ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Tue, 26 Sep 2023 09:36:59 GMT2023-09-26T09:36:59Z50191- Curves generated by a three-term difference algorithmhttps://scholarbank.nus.edu.sg/handle/10635/103092Title: Curves generated by a three-term difference algorithm
Authors: Qu, R.
Abstract: Algorithms for the generation of curves and surfaces play a very important role in shape design and modelling in CAD/CAM systems. In this paper, a simple three term difference algorithm is studied in detail and it is concluded that this algorithm could generate both conic curves, general monomial curves, and exponential spiral curves which interpolate the initial points. The geometric constructions of such curves and their properties are also obtained. Two shape control parameters are provided so that the shape of the generated curve can be adjusted according to requirements. An immediate generalization of the method is the generation of uniform surface data in Rd, d ≥ 3. © 1995.
Tue, 01 Aug 1995 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030921995-08-01T00:00:00Z
- Initialization and inner product computations of wavelet transform by interpolatory subdivision schemehttps://scholarbank.nus.edu.sg/handle/10635/103428Title: Initialization and inner product computations of wavelet transform by interpolatory subdivision scheme
Authors: Wang, Y.-P.; Qu, R.
Abstract: The initialization of wavelet transforms and the inner product computations of wavelets with their derivatives are very important in many applications. In this correspondence, the interpolatory subdivision scheme (ISS) is proposed to solve these problems efficiently. We introduce a general procedure to compute the exact values of derivatives of the interpolatory fundamental function and then derive a fast recursive algorithm for the realization of the initialization and inner product evaluations. Error analysis of the algorithm and its comparison with other approaches are discussed. Numerical experiments demonstrate high performance of the algorithm. Index Terms - Interpolatory subdivision scheme, wavelet transform, wavelet-Galerkin algorithm. © 1999 IEEE.
Fri, 01 Jan 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034281999-01-01T00:00:00Z
- Improved error bounds for freezing solutions of linear boundary value problemshttps://scholarbank.nus.edu.sg/handle/10635/103406Title: Improved error bounds for freezing solutions of linear boundary value problems
Authors: Qu, Ruibin; Agarwal, Ravi P.
Abstract: For the error in the freezing solutions of linear boundary value problems we obtain a bound which is sharper than that obtained recently by Shahruz and Schwartz [Appl. Math. Comput. 60 (1994) 285; Comput. Math. Appl. 28 (1994) 75]. A different freezing technique, 'global freezing', is also proposed. It is shown that this new technique is easy to implement for numerical computation of the solutions. Moreover, the corresponding solution has an error bound similar to that of the freezing method.
Thu, 01 Jan 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034061998-01-01T00:00:00Z
- Efficient approximation of minimum energy curves with interpolatory constraintshttps://scholarbank.nus.edu.sg/handle/10635/103180Title: Efficient approximation of minimum energy curves with interpolatory constraints
Authors: Qu, R.; Ye, J.
Abstract: Different methods for the approximation of a set of data points with interpolatory property and appropriate boundary conditions are investigated with respect to the exact energy value. It is found that for a given set of data points on a plane, the 6-point interpolatory subdivision method could be the best choice among the current widely used methods such as cubic splines and exponential splines due to its simplicity, locality, efficiency and most of all, its near-minimum energy property. Examples and graphics are provided to show these properties of the curves produced by the subdivision algorithm. © 2000 Elsevier Science Inc. All rights reserved.
Wed, 15 Mar 2000 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031802000-03-15T00:00:00Z
- Nonuniform corner cuttinghttps://scholarbank.nus.edu.sg/handle/10635/103633Title: Nonuniform corner cutting
Authors: Gregory, J.A.; Qu, R.
Abstract: The convergence of a nonuniform corner cutting process is investigated. It is shown that the limit curve will be differentiable provided the proportions of the corner cuts are kept within appropriate constraints.
Fri, 01 Nov 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036331996-11-01T00:00:00Z
- Solving two point boundary value problems by interpolatory subdivision algorithmshttps://scholarbank.nus.edu.sg/handle/10635/104153Title: Solving two point boundary value problems by interpolatory subdivision algorithms
Authors: Qu, R.; Agarwal, R.P.
Abstract: Smooth interpolatory subdivision algorithms for the generation of curves are used to solve two point boundary value problems. A method of collocation is formulated for linear second order two point boundary value problems. It is proved that the algorithms produce smooth continuous solutions provided the algorithms are chosen appropriately. Error estimates for uniform partitions are also investigated. Finally, some numerical examples are given to show the convergence of the algorithms.
Mon, 01 Jan 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1041531996-01-01T00:00:00Z
- Generalized conic curves and their applications in curve approximationhttps://scholarbank.nus.edu.sg/handle/10635/103325Title: Generalized conic curves and their applications in curve approximation
Authors: Qu, R.
Abstract: The appriximation properties of generalized conic curves are studied in this paper. A generalized conic curve is defined as one of the following curves or their affine and translation equivalent curves: (i) conic curves, including parabolas, hyperbolas and ellipses; (ii) generalized monomial curves, including curves of the form x=yγ, γ∈R, γ≠0,1, in the x-y Cartesian coordinate system; (iii) exponential spiral curves of the form ρ(θ{symbol})=Aeγθ{symbol}, A>0, γ≠0, in the ρ-θ{symbol} polar coordinate system. This type of curves has many important properties such as convexity, approximation property, effective numerical computation property and the subdivision property etc. Applications of these curves in both interpolation and approximations using piecewise generalized conic segment are also developed. It is shown that these generalized conic splines are very similar to the cubic polynomial splines and the best error of approximation is O(h5) or at least O(h4) in general provided appropriate procedures are used. Finally some numerical examples of interpolation and approximations with generalized conic splines are given. © 1997 Springer.
Mon, 01 Dec 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033251997-12-01T00:00:00Z
- A collocation method for solving a class of singular nonlinear two-point boundary value problemshttps://scholarbank.nus.edu.sg/handle/10635/102617Title: A collocation method for solving a class of singular nonlinear two-point boundary value problems
Authors: Qu, R.; Agarwal, R.P.
Abstract: In this paper, an iterative algorithm for solving singular nonlinear two-point boundary value problems is formulated. This method is basically a collocation method for nonlinear second-order two-point boundary value problems with singularities at either one or both of the boundary points. It is proved that the iterative algorithm converges to a smooth approximate solution of the BVP provided the boundary value problem is well posed and the algorithm is applied appropriately. Error estimates for uniform partitions are also investigated. It has been shown that, for sufficiently smooth solutions, the method produces order h4 approximations. Numerical examples are provided to show the effectiveness of the algorithm.
Tue, 07 Oct 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1026171997-10-07T00:00:00Z
- A subdivision approach to the construction of approximate solutions of boundary-value problems with deviating argumentshttps://scholarbank.nus.edu.sg/handle/10635/102768Title: A subdivision approach to the construction of approximate solutions of boundary-value problems with deviating arguments
Authors: Qu, R.; Agarwal, R.P.
Abstract: Using the ideas employed in the construction of subdivision algorithms, we offer here a high-accuracy algorithm to compute numerical solutions for two point boundary-value problems of differential equations with deviating arguments. Numerical examples are included to demonstrate the fast convergence and high accuracy of the algorithm. This paper is a further development to our previous works for solving various types of boundary-value problems. © 1998 Elsevier Science Ltd. All rights reserved.
Mon, 01 Jun 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1027681998-06-01T00:00:00Z
- An iterative scheme for solving nonlinear two point boundary value problemshttps://scholarbank.nus.edu.sg/handle/10635/102846Title: An iterative scheme for solving nonlinear two point boundary value problems
Authors: Qu, R.; Agarwal, R.P.
Abstract: In this paper, by using the ideas employed in the analysis of interpolatory subdivision algorithms for the generation of smooth curves, an iterative scheme for solving nonlinear two point boundary value problems is formulated. This method is basically a collocation method for nonlinear second order two point boundary value problems. It is proved that the iterative algorithm converges to a smooth approximate solution provided the boundary value problem is well posed and the algorithm is applied appropriately. Error estimates in the case of uniform partitions are also investigated. Some numerical examples are included to show the convergence of the proposed algorithm.
Wed, 01 Jan 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028461997-01-01T00:00:00Z
- Approximate solutions of general nonlinear boundary value problems using subdivision techniqueshttps://scholarbank.nus.edu.sg/handle/10635/102871Title: Approximate solutions of general nonlinear boundary value problems using subdivision techniques
Authors: Qu, R.
Abstract: A special class of basis functions generated by uniform subdivision algorithms is used to formulate a high accuracy algorithm for the computation of approximate solutions of general two point boundary value problems of differential equations with or without deviating arguments. This approach, which is different from the traditional finite difference or finite element method, produces non-polynomial/non-spline type, but continuous and differentiable approximate solutions to the boundary value problems provided the parameters of the algorithm are chosen appropriately. The main ideas of the method are generation of basis functions, node collocation, and boundary treatments. Numerical examples of various types of non-linear two-point boundary value problems are included to show the fast convergence and high accuracy of the algorithm. This paper is a further development of our previous work for solving linear boundary value problems and boundary value problems with deviating arguments.
Mon, 01 Dec 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028711997-12-01T00:00:00Z
- Approximation of minimum energy curveshttps://scholarbank.nus.edu.sg/handle/10635/102878Title: Approximation of minimum energy curves
Authors: Qu, R.; Ye, J.
Abstract: The problem of interpolating or approximating a given set of data points obtained empirically by measurement frequently arises in a vast number of scientific and engineering applications, for example, in the design of airplane bodies, cross sections of ship hull and turbine blades, in signal processing or even in less classical things like flow lines and moving boundaries from chemical processes. All these areas require fast, efficient, stable and flexible algorithms for smooth interpolation and approximation to such data. Given a set of empirical data points in a plane, there are quite a few methods to estimate the curve by using only these data points. In this paper, we consider using polynomial least squares approximation, polynomial interpolation, cubic spline interpolation, exponential spline interpolation and interpolatory subdivision algorithms. Through the investigation of a lot of examples, we find a 'reasonable good' fitting curve to the data. © 2000 Elsevier Science Inc. All rights reserved.
Tue, 15 Feb 2000 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028782000-02-15T00:00:00Z
- Smooth curve interpolation with generalized conicshttps://scholarbank.nus.edu.sg/handle/10635/104138Title: Smooth curve interpolation with generalized conics
Authors: Qu, R.
Abstract: Efficient algorithms for shape preserving approximation to curves and surfaces are very important in shape design and modelling in CAD/CAM systems. In this paper, a local algorithm using piecewise generalized conic segments is proposed for shape preserving curve interpolation. It is proved that there exists a smooth piecewise generalized conic curve which not only interpolates the data points, but also preserves the convexity of the data. Furthermore, if the data is strictly convex, then the interpolant could be a locally adjustable GC2 curve provided the curvatures at the data points are well determined. It is also shown that the best approximation order is script O sign(h6). An efficient algorithm for the simultaneous computation of points on the curve is derived so that the curve can be easily computed and displayed. The numerical complexity of the algorithm for computing N points on the curve is about 2N multiplications and N additions. Finally, some numerical examples with graphs are provided and comparisons with both quadratic and cubic spline interpolants are also given.
Mon, 01 Apr 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1041381996-04-01T00:00:00Z
- Smooth surface interpolation to scattered data using interpolatory subdivision algorithmshttps://scholarbank.nus.edu.sg/handle/10635/104139Title: Smooth surface interpolation to scattered data using interpolatory subdivision algorithms
Authors: Qu, R.; Agarwal, R.P.
Abstract: In this paper, a smooth interpolatory subdivision algorithm for the generation of interpolatory surfaces (GC1) over arbitrary triangulations is constructed and its convergence properties over nonuniform triangulations studied. An immediate application of this algorithm to surface interpolation to scattered data in Rn, n ≥ 3 is also studied. For uniform data, this method is a generalization of the analyses for univariate subdivision algorithms, and for nonuniform data, an extraordinary point analysis is proposed and a local subdivision matrix analysis presented. It is proved that the subdivision algorithm produces smooth surfaces over arbitrary networks provided the shape parameters of the algorithm are kept within an appropriate range. Some error estimates for both uniform and nonuniform triangulations are also investigated. Finally, three graphical examples of surface interpolations over nonuniform data are given to show the smoothing interpolating process of the algorithm.
Thu, 01 Aug 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1041391996-08-01T00:00:00Z
- Fairing of parametric cubic splineshttps://scholarbank.nus.edu.sg/handle/10635/103265Title: Fairing of parametric cubic splines
Authors: Ye, J.; Qu, R.
Abstract: The generation of smooth curves from a given set of data points is a very important problem in the field of Computer Aided Geometric Design. Cubic splines are widely used in practical applications, because of its simple mathematics and computation. However, these curves could sometimes oscillate and introduce unwanted inflexions. This paper develops an iterative method similar to Kjellander's to eliminate these oscillations and inflexions.
Wed, 01 Sep 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1032651999-09-01T00:00:00Z
- Approximation of minimum energy surfaces using optimal twistshttps://scholarbank.nus.edu.sg/handle/10635/102879Title: Approximation of minimum energy surfaces using optimal twists
Authors: Qu, R.; Ye, J.
Abstract: This paper gives a method for specifying the optimal 'twist vectors' at grid points for an interpolating surface to a rectangular network of curves. These twists are uniquely determined by minimizing an approximate energy form of the surface and can be obtained by solving a well-defined linear system.
Tue, 01 Dec 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028791998-12-01T00:00:00Z
- A generalization of cubic curves and their bezier representationshttps://scholarbank.nus.edu.sg/handle/10635/102654Title: A generalization of cubic curves and their bezier representations
Authors: Qu, R.; Gong, W.
Abstract: In this paper, the relation between difference algorithms and the representation of parametric curves is studied in detail. It is shown that stationary difference algorithms could generate a class of curves, the so-called D-curves, that are suitable in free-form curve and surface modelling and design. The corresponding D-Bezier curves are also constructed and their properties studied. This generalizes our findings in the study of a simple three-term difference algorithm in which it has been concluded that a simple three-term difference algorithm could generate both conic curves, general monomial curves, and exponential spiral curves.
Wed, 01 Jul 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1026541998-07-01T00:00:00Z
- Fast implementation of scale-space by interpolatory subdivision schemehttps://scholarbank.nus.edu.sg/handle/10635/103266Title: Fast implementation of scale-space by interpolatory subdivision scheme
Authors: Wang, Y.-P.; Qu, R.
Abstract: While the scale-space approach has been widely used in computer vision, there has been a great interest in fast implementation of scale-space filtering. In this paper, we introduce an interpolatory subdivision scheme (ISS) for this purpose. In order to extract the geometric features in a scale-space representation, discrete derivative approximations are usually needed. Hence, a general procedure is also introduced to derive exact formulae for numerical differentiation with respect to this ISS. Then, from ISS, an algorithm is derived for fast approximation of scale-space filtering. Moreover, the relationship between the ISS and the Whittaker-Shannon sampling theorem and the commonly used spline technique is discussed. As an example of the application of ISS technique, we present some examples on fast implementation of λτ-spaces as introduced by Gokmen and Jain, which encompasses various famous edge detection filters. It is shown that the ISS technique demonstrates high performance in fast implementation of the scale-space filtering and feature extraction.
Wed, 01 Sep 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1032661999-09-01T00:00:00Z
- A new approach to numerical differentiation and integrationhttps://scholarbank.nus.edu.sg/handle/10635/102688Title: A new approach to numerical differentiation and integration
Authors: Qu, R.
Abstract: Some new formulae for numerical differentiation and integration are derived by using interpolatory subdivision algorithms. These interpolatory subdivision algorithms are originally designed for the generation of smooth curves. The main advantage of these numerical formulae is that they produce better numerical results if the data comes from functions with fractal-like derivatives. The main disadvantage of these formulae is that they normally do not have the best approximation orders. By using different interpolatory subdivision algorithms, higher order approximation formulae can be obtained. Some numerical examples are given to compare these formulae with the traditional high accuracy formulae.
Fri, 01 Nov 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1026881996-11-01T00:00:00Z