Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin March 2007 Topology Based Selection and Curation of Level Sets Andrew Gillette Joint work with Chandrajit Bajaj and Samrat Goswami

Center for Computational Visualization University 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 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 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 Visualization University 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 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 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 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 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 Visualization University 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 Visualization 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 

Center for Computational Visualization University 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 V p facets –Facets = dual to V p edges –Tetrahedra = centered at Vor P vertices Vor P Del P

Center for Computational Visualization University of Texas at Austin 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:

Center for Computational Visualization University of Texas at Austin March 2007 Critical points of h P by analogy hShS hPhP SmoothNot smooth GradientFlow Gradient = 0Intersection of Vor P and Del P MinimumPoint of P Index 1 saddleIntersection of Vor P facet and Del P edge Index 2 saddleIntersection of Del P facet and Vor P edge MaximumVertex of Vor P

Center for Computational Visualization University of Texas at Austin March 2007 Flow Minimum Saddle Maximum 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 Visualization University of Texas at Austin March 2007 Stable Manifolds Given a critical value c of h P, 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… MaxIndex 2 saddle Index 1 saddle 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 1.Find critical points of distance function h P 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:

Center for Computational Visualization University of Texas at Austin March 2007 Algorithm in pictures Void: Pocket: Tunnel:

Center for Computational Visualization 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 BC1- complex)

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 pre- and post-curation surfaces.

Center for Computational Visualization University of Texas at Austin March 2007 Thank you! (Danke)