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International Conference on Cyberworlds (CW)

作者： Iglesias, Andres Galvez, Akemi Collantes, Marta Univ Cantabria ETSI Caminos Canales & Puertos Dept Appl Math & Computat Sci Avda Castros S-N E-39005 Santander Spain Toho Univ Fac Sci Dept Informat Sci Funabashi Chiba Japan

ISBN: (纸本)9781467394031

A very important issue in many applied fields is to construct the fitting curve that approximates a given set of data points optimally in the sense of least-squares. This problem arises in a number of areas, such as computer-aided design and manufacturing (CAD/CAM), virtual reality, medical imaging, computer graphics, computer animation, and many others. Since the problem consists of minimizing the least squares fitting error, it can be formulated as an optimization problem. Unfortunately, it is also a highly nonlinear, over determined, multivariate continuous optimization problem. As a consequence, classical mathematical methods cannot solve it in its generality. Clearly, there is a need for more general techniques to tackle this issue. A critical step in this process is to obtain a suitable parameterization of the data points. In this context, this paper introduces a new method to obtain an optimal solution for the parameterization problem of the least-squares fitting Bezier curve. Our approach is based on a powerful nature-inspired optimization method called bat algorithm, which has been recently introduced to solve hard continuous optimization problems. In spite of these remarkable features for optimization, the bat algorithm has never been applied in the context of data fitting for geometric modeling or computer graphics. To analyze the performance of this approach, it has been applied to some simple yet illustrative examples of Bezier curves with satisfactory results.

关键词： Nature-inspired techniques bat algorithm data fitting reverse engineering curve parameterization Bezier curve

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Four Adaptive Memetic Bat Algorithm Schemes for Bezier curve parameterization

Four Adaptive Memetic Bat Algorithm Schemes for Bezier Curve...

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15th International Conference on Cyberworlds

作者： Iglesias, Andres Galvez, Akemi Collantes, Marta Univ Cantabria Dept Appl Math & Computat Sci Avda Castros S-N E-39005 Santander Spain Toho Univ Fac Sci Dept Informat Sci Narashino Campus2-2-1 Miyama Funabashi Chiba 2748510 Japan

ISBN: (纸本)9783662530900;9783662530894

This paper is an extension of a previous one presented at the conference Cyberworlds 2015. In that work we addressed the problem to fit a given set of data points in the least-square sense by using a polynomial Bezier curve. This problem arises in many scientific and industrial domains, such as numerical analysis, statistical regression, computer-aided design and manufacturing, computer graphics, virtual reality, etc. A critical issue to address this least-squares minimization problem is that of curve parameterization. In our previous work we solve it by applying a powerful nature-inspired optimization method called the bat algorithm. Although we obtained pretty good results on a number of examples, the method can still be further improved by considering a memetic approach, in which the global search bat algorithm is hybridized with a local search procedure to enhance the exploitation phase of the minimization process. In this paper we extend our previous method through two local search strategies: Luus-Jaakola and ASSRS. In both cases, the adaptive and self-adaptive versions are considered, leading to four memetic schemes. A comparative analysis of our results on the previous benchmark for these four memetic schemes and our previous method has been carried out. It shows that the memetic approaches improve the efficiency of the previous method at different extent for all instances in our benchmark.

关键词： Memetic approach Local search procedure Data fitting Bat algorithm curve parameterization Self-adaptive approach

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parameterization of planar curves immersed in triangulations with application to finite elements

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2011年第6期88卷 556-585页

作者： Rangarajan, Ramsharan Lew, Adrian J. Stanford Univ Dept Mech Engn Stanford CA 94305 USA

We construct a method for the parameterization of a class of planar piecewise C-2-curves over a collection of edges in an ambient triangulation. The map from the collection of edges to the curve is the closest-point projection. A distinguishing feature of the method is that edges in the ambient triangulation need not interpolate the curve. We formulate conditions on the ambient triangulations so that the resulting parameterization over its selected edges is (i) bijective, (ii) maps simple, connected collection of edges to simple, connected components of the curve, and (iii) is C-1 within each edge of the collection. These properties of the parameterization make it particularly useful in the construction of high-order finite element approximation spaces on planar curves immersed in triangulations. We discuss this application and illustrate it with numerical examples. The parameterization method applies to a large class of planar curves, including most ones of interest in engineering and computer graphics applications, and to a large family of triangulations, including acute-angled triangulations. Copyright (C) 2011 John Wiley & Sons, Ltd.

关键词： immersed boundary curve parameterization closest-point projection high-order finite elements

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ANALYSIS OF A METHOD TO PARAMETERIZE PLANAR curveS IMMERSED IN TRIANGULATIONS

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2013年第3期51卷 1392-1420页

作者： Rangarajan, Ramsharan Lew, Adrian J. Stanford Univ Dept Mech Engn Stanford CA 94305 USA

We prove that a planar C-2-regular boundary Gamma can always be parameterized with its closest point projection pi over a certain collection of edges Gamma(h) in an ambient triangulation by making simple assumptions on the background mesh. For Gamma(h), we select the edges that have both vertices on one side of Gamma and belong to a triangle that has a vertex on the other side. By imposing restrictions on the size of triangles near the curve and by requesting that certain angles in the mesh be strictly acute, we prove that pi : Gamma(h) -> Gamma is a homeomorphism and that it is C-1 on each edge in Gamma(h), and we provide bounds for the Jacobian of the parameterization. The assumptions on the background mesh are both easy to satisfy in practice and conveniently verified in computer implementations. The parameterization analyzed here was previously proposed by the authors and applied to the construction of high-order curved finite elements on a class of planar piecewise C-2-curves.

关键词： curve parameterization closest point projection curved finite elements

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Chordal cubic spline interpolation is fourth-order accurate

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IMA JOURNAL OF NUMERICAL ANALYSIS 2006年第1期26卷 25-33页

作者： Floater, MS Univ Oslo Dept Informat Ctr Math Applicat N-0316 Oslo Norway

It is well known that complete cubic spline interpolation of functions with four continuous derivatives is fourth-order accurate. In this paper we show that this kind of interpolation, when used to construct parametric spline curves through sequences of points in any space dimension, is again fourth-order accurate if the parameter intervals are chosen by chord length. We also show how such chordal spline interpolants can be used to approximate the arc-length derivatives of a curve and its length.

关键词： curve parameterization arc length spline interpolation approximation order

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A complete, exact and efficient implementation for computing the edge-adjacency graph of an arrangement of quadrics

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JOURNAL OF SYMBOLIC COMPUTATION 2011年第4期46卷 467-494页

作者： Hemmer, Michael Dupont, Laurent Petitjean, Sylvain Schoemer, Elmar Max Planck Inst Informat Saarbrucken Germany Nancy Univ LORIA Nancy France LORIA INRIA Nancy Grand Est Nancy France Johannes Gutenberg Univ Mainz Inst Informat Mainz Germany

We present a complete, exact and efficient implementation to compute the edge-adjacency graph of an arrangement of quadrics, i.e. surfaces of algebraic degree 2. This is a major step towards the computation of the full 3D arrangement. We enhanced an implementation for an exact parameterization of the intersection curves of two quadrics, such that we can compute the exact parameter value for intersection points and from that the edge-adjacency graph of the arrangement. Our implementation is complete in the sense that it can handle all kinds of inputs including all degenerate ones, i.e. singularities or tangential intersection points. It is exact in that it always computes the mathematically correct result. It is efficient measured in running times, i.e. it compares favorably to the only previous implementation. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： Arrangement Intersection of surfaces Quadrics Pencils of quadrics curve parameterization

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Near-optimal parameterization of the intersection of quadrics: III. Parameterizing singular intersections

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JOURNAL OF SYMBOLIC COMPUTATION 2008年第3期43卷 216-232页

作者： Dupont, Laurent Lazard, Daniel Lazard, Sylvain Petitjean, Sylvain Univ Nancy CNRS INRIA LORIA F-54602 Nancy France VEGAS Project INRIA Nancy Grand E F-54602 Nancy France Univ Paris 06 CNRS LIP6 F-75016 Paris France INRIA Paris Rocquencourt F-75016 Paris France

We conclude, in this third part, the presentation of an algorithm for computing an exact and proper parameterization of the intersection of two quadrics. The coordinate functions of the parameterizations in projective space are polynomial, whenever it is possible. They are also near-optimal in the sense that the number of distinct square roots appearing in the coefficients of these functions is minimal except in a small number of cases (characterized by the real type of the intersection) where there may be an extra square root. Our algorithm builds on the classification of pencils of quadrics of P-3 (R) over the reals presented in Part II and the type-detection algorithm that we deduced from this classification. Moreover, since the algorithm presented in Part I is near-optimal when the intersection is a non-singular quartic, we focus here on the case where the intersection is singular and present, for all possible real types of intersection, algorithms for computing near-optimal rational parameterizations. We also give examples covering all the possible situations, in terms of both the real type of intersection and the number and depth of square roots appearing in the coefficients of the parameterizations. (c) 2007 Elsevier Ltd. All rights reserved.

关键词： intersection of surfaces pencils of quadrics curve parameterization singular intersections

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curve fitting when the curve may not be a function

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JOURNAL OF INTERDISCIPLINARY MATHEMATICS 2006年第3期9卷 551-568页

作者： Pandit, S. N. N. Ramamurthy, S. Gandhi, B. Krishna Osmania Univ Ctr Quantitat Methods Hyderabad Andhra Pradesh India Gokaraju Rangaraju Inst Engn & Technol Hyderabad Andhra Pradesh India Jawaharlal Nehru Technol Univ Dept Math Hyderabad Andhra Pradesh India

A function in general is a many-to-one mapping of, say, x on y. However, sometimes, curves may not be acceptable as functions in this sense: non-monotonic and closed curves are examples of this type. If points on such curves are given, the usual curve fitting methods cannot work. In such cases, parametric form of curve representation appears the only way out. However, this approach can be used only when a parameter suitable for the problem can be introduced and this itself can be a very challenging task, particularly when the genesis of the data is unknown. Another aspect of the problem is to induce an ordering among the data points. In this paper, we report a novel method in which we achieve both the goals;inducing an ordering among the points and defining a parameter for the ordered points to enable a good enough approximation of the given data. Our parameterization scheme appears to be working fairly satisfactorily, albeit for one class of curves. At the heart of this work are the twin matrix operations: MINMAXION and MINADDITION.

关键词： curve approximation MINMAXION MINADDITION ordering of points curve parameterization

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Near-optimal parameterization of the intersection of quadrics: II. A classification of pencils

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JOURNAL OF SYMBOLIC COMPUTATION 2008年第3期43卷 192-215页

作者： Dupont, Laurent Lazard, Daniel Lazard, Sylvain Petitjean, Sylvain Univ Nancy CNRS INRIA LORIA F-54602 Nancy France VEGAS Project INRIA Nancy Grand E F-54602 Nancy France Univ Paris 06 LIP6 F-75016 Paris France SALSA Project INRIA Paris Rocquencourt F-75016 Paris France

We present here the first classification of pencils of quadrics based on the type of their intersection in real projective space and we show how this classification can be used to compute efficiently the type of the real intersection. This classification is at the core of the design of the algorithms, presented in Part III, for computing, in all cases of singular intersection, a near-optimal parameterization with polynomial functions, that is a parameterization in projective space whose coordinate functions are polynomial and such that the number of distinct square roots appearing in the coefficients is at most one away from the minimum. (c) 2007 Elsevier Ltd. All rights reserved.

关键词： intersection of surfaces pencils of quadrics classification curve parameterization

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Near-optimal parameterization of the intersection of quadrics: I. The generic algorithm

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JOURNAL OF SYMBOLIC COMPUTATION 2008年第3期43卷 168-191页

作者： Dupont, Laurent Lazard, Daniel Lazard, Sylvain Petitjean, Sylvain Univ Nancy CNRS INRIA LORIA F-54602 Nancy France VEGAS Project INRIA Nancy Grand E F-54602 Nancy France Univ Paris 06 LIP6 F-75016 Paris France SALSA Project INRIA Paris Rocquencourt F-75016 Paris France

We present an exact and efficient algorithm for computing a proper parametric representation of the intersection of two quadrics in three-dimensional real space given by implicit equations with rational coefficients. The output functions parameterizing the intersection in projective space are polynomial, whenever it is possible, which is the case when the intersection is not a smooth quartic (for example, a singular quartic, a cubic and a line, and two conics). Furthermore, the parameterization is near-optimal in the sense that the number of distinct square roots appearing in the coefficients of these functions is minimal, except in a small number of well-identified cases where there may be an extra square root. In addition, the algorithm is practical: a complete and efficient C++ implementation is described in Lazard et al. [Lazard, S., Pefiaranda, L.M., Petitjean, S., 2006. Intersecting quadrics: An efficient and exact implementation. In: 20th ACM Symposium on Computational Geometry, 2004. Computational Geometry: Theory and Applications 35 (1-2), 74-99 (special issue)]. In Part 1, we present an algorithm for computing a parameterization of the intersection of two arbitrary quadrics which we prove to be near-optimal in the generic, smooth quartic, case. Parts II and III treat the singular cases. We present in Part 11 the first classification of pencils of quadrics according to the real type of the intersection and we show how this classification can be used to efficiently determine the type of the real part of the intersection of two arbitrary quadrics. This classification is at the core of the design of our algorithms for computing near-optimal parameterizations of the real part of the intersection in all singular cases. We present these algorithms in Part III and give examples covering all the possible situations in terms of both the real type of intersection and the number and depth of square roots appearing in the coefficients. (c) 2007 Elsevier Ltd. All rights

关键词： intersection of surfaces quadrics pencils of quadrics curve parameterization

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