Cheol-Hyung Cho

Three-dimensional beta shapes

The Voronoi diagram of a point set has been extensively used in various disciplines ever since it was first proposed. Its application realms have been even further extended to estimate the shape of point clouds when Edelsbrunner and Mucke introduced the concept of @a-shape based on the Delaunay triangulation of a point set. In this paper, we present the theory of @b-shape for a set of three-dimensional spheres as the generalization of the well-known @a-shape for a set of points. The proposed @b-shape fully accounts for the size differences among spheres and therefore it is more appropriate for the efficient and correct solution for applications in biological systems such as proteins. Once the Voronoi diagram of spheres is given, the corresponding @b-shape can be efficiently constructed and various geometric computations on the sphere complex can be efficiently and correctly performed. It turns out that many important problems in biological systems such as proteins can be easily solved via the Voronoi diagram of atoms in proteins and @b-shapes transformed from the Voronoi diagram.

Recognition of docking sites on a protein using β-shape based on Voronoi diagram of atoms

A protein consists of atoms. Given a protein, the automatic recognition of depressed regions on the surface of the protein, often called docking sites or pockets, is important for the analysis of interaction between a protein and a ligand and facilitates fast development of new drugs. Presented in this paper is a geometric approach for the detection of docking sites using @b-shape which is based on the Voronoi diagram for atoms in Euclidean distance metric. We first propose a geometric construct called a @b-shape which represents the proximity among atoms on the surface of a protein. Then, using the @b-shape, which takes the size differences among different atoms into account, we present an algorithm to extract the pockets for the possible docking site on the surface of a protein.

Pocket recognition on a protein using euclidean voronoi diagram of atoms

Proteins consist of atoms. Given a protein, the automatic recognition of depressed regions, called pockets, on the surface of the protein is important for protein-ligand docking and facilitates fast development of new drugs. Recently, computational approaches for the recognition of pockets have emerged. Presented in this paper is a geometric method for the pocket recognition based on Voronoi diagram for atoms in Euclidean distance metric.

Visualization and analysis of protein structures using euclidean voronoi diagram of atoms

Protein consists of amino acids, and an amino acid consists of atoms. Given a protein, understanding its functions is critical for various reasons for designing new drugs, treating diseases, and so on. Due to recent researches, it is now known that the structure of protein directly influences its functions. Hence, there have been strong research trends towards understanding the geometric structure of proteins. In this paper, we present a Euclidean Voronoi diagram of atoms constituting a protein and show how this computational tool can effectively and efficiently contribute to various important problems in biology. Some examples, among others, are the computations for molecular surface, solvent accessible surface, extraction of pockets, interaction interface, convex hull, etc.

A β-shape from the Voronoi diagram of atoms for protein structure analysis

In this paper, we present a β-shape and a β-complex for a set of atoms with arbitrary sizes for a faster response to the topological queries among atoms. These concepts are the generalizations of the well-known α-shape and α-complex (and their weighted counterparts as well). To compute a β-shape, we first compute the Voronoi diagram of atoms and then transform the Voronoi diagram to a quasi-triangulation which is the topological dual of the Voronoi diagram. Then, we compute a β-complex from the quasi-triangulation by analyzing the valid intervals for each simplex in the quasi-triangulation. It is shown that a β-complex can be computed in O(m) time in the worst case from the Voronoi diagram of atoms, where m is the number of simplices in the quasi-triangulation. Then, a β-shape for a particular β consisting of k simplices can be located in O(log m + k) time in the worst case from the simplicies in the β-complex sorted according to the interval values.

Plane-Sweep Algorithm of O(nlogn) for the Inclusion Hierarchy among Circles

Suppose that we have a number of circles in a plane and some of them may contain other circles. Then, finding the hierarchy among circles is of important for various applications such as the simulation of emulsion. In this paper, we present a standard plane-sweep algorithm which can identify the inclusion relationship among the circles in O(nlogn) time in the worst-case. The algorithm uses a number of sweep-lines and a red-black tree for the efficient computation.

The B-shape and B-complex for three-dimensional spheres

In recent years, there have been extensive studies on biological systems such as proteins. Being one of the most important aspects, the geometry has been more important since the morphology of a molecular system is known to determine the critical functions of the molecule. In the study of the shape and the structure of a molecule, the representation of proximity information among atoms in the molecule is the most fundamental research issue. In this paper, we present a The beta-shape and beta-complex for three-dimensional spheres-shape and a The beta-shape and beta-complex for three-dimensional spheres-complex for a set of atoms with arbitrary sizes for a faster response to the topological queries among atoms. These concepts are the generalizations of the well-known a-shape and a- complex (and their weighted counterparts as well). To compute a The beta-shape and beta-complex for three-dimensional spheres-shape, we first compute the Voronoi diagram of atoms and then transform the Voronoi diagram to a quasi- triangulation which is the topological dual of the Voronoi diagram. Then, we compute a The beta-shape and beta-complex for three-dimensional spheres-complex from the quasi- triangulation by analyzing the valid intervals for each simplex in the quasi-triangulation. It is shown that a The beta-shape and beta-complex for three-dimensional spheres-complex can be computed in O(m) time in the worst case from the Voronoi diagram of atoms, where m is the number of simplices in the quasi-triangulation. Then, a The beta-shape and beta-complex for three-dimensional spheres-shape for a particular The beta-shape and beta-complex for three-dimensional spheres consisting of k simplices can be located O(log m + k) time in the worst case from the simplicies in the The beta-shape and beta-complex for three-dimensional spheres-complex sorted according to the interval values.

Probability distribution of the five op-codes in Edgebreaker

The rapid Internet transmission of 3-D mesh models is becoming increasingly important and considerable research effort has been expended on the compression of mesh models. Although a mesh model usually consists of the coordinates of the vertices and properties such as colors and normals, topology plays the most important role in the compression of other information in the models. Despite extensive studies on Edgebreaker, the most frequently used and rigorously evaluated topology compressor, the probability distribution of its five op-codes, C, R, E, S and L, has to date not been rigorously analyzed. In this paper, we present the limiting probability distribution of the op-codes via the concept of triangle subdivision. The presented probability distribution is useful in the optimization of the compression performance of Edgebreaker.

Probability Distribution of Op-Codes in Edgebreaker

Rapid transmission of 3D mesh models has become important with the use of Internet and with increased number of studies on the compression of various aspects for mesh models. Despite the extensive studies on Edgebreaker for the compression of topology for meshes, the probability distribution of its five op-codes, C, R, E, S, and L, has not yet been rigorously analyzed. In this paper we present the probability distribution of the op-codes which is useful for both the optimization of the compression performances and a priori estimation of compressed file size.