske Iron-Sulfur
Protein of the bovine
heart mitochondrial
cytochrome BC1complex)
Center for Computational Visualization
University 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 Visualization
University 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 preand post-curation surfaces.
Center for Computational Visualization
University of Texas at Austin
March 2007
Thank you!
(Danke)
Center for Computational Visualization
University of Texas at Austin
March 2007