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.
205873026
Control point adjustment for B-spline curve approximation
Pottmann et al. propose an iterative optimization scheme for approximating a target curve with a B-spline curve based on square distance minimization, or SDM. The main advantage of SDM is that it does not need a parameterization of data points on the target curve. Starting with an initial B-spline curve, this scheme makes an active B-spline curve converge faster towards the target curve and produces a better approximating B-spline curve than existing methods relying on data point parameterization. However, SDM is sensitive to the initial B-spline curve due to its local nature of optimization. To address this, we integrate SDM with procedures for automatically adjusting both the number and locations of the control points of the active spline curve. This leads to a method that is more robust and applicable than SDM used alone. Furthermore, it is observed that the most time consuming part of SDM is the repeated computation of the foot-point on the target curve of a sample point on the active B-spline curve. In our implementation, we speed up the foot-point computation by pre-computing the distance field of the target curve using the Fast Marching Method. Experimental examples are presented to demonstrate the effectiveness of our method. Problems for further research are discussed. q 2003 Elsevier Ltd. All rights reserved.
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.
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.
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.
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.
119016
Towards automatic clustering of protein sequences
Analyzing protein sequence data becomes increasingly important recently. Most previous work on this area has mainly focused on building classification models. In this paper we investigate in the problem of automatic clustering of unlabeled protein sequences. As a widely recognized technique in statistics and computer science, clustering has been proven very useful in detecting unknown object categories and revealing hidden correlations among objects. One difficulty, that prevents clustering from being performed directly on protein sequence is the lack of an effective similarity measure that can be computed efficiently. Therefore, we propose a novel model for protein sequence cluster by exploring significant statistical properties possessed by the sequences. The concept of imprecise probabilities are introduced to the original probabilistic suffix tree to monitor the convergence of the empirical measurement and to guide the clustering process. It is demonstrated that the proposed method can successfully discover meaningful families without the necessity of learning models of different families from pre-labeled "training data".
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.
9604504
Bloom Histogram: Path Selectivity Estimation for XML Data with Updates
Cost-based XML query optimization calls for accurate estimation of the selectivity of path expressions. Some other interactive and internet applications can also benefit from such estimations. While there are a number of estimation techniques proposed in the literature, almost none of them has any guarantee on the estimation accuracy within a given space limit. In addition, most of them assume that the XML data are more or less static, i.e., with few updates. In this paper, we present a framework for XML path selectivity estimation in a dynamic context. Specifically, we propose a novel data structure, bloom histogram, to approximate XML path frequency distribution within a small space budget and to estimate the path selectivity accurately with the bloom histogram. We obtain the upper bound of its estimation error and discuss the trade-offs between the accuracy and the space limit. To support updates of bloom histograms efficiently when underlying XML data change, a dynamic summary layer is used to keep exact or more detailed XML path information. We demonstrate through our extensive experiments that the new solution can achieve significantly higher accuracy with an even smaller space than the previous methods in both static and dynamic environments.
62579240
A Selective Rendering Method for Data Visualization
Author(s): Wang, Wencheng; Wu, En-Hua; Max, Nelson | Abstract: Selective visualization is a solution for visualizing data of large size and dimensionality. In this paper a new method is proposed for effectively rendering certain chosen parts among the full set of data in terms of a colour buffer, referred to as the virtual plane, for storing intermediate results. By this method, scientists may concentrate their attention on the contents of data in which they are interested. Besides, the method could be easily integrated with all the current direct volume rendering techniques, especially progressive refinement methods and selective methods.
14611333
TAR: temporal association rules on evolving numerical attributes
Data mining has been an area of increasing interest. The association rule discovery problem in particular has been widely studied. However, there are still some unresolved problems. For example, research on mining patterns in the evolution of numerical attributes is still lacking. This is both a challenging problem and one with significant practical applications in business, science, and medicine. In this paper we present a temporal association rule model for evolving numerical attributes. Metrics for qualifying a temporal association rule include the familiar measures of support and strength used in traditional association rule mining and a new metric called density. The density metric not only gives us a way to extract the rules that best represent the data, but also provides an effective mechanism to prune the search space. An efficient algorithm is devised for mining temporal association rules, which utilizes all three thresholds (especially the strength) to prune the search space drastically. Moreover, the resulting rules are represented in a concise manner via rule sets to reduce the output size. Experimental results on real and synthetic data sets demonstrate the efficiency of our algorithm.