Topology Based Selection and Curation of Level Sets

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.


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

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:


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


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

• 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


Mathematical Preliminaries

A. Contour Tree

B. Voronoi / Delaunay Triangulation

C.Distance Function and Stable Manifolds


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)


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


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





isovalue selection


Isosurface





isovalue selection


Isosurface





isovalue selection


Isosurface





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

• 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


The distance function

• Let S be a surface smoothly embedded in

• Let P be a finite sampling of points on S. Then we approximate:


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


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.


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)


Algorithm and Results

A. Description of Algorithm

B. Results

C.Future Work


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:


Algorithm in pictures

1 2 3 4 5

Void:

Pocket:

Tunnel:


Results


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)

Documents

Topology Based Selection and Curation of Level Sets