14919460
Interpolation on quadric surfaces with rational quadratic spline curves
Abstract Given a sequence of points { X i } i =1 n on a regular quadric S : X T AX = 0 ⊂ E d , d ⩾ 3, we study the problem of constructing a G 1 rational quadratic spline curve lying on S that interpolates { X i } i =1 n . It is shown that a necessary condition for the existence of a nontrivial interpolant is ( X 1 T AX 2 )( X i T AX i +1 ) > 0, i = 1,2…, n − 1. Also considered is a Hermite interpolation problem on the quadric S : a biarc consisting of two conic arcs on S joined with G 1 continuity is used to interpolate two points on S and two associated tangent directions, a method similar to the biarc scheme in the plane (Bolton, 1975) or space (Sharrock, 1987). A necessary and sufficient condition is obtained on the existence of a biarc whose two arcs are not major elliptic arcs. In addition, it is shown that this condition is always fulfilled on a sphere for generic interpolation data.
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.
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.
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.
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.
62615149
Mining Asynchronous Periodic Patterns in Time Series Data
Periodicy detection in time series data is a challenging problem of great importance in many applications. Most previous work focused on mining synchronous periodic patterns and did not recognize the misaligned presence of a pattern due to the intervention of random noise. In this paper, we propose a more flexible model of asynchronous periodic pattern that may be present only within a subsequence and whose occurrences may be shifted due to disturbance. Two parameters min/spl I.bar/rep and max/spl I.bar/dis are employed to specify the minimum number of repetitions that is required within each segment of nondisrupted pattern occurrences and the maximum allowed disturbance between any two successive valid segments. Upon satisfying these two requirements, the longest valid subsequence of a pattern is returned. A two-phase algorithm is devised to first generate potential periods by distance-based pruning followed by an iterative procedure to derive and validate candidate patterns and locate the longest valid subsequence. We also show that this algorithm cannot only provide linear time complexity with respect to the length of the sequence but also achieve space efficiency.
5467929
Some sufficient conditions for a planar graph of maximum degree six to be Class 1
Let G be a planar graph of maximum degree 6. In this paper we prove that if G does not contain either a 6-cycle, or a 4-cycle with a chord, or a 5- and 6-cycle with a chord, then @g^'(G)=6, where @g^'(G) denotes the chromatic index of G.
2548187
Mining asynchronous periodic patterns in time series data
Periodicy detection in time series data is a challenging problem of great importance in many applications. Most previous work focused on mining synchronous periodic patterns and did not recognize the misaligned presence of a pattern due to the intervention of random noise. In this paper, we propose a more flexible model of asynchronous periodic pattern that may be present only within a subsequence and whose occurrences may be shifted due to disturbance. Two parameters min rep and max dis are employed to specify the minimum number of repetitions that is required within each segment of nondisrupted pattern occurrences and the maximum allowed disturbance between any two successive valid segments. Upon satisfying these two requirements, the longest valid subsequence of a pattern is returned. A two-phase algorithm is devised to first generate potential periods by distance-based pruning followed by an iterative procedure to derive and validate candidate patterns and locate the longest valid subsequence. We also show that this algorithm cannot only provide linear time complexity with respect to the length of the sequence but also achieve space
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".
34993492
Microarray image compression: SLOCO and the effect of information loss
Microarray image technology is a powerful tool for monitoring the expression of thousands of genes simultaneously. Each microarray experiment produces immense amounts of image data, and efficient storage and transmission require compression that takes advantage of microarray image structure. In this paper we develop a compression scheme for microarray images which can be either lossless or lossy with successive refinements. Existing measures of distortion such as mean squared pixel-wise error and visual fidelity are not appropriate for microarray images. We introduce a new measure of distortion for lossy compression: the sensitivity of microarray information extraction to compression loss. Furthermore, our scheme has a coded data structure that allows fast decoding and reprocessing of image sub-blocks, and includes summary statistics and image segmentation information. The average lossless compression ratio is 1.83:1 for our cDNA test images and 2.43:1 for our inkjet test images, comparable or better than state-of-the-art lossless schemas, yet with additional structure and information. At an average lossy compression ratio of 8:1 for cDNA microarrays, we find that our scheme minimizes the effects of compression loss compared to other algorithms. We show that the variability in differential gene expression levels extracted from lossily vs. losslessly compressed microarray images is less than both the variability between different arrays and the variability between different extraction algorithms. In fact, lossy compression can improve the estimation of gene expression levels for cDNA images.