Showing papers by "Herbert Edelsbrunner published in 2004"

A topological hierarchy for functions on triangulated surfaces

Abstract: We combine topological and geometric methods to construct a multiresolution representation for a function over a two-dimensional domain. In a preprocessing stage, we create the Morse-Smale complex of the function and progressively simplify its topology by cancelling pairs of critical points. Based on a simple notion of dependency among these cancellations, we construct a hierarchical data structure supporting traversal and reconstruction operations similarly to traditional geometry-based representations. We use this data structure to extract topologically valid approximations that satisfy error bounds provided at runtime.

Loops in Reeb Graphs of 2-Manifolds

Abstract: Given a Morse function f over a 2-manifold with or without boundary, the Reeb graph is obtained by contracting the connected components of the level sets to points. We prove tight upper and lower bounds on the number of loops in the Reeb graph that depend on the genus, the number of boundary components, and whether or not the 2-manifold is orientable. We also give an algorithm that constructs the Reeb graph in time O(n log n), where n is the number of edges in the triangulation used to represent the 2-manifold and the Morse function.

Time-varying reeb graphs for continuous space-time data

Abstract: We study the evolution of the Reeb graph of a time-varying continuous function defined in three-dimensional space. While maintaining the Reeb graph, we compress the evolving sequence into a single, partially persistent data structure. We envision this data structure as a useful tool in visualizing real-valued space-time data obtained from computational simulations of physical processes.

Extreme elevation on a 2-manifold

Abstract: Given a smoothly embedded 2-manifold in R3, we define the elevation of a point as the height difference to a canonically defined second point on the same manifold. Our definition is invariant under rigid motions and can be used to define features such as lines of discontinuous or continuous but non-smooth elevation. We give an algorithm for finding points of locally maximum elevation, which we suggest mark cavities and protrusions and are useful in matching shapes as for example in protein docking.

Local and Global Comparison of Continuous Functions

Abstract: We introduce local and global comparison measures for a collection of k /spl les/ d real-valued smooth functions on a common d-dimensional Riemannian manifold. For k = d = 2 we relate the measures to the set of critical points of one function restricted to the level sets of the other. The definition of the measures extends to piecewise linear functions for which they are easy to compute. The computation of the measures forms the centerpiece of a software tool which we use to study scientific datasets.

Interface surfaces for protein-protein complexes

Abstract: Protein-protein interactions, which form the basis for most cellular processes, result in the formation of protein interfaces. Believing that the local shape of proteins is crucial, we take a geometric approach and present a definition of an interface surface formed by two or more proteins. We also present an algorithm and study the geometric and topological properties of these surfaces, thus paving the way for future biochemical studies of protein-protein interactions.

Coarse and reliable geometric alignment for protein docking.

Abstract: We present an efficient algorithm for generating a small set of coarse alignments be- tween interacting proteins using meaningful features on their surfaces. The proteins are treated as rigid bodies, but the results are more generally useful as the produced configurations can serve as input to local improvement algorithms that allow for protein flexibility. We apply our algorithm to a diverse set of protein complexes from the Protein Data Bank, demonstrating the effectivity of our algorithm, both for bound and for unbound protein docking problems.

Simplification of three-dimensional density maps

Abstract: We consider scientific data sets that describe density functions over three-dimensional geometric domains. Such data sets are often large and coarsened representations are needed for visualization and analysis. Assuming a tetrahedral mesh representation, we construct such representations with a simplification algorithm that combines three goals: the approximation of the function, the preservation of the mesh topology, and the improvement of the mesh quality. The third goal is achieved with a novel extension of the well-known quadric error metric. We perform a number of computational experiments to understand the effect of mesh quality improvement on the density map approximation. In addition, we study the effect of geometric simplification on the topological features of the function by monitoring its critical points.

The Area Derivative of a Space-Filling Diagram

Abstract: The motion of a biomolecule greatly depends on the engulfing solution, which is mostly water. Instead of representing individual water molecules, it is desirable to develop implicit solvent models that nevertheless accurately represent the contribution of the solvent interaction to the motion. In such models, hydrophobicity is expressed as a weighted sum of atomic surface areas. The derivatives of these weighted areas contribute to the force that drives the motion. In this paper we give formulas for the weighted and unweighted area derivatives of a molecule modeled as a space-filling diagram made up of balls in motion. Other than the radii and the centers of the balls, the formulas are given in terms of the sizes of circular arcs of the boundary and edges of the power diagram. We also give inclusion---exclusion formulas for these sizes.

Computing the Writhing Number of a Polygonal Knot

Abstract: The writhing number measures the global geometry of a closed space curve or knot. We show that this measure is related to the average winding number of its Gauss map. Using this relationship, we give an algorithm for computing the writhing number for a polygonal knot with n edges in time roughly proportional to n1.6. We also implement a different, simple algorithm and provide experimental evidence for its practical efficiency.

Local Search Heuristic for Rigid Protein Docking

Abstract: We give an algorithm that locally improves the fit between two proteins modeled as space-filling diagrams. The algorithm defines the fit in purely geometric terms and improves by applying a rigid motion to one of the two proteins. Our implementation of the algorithm takes between three and ten seconds and converges with high likelihood to the correct docked configuration, provided it starts at a position away from the correct one by at most 18 degrees of rotation and at most 3.0A of translation. The speed and convergence radius make this an attractive algorithm to use in combination with a coarse sampling of the six-dimensional space of rigid motions.

Topological analysis of scalar functions for scientific data visualization

Abstract: Scientists attempt to understand physical phenomena by studying various quantities measured over the region of interest. A majority of these quantities are scalar (real-valued) functions. These functions are typically studied using traditional visualization techniques like isosurface extraction, volume rendering etc. As the data grows in size and becomes increasingly complex, these techniques are no longer effective. State of the art visualization methods attempt to automatically extract features and annotate a display of the data with a visualization of its features. In this thesis, we study and extract the topological features of the data and use them for visualization. We have three results: (1) An algorithm that simplifies a scalar function defined over a tetrahedral mesh. In addition to minimizing the error introduced by the approximation of the function, the algorithm improves the mesh quality and preserves the topology of the domain. We perform an extensive set of experiments to study the effect of requiring better mesh quality on the approximation error and the level of simplification possible. We also study the effect of simplification on the topological features of the data. (2) An extension of three-dimensional Morse-Smale complexes to piecewise linear 3-manifolds and an efficient algorithm to compute its combinatorial analog. Morse-Smale complexes partition the domain into regions with similar gradient flows. Letting n be the number of vertices in the input mesh, the running time of the algorithm is proportional to n log(n) plus the total size of the input mesh plus the total size of the output. We develop a visualization tool that displays different substructures of the Morse-Smale complex. (3) A new comparison measure between k functions defined on a common d-manifold. For the case d = k = 2, we give alternative formulations of the definition based on a Morse theoretic point of view. We also develop visualization software that performs local comparison between pairs of functions in datasets containing multiple and sometimes time-varying functions. We apply our methods to data from medical imaging, electron microscopy, and x-ray crystallography. The results of these experiments provide evidence of the usability of our methods.

Geometric and topological methods in protein structure analysis

Abstract: With the recent success of the Human Genome Project, one of the main challenges in molecular biology in this post-genomic era is the determination and exploitation of the three-dimensional structure of proteins and their function. The ability for proteins to perform their numerous functions is made possible by the diversity of their three-dimensional structures. Hence, to attack the key problems involved, such as protein folding and docking, geometry and topology become important tools. Despite their essential roles, geometric and topological methods are relatively uncommon in computational biology, partly due to a number of modeling and algorithmic challenges. This thesis describes efficient computational methods for describing and comparing molecular structures by combining both geometric and topological approaches. In particular, in the first part of the thesis, we study three geometric descriptions: (i) the writhing number of protein backbones, which measures how many times a backbone coils around itself; (ii) the level-of-details representation of protein backbones via simplification, which helps to extract main features of backbones; and (iii) the elevation of molecular surfaces, which we propose to identify geometric features such as protrusions and cavities from protein surfaces. We develop efficient algorithms for computing these descriptions. The second part of the thesis focuses on molecular shape matching algorithms. By modeling a molecule as the union of balls, we propose algorithms to compute the similarity between two such unions by (variants of) the widely used Hausdorff distance. We also study the protein docking problem, which, from a geometric perspective, can be considered as the problem of searching for configurations with maximum complementarity between two molecular surfaces. Using the feature information computed from the elevation function, we describe an efficient algorithm to find promising initial relative placements of the proteins.