Topology Based Selection and Curation of Level Sets

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Topology Based Selection and Curation of Level
Sets

• Andrew Gillette
• Joint work with
• Chandrajit Bajaj and Samrat Goswami

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Problem Statement

• Given a trivariate function
  we want to select a level set L(r)
  with the following properties
• L(r) is a single, smooth component.
• 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.

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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
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Computational Pipeline
Atomic Data (e.g. pdb files for proteins)
Physical Observation
Gaussian Decay Model
Volumetric Data (e.g. cryo-EM for viruses)
Trivariate Electron Density Function
Our algorithm
Level Set (isosurface) Selection
Level Set (isosurface) Curation
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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

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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

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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

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Mathematical Preliminaries

1. Contour Tree
2. Voronoi / Delaunay Triangulation
3. Distance Function and Stable Manifolds

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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)

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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

Isosurface with three contours
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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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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
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Isosurface ? (from 1AOR pdb Hyperthormophilic
Tungstopterin Enzyme, Aldehyde Ferredoxin
Oxidoreductase) Bar below green square
indicates isovalue selection ?
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Isosurface ? (from 1AOR pdb Hyperthormophilic
Tungstopterin Enzyme, Aldehyde Ferredoxin
Oxidoreductase) Bar below green square
indicates isovalue selection ?
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Isosurface ? (from 1AOR pdb Hyperthormophilic
Tungstopterin Enzyme, Aldehyde Ferredoxin
Oxidoreductase) Bar below green square
indicates isovalue selection ?
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Isosurface ? (from 1AOR pdb Hyperthormophilic
Tungstopterin Enzyme, Aldehyde Ferredoxin
Oxidoreductase) Bar below green square
indicates isovalue selection ?
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Isosurface ? (from 1AOR pdb Hyperthormophilic
Tungstopterin Enzyme, Aldehyde Ferredoxin
Oxidoreductase) Bar below green square
indicates isovalue selection ?
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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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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

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
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Flow
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.

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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)
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Algorithm and Results

1. Description of Algorithm
2. Results
3. Future Work

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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

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Algorithm in pictures
1 2 3 4
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Void
Pocket
Tunnel
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Results
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Results
From 1RIE pdb (Rieske Iron-Sulfur Protein of the
bovine heart mitochondrial cytochrome BC1-complex)
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Results

• The chaperon GroEL generated from cryo-EM
  density map.
• The large tunnel is used for forming and folding
  proteins.

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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.

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Thank you!
(Danke)