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. Problem Statement. Given a trivariate function we want to select a level set L(r) = with the following properties:

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### 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:

• 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.
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

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

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
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
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
Mathematical Preliminaries
• Contour Tree
• Voronoi / Delaunay Triangulation
• Distance Function and Stable Manifolds
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)

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

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

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

Isosurface 

(from 1AOR pdb: Hyperthormophilic Tungstopterin Enzyme, Aldehyde Ferredoxin Oxidoreductase)

Bar below green square indicates isovalue selection

Isosurface 

(from 1AOR pdb: Hyperthormophilic Tungstopterin Enzyme, Aldehyde Ferredoxin Oxidoreductase)

Bar below green square indicates isovalue selection

Isosurface 

(from 1AOR pdb: Hyperthormophilic Tungstopterin Enzyme, Aldehyde Ferredoxin Oxidoreductase)

Bar below green square indicates isovalue selection

Isosurface 

(from 1AOR pdb: Hyperthormophilic Tungstopterin Enzyme, Aldehyde Ferredoxin Oxidoreductase)

Bar below green square indicates isovalue selection

Isosurface 

(from 1AOR pdb: Hyperthormophilic Tungstopterin Enzyme, Aldehyde Ferredoxin Oxidoreductase)

Bar below green square indicates isovalue selection

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 
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

The distance function
• Let S be a surface smoothly embedded in
• Let P be a finite sampling of points on S. Then we approximate:

Minimum

Maximum

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.
Stable Manifolds

Given a critical value c of hP, the stable manifold of c is the set of points whose orbits end at c.

Algorithm and Results
• Description of Algorithm
• Results
• Future Work
Algorithm in words

Given an isosurface S sampled by pointset P:

• Find critical points of distance function hP
• Classify critical points exterior to S as max, saddle, or saddle incident on infinity
• Cluster points based on stable manifolds
• Classify clusters based on number of mouths
• Rank clusters based on geometric significance
Algorithm in pictures

1 2 3 4 5

Void:

Pocket:

Tunnel:

Results

From 1RIE pdb

(Rieske Iron-Sulfur Protein of the bovine heart mitochondrial cytochrome BC1-complex)

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

(Danke)