Jeongyeon Seo

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.

A Tabu Search Algorithm using the Voronoi Diagram for the Capacitated Vehicle Routing Problem

We consider the capacitated vehicle routing problem that determines the routes of vehicles in such a way that each customer can be visited exactly once by one vehicle, starting and terminating at the depot while the vehicle capacity and the travel time constraints must be satisfied. The objective is to minimize the total traveling cost. Due to the complexity of the problem, we suggest a tabu search algorithm that combines the features of the exiting local search heuristics. In particular, our tabu search algorithm incorporates the method to reduce the neighborhoods using the proximity information of the Voronoi diagram corresponding to each problem instance. Computational experiments are done on the benchmark problems and the test results are reported.In this paper, we give an algorithm for the analysis and correction of the distorted QR barcode (QR-code) image. The introduced algorithm is based on the code area finding by four corners detection for 2D barcode. We combine Canny edge detection with contours finding algorithms to erase noises and reduce computation and utilize two tangents to approximate the right-bottom point. Then, we give a detail description on how to use inverse perspective transformation in rebuilding a QR-code image from a distorted one. We test our algorithm on images taken by mobile phones. The experiment shows that our algorithm is effective.

An efficient algorithm for three-dimensional β-complex and β-shape via a quasi-triangulation

The concept of a β-shape has been recently proposed by extending the concept of the well-known α-shape. Since the β-shape takes full consideration of the Euclidean geometry of spherical particles, it is better suited than the (weighted) α-shape for applications using spatial queries on the system of variable sized spheres based on the Euclidean distance metric. In this paper, we present an efficient and elegant algorithm which computers a β-shape from a quasi-triangulation in O(log m + k) time in the worst case, where the quasi-triangulation has m simplicies and the boundary of β-shape consists of k simplicies. We believe that the β-shape and β-complex for a set of variable sized spheres (such as the atoms in a protein) will be very useful in the near future since the precise and efficient analysis of molecular structure can be conveniently facilitated by using these structures.

\beta-shape Based Computation of Blending Surfaces on a Molecule

It has been generally accepted that the structure of molecule is one of the most important factors which determine the functions of a molecule. Hence, studies have been conducted to analyze the structure of a molecule. Molecular surface is an important example of molecular structure. Given a molecular surface, the area and volume of the molecule can be computed to facilitate problems such as protein docking and folding. Therefore, it is important to compute a molecular surface precisely and efficiently. This paper presents an algorithm for correctly and efficiently computing the blending surfaces of a protein which is an important part of the molecular surface. Assuming that the Voronoi diagram of atoms of a protein is given, we first compute the beta-shape of the protein corresponding to a solvent probe. Then, we use a search space reduction technique for the intersection tests while the link blending surface is computed. Once a beta-shape is obtained, the blending surfaces corresponding to a given solvent probe can be computed in O(n) in the worst case, where n is the number of atoms. The correctness and efficiency of the algorithm stem from the powerful properties of beta-shape, quasi-triangulation, and the interworld data structure.

The β-Shape and β-Complex for Analysis of Molecular Structures

The topology among particles frequently plays a core role in many applications. One of the emerging application areas of particle systems is the analysis of molecular structures since the morphology of a molecule has been recognized as one of the most important factors which determines the functions of the molecule.

Real-time triangulation of molecular surfaces

Protein consists of a set of atoms. Given a protein, the molecular surface of the protein is defined with respect to a probe approximating a solvent molecule. This paper presents an efficient, as efficient as the realtime, algorithm to triangulate the blending surfaces which is the most critical subset of a molecular surface. For the quick evaluation of points on the surface, the proposed algorithm uses masks which are similar in their concepts to those in subdivision surfaces. More fundamentally, the proposed algorithm takes advantage of the concise representation of topology among atoms stored in the β-shape which is indeed used in the computation of the blending surface itself. Given blending surfaces and the corresponding β-shape, the proposed algorithm triangulates the blending surfaces in O(c ċ m) time in the worst case, where m is the number of boundary atoms in the protein and c is the number of point evaluations on a patch in the blending surface.

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.

BetaMDGP: Protein Structure Determination Algorithm Based on the Beta-complex

The molecular distance geometry problem (MDGP) is a fundamental problem in determining molecular structures from the NMR data. We present a heuristic algorithm, the BetaMDGP, which outperforms existing algorithms for solving the MDGP. The BetaMDGP algorithm is based on the beta-complex, which is a geometric construct extracted from the quasi-triangulation derived from the Voronoi diagram of atoms. Starting with an initial tetrahedron defined by the centers of four closely located atoms, the BetaMDGP determines a molecular structure by adding one shell of atoms around the currently determined substructure using the beta-complex. The proposed algorithm has been entirely implemented and tested with atomic arrangements stored in an NMR format created from PDB files. Experimental results are also provided to show the powerful capability of the proposed algorithm.

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.

Regrouping service sites: a genetic approach using a voronoi diagram

In this paper, we consider the problem of regrouping service sites into a smaller number of service sites called centers. Each service site is represented as a point in the plane and has service demand. We aim to group the sites so that each group has balanced service demand and the sum of distances between sites and their corresponding center is minimized. By using Voronoi diagrams, we obtain topological information among the sites and based on this, we define a mutation operator of a genetic algorithm. The experimental results show improvements in running time as well as cost optimization. We also provide a variety of empirical results by changing the relative importance of the two criteria, which involve service demand and distances, respectively.