15982467
On intrinsic representations of 3D polygons for shape blending
Abstract An intrinsic representation of a geometric object comprises a set of parameters that are invariant under Euclidean transformations and determine the shape and size of the object uniquely. Unlike the case of a two-dimensional polygon which has essentially only one intrinsic representation, a three-dimensional (3D) polygon allows several variations. We study three different intrinsic representations of a 3D polygon, focusing on properties regarding their applications in shape blending. We show that only one of these representations, which is a natural extension to the intrinsic functions of a 3D differentiable curve, is suitable for shape blending, while the others lead to discontinuous shape transformation. Detailed discussions about the evaluation criteria, numerical sensitivity, and an inherent difficulty in shape blending of 3D polygons are also presented.
14471402
Computing quadric surface intersections based on an analysis of plane cubic curves
Computing the intersection curve of two quadrics is a fundamental problem in computer graphics and solid modeling. We present an algebraic method for classifying and parameterizing the intersection curve of two quadric surfaces. The method is based on the observation that the intersection curve of two quadrics is birationally related to a plane cubic curve. In the method this plane cubic curve is computed first and the intersection curve of the two quadrics is then found by transforming the cubic curve by a rational quadratic mapping. Topological classification and parameterization of the intersection curve are achieved by invoking results from algebraic geometry on plane cubic curves. 2003 Elsevier Science (USA). All rights reserved.
45794206
Rational Quadratic Parameterizations of Quadrics
Every irreducible quadric in E3 has infinitely many different rational quadratic parameterizations. These parameterizations and the relationships between them are investigated. It is shown that every faithful rational quadratic parameterization of a quadric can be generated by a stereographic projection from a point on the quadric, called the center of projection (COP). Two such parameterizations for the same quadric are related by a rational linear reparameterization if they have the same COP; otherwise they are related by a rational quadratic reparameterization. We also consider unfaithful parameterizations for which, in general, a one-to-one correspondence between points on the surface and parameters in the plane does not exist. It is shown that all unfaithful rational quadratic parameterizations of a properly degenerate quadric can be characterized by a simple canonical form, and there exist no unfaithful rational quadratic parameterizations for a nondegenerate quadric. In addition, given a faithful rational quadratic parameterization of a quadric, a new technique is presented to compute its base points and inversion formula. These results are applied to solve the problems of parameterizing the intersection of two quadrics and reparameterizing a given quadric parameterization with respect to a different COP without implicitization.
2992545
Constrained 3D shape reconstruction using a combination of surface fitting and registration
We investigate 3D shape reconstruction from measurement data in the presence of constraints. The constraints may fix the surface type or set geometric relations between parts of an object's surface, such as orthogonality, parallelity and others. It is proposed to use a combination of surface fitting and registration within the geometric optimization framework of squared distance minimization (SDM). In this way, we obtain a quasi-Newton like optimization algorithm, which in each iteration simultaneously registers the data set with a rigid motion to the fitting surface and adapts the shape of the fitting surface. We present examples to show the applicability of our method to constrained 3D shape fitting for reverse engineering of CAD models and to high accuracy fitting with kinematic surfaces, which include surfaces of revolution (reconstructed from fragments of archeological pottery) and spiral surfaces, which are fitted to 3D measurement data of shells. Our optimization algorithm can combine registration of multiple scans of an object and model fitting into a single optimization process which is shown to be superior to the traditional procedure, which first registers the data and then fits a model to it.
7101460
On the Difference Method for Drawing Conic Arcs
Abstract The difference method is an efficient way to generate an inscribed polygon of a conic section. We address the implementation issues of the difference method, that is, how to choose the difference parameter and initial points so that the inscribed polygon has a minimal number of line segments and is within a given tolerance of the conic arc. We also discuss how the difference method can be used to draw conic arcs represented by rational quadratic Bezier curves.
16675497
Revisiting the [mu]-basis of a rational ruled surface
The µ-basis of a rational ruled surface P(s,t) = P0(s) + tP1(s) is defined in Chen et al. (Comput. Aided Geom. Design 18 (2001) 61) to consist of two polynomials p(x, y, z, s) and q(x, y, z, s) that are linear in x, y, z. It is shown there that the resultant of p and q with respect to s gives the implicit equation of the rational ruled surface; however, the parametric equation P(s, t) of the rational ruled surface cannot be recovered from p and q. Furthermore, the µ-basis thus defined for a rational ruled surface does not possess many nice properties that hold for the µ-basis of a rational planar curve (Comput. Aided Geom. Design 18 (1998) 803). In this paper, we introduce another polynomial r(x, y, z, s, t) that is linear in x, y, z and t such that p, q, r can be used to recover the parametric equation P(s, t) of the rational ruled surface; hence, we redefine the µ-basis to consist of the three polynomials p, q, r. We present an efficient algorithm for computing the newly-defined µ-basis, and derive some of its properties. In particular, we show that the new µ-basis serves as a basis for both the moving plane module and the moving plane ideal corresponding to the rational ruled surface.
1166455
Comparing Graph Representations of Protein Structure for Mining Family-Specific Residue-Based Packing Motifs
We find recurring amino-acid residue packing patterns, or spatial motifs, that are characteristic of protein structural families, by applying a novel frequent subgraph mining algorithm to graph representations of protein three-dimensional structure. Graph nodes represent amino acids, and edges are chosen in one of three ways: first, using a threshold for contact distance between residues; second, using Delaunay tessellation; and third, using the recently developed almost-Delaunay edges. For a set of graphs representing a protein family from the Structural Classification of Proteins (SCOP) database, subgraph mining typically identifies several hundred common subgraphs corresponding to spatial motifs that are frequently found in proteins in the family but rarely found outside of it. We find that some of the large motifs map onto known functional regions in two protein families explored in this study, i.e., serine proteases and kinases. We find that graphs based on almost-Delaunay edges significantly reduce the number of edges in the graph representation and hence present computational advantage, yet the patterns extracted from such graphs have a biological interpretation approximately equivalent to that of those extracted from distance based graphs.
884789
A Fast Algorithm for Approximate Quantiles in High Speed Data Streams
We present a fast algorithm for computing approximate quantiles in high speed data streams with deterministic error bounds. For data streams of size N where N is unknown in advance, our algorithm partitions the stream into sub-streams of exponentially increasing size as they arrive. For each sub-stream which has a fixed size, we compute and maintain a multi-level summary structure using a novel algorithm. In order to achieve high speed performance, the algorithm uses simple block-wise merge and sample operations. Overall, our algorithms for fixed-size streams and arbitrary-size streams have a computational cost of O(N log(1/epsivlogepsivN)) and an average per-element update cost of O(log logN) if epsiv is fixed.
3845821
Performance Analysis of Three Text-Join Algorithms
When a multidatabase system contains textual database systems (i.e., information retrieval systems), queries against the global schema of the multidatabase system may contain a new type of joins-joins between attributes of textual type. Three algorithms for processing such a type of joins are presented and their I/O costs are analyzed in this paper. Since such a type of joins often involves document collections of very large size, it is very important to find efficient algorithms to process them. The three algorithms differ on whether the documents themselves or the inverted files on the documents are used to process the join. Our analysis and the simulation results indicate that the relative performance of these algorithms depends on the input document collections, system characteristics, and the input query. For each algorithm, the type of input document collections with which the algorithm is likely to perform well is identified. An integrated algorithm that automatically selects the best algorithm to use is also proposed.
29035045
Genetic algorithms for determining fuzzy measures from data
A synthetic evaluation of a given object in terms of multiple factors that contribute to some feature of the object (quality, performance, etc.) may be regarded as a system with multiple inputs and one output. Traditionally, the output is expressed as the weighted average of the inputs. Unfortunately, this method is severely limited as it cannot capture any inherent relation among the factors involved. This limitation can be overcome by using the Choquet integral or the fuzzy integral with respect to a fuzzy measure that captures the relation among the factors. The crux of this method is to determine the right fuzzy measure. In this paper, we describe an efficient genetic algorithm for constructing a suitable fuzzy measure from relevant input-output data. This algorithm has a broad applicability in various problem areas, such as decision making, cluster analysis, pattern recognition, image and speech processing, and expert systems.
8353152
An Efficient Algorithm for Mining Coherent Patterns from Heterogeneous Microarrays
DNA microarray techniques present a novel way for geneticists to monitor interactions among tens of thousands of genes simultaneously, and have become standard lab routines in gene discovery, disease diagnosis, and drug design. There has been extensive research on coherent subspace clustering of gene expressions measured under consistent experimental settings. This implies that all experiments are run using the same batch of microarray chips with similar characteristics of noise. Algorithms developed under this assumption may not be applicable for analyzing data collected from heterogeneous settings, where the set of genes being monitored may be different and expression levels may be not directly comparable even for the same gene. In this paper, we propose a model, F-cluster, for mining subspace coherent patterns from heterogeneous gene expression data, which is shown effective for revealing truthful patterns and reducing spurious ones. We also develop an efficient and scalable hybrid approach that combines gene-pair based and sample-pair based pruning to generate F-clusters from multiple gene expression matrices simultaneously. The experimental results demonstrate that our model can discover significant clusters that may not be identified by previous models.