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Spectral Methods of Polynomial interpolation and approximation

COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS 2025年第2期65卷 224-235页

作者： Varin, V. P. Russian Acad Sci Fed Res Ctr Keldysh Inst Appl Math Moscow 125047 Russia

The classical problem of interpolation and approximation of functions by polynomials is considered here as a special case of spectral representation of functions. We have previously developed this approach for the Legendre and Chebyshev orthogonal polynomials. Here we use fundamental Newton polynomials as basis functions. It is shown that the spectral approach has computational advantages over the divided difference method. In a number of problems, Newton and Hermite interpolations are indistinguishable with our approach and are calculated by the same formulas. Also, the computational algorithms we proposed earlier using orthogonal polynomials are transferred without changes to Newton and Hermite polynomials.

关键词： spectral methods Newton and Hermite polynomials interpolation and approximation

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Constrained least square progressive and iterative approximation (CLSPIA) for B-spline curve and surface fitting

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VISUAL COMPUTER 2024年第6期40卷 4427-4439页

作者： Chang, Qingjun Ma, Weiyin Deng, Chongyang Univ Svizzera italiana Fac Informat CH-6900 Lugano Switzerland Hangzhou Dianzi Univ Sch Sci Hangzhou 310018 Peoples R China City Univ Hong Kong Dept Mech Engn Kowloon Hong Kong Peoples R China

Combining the Lagrange multiplier method, the Uzawa algorithm, and the least square progressive and iterative approximation (LSPIA), we proposed the constrained least square progressive and iterative approximation (CLSPIA) to solve the problem of B-spline curve and surface fitting with constraint on data interpolation, i.e., computing the control points of a B-spline curve or surface which interpolates one set of input points while approximating the other set of given points. Compared with the method of solving the linear system directly, CLSPIA has some advantages as it inherits all the nice properties of LSPIA. Because of the data reuse property of LSPIA, CLSPIA reduces a great amount of computation. Using the local property of LSPIA, we can get shape preserving fitting curves by CLSPIA. CLSPIA is efficient for fitting large-scale data sets due to the fact that its computational complexity is linear to the scale of the input data. The many numerical examples in this paper show the efficiency and effectiveness of CLSPIA.

关键词： B-spline interpolation and approximation Data fitting Progressive and iterative approximation (PIA) Least square progressive and iterative approximation (LSPIA)

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approximation of fuzzy functions by fuzzy interpolating bicubic splines: 2018 CMMSE conference

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JOURNAL OF MATHEMATICAL CHEMISTRY 2019年第5期57卷 1252-1267页

作者： Gonzalez, P. Idais, H. Pasadas, M. Yasin, M. Univ Granada Dept Matemat Aplicada E-18071 Granada Spain

One of the most interesting and important techniques in applied mathematics is interpolation and approximation. It can be also shown that basic theory of fuzzy sets and operations with fuzzy numbers can be used, and result very convenient, for solving many analytical chemistry problems: as depth profile comparison or calibration with errors in signals and/or concentrations, for spectra interpretation, or even for automatic qualitative analysis or expert systems with X-ray spectroscopy. In this special context, we introduce in this paper a new fuzzy interpolation method of fuzzy data or functions, specially indicated in all these chemical applications. We will use bicubic splines (cubic splines of two variables) of class C2 as linear combinations of a basis constructed by a tensor product of univariate B-splines in each of these variables. First we establish a bicubic interpolation spline method of a given fuzzy data set or a given fuzzy bivariate function. We proof the existence of a unique solution of this problem and we show a convergence result. Finally, we test the effectiveness of this method by some numerical and graphical examples.

关键词： interpolation and approximation Fuzzy data Bicubic splines Analytical chemistry Calibration Spectroscopy

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Bandlimited Reconstruction of Multidimensional Images from Irregular Samples

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IEEE TRANSACTIONS ON IMAGE PROCESSING 2013年第10期22卷 3950-3960页

作者： Xu, Xie Ye, Wenxing Entezari, Alireza Univ Florida Dept Comp & Informat Sci & Engn Gainesville FL 32611 USA Univ Florida Dept Elect & Comp Engn Gainesville FL 32611 USA

We examine different sampling lattices and their respective bandlimited spaces for reconstruction of irregularly sampled multidimensional images. Considering an irregularly sampled dataset, we demonstrate that the non-tensor-product bandlimited approximations corresponding to the body-centered cubic and face-centered cubic lattices provide a more accurate reconstruction than the tensor-product bandlimited approximation associated with the commonly-used Cartesian lattice. Our practical algorithm uses multidimensional sinc functions that are tailored to these lattices and a regularization scheme that provides a variational framework for efficient implementation. Using a number of synthetic and real data sets we record improvements in the accuracy of reconstruction in a practical setting.

关键词： Irregular sampling sinc interpolation and approximation multidimensional signal processing

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Reconstruction of Irregularly-Sampled Volumetric Data in Efficient Box Spline Spaces

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IEEE TRANSACTIONS ON MEDICAL IMAGING 2012年第7期31卷 1472-1480页

作者： Xu, Xie Alvarado, Alexander Singh Entezari, Alireza Univ Florida Dept Comp & Informat Sci & Engn Gainesville FL 32611 USA Univ Florida Dept Elect & Comp Engn Gainesville FL 32611 USA

We present a variational framework for the reconstruction of irregularly-sampled volumetric data in, non-tensor-product, spline spaces. Motivated by the sampling-theoretic advantages of body centered cubic (BCC) lattice, this paper examines the BCC lattice and its associated box spline spaces in a variational setting. We introduce a regularization scheme for box splines that allows us to utilize the BCC lattice in a variational reconstruction framework. We demonstrate that by choosing the BCC lattice over the commonly-used Cartesian lattice, as the shift-invariant representation, one can increase the quality of signal reconstruction. Moreover, the computational cost of the reconstruction process is reduced in the BCC framework due to the smaller bandwidth of the system matrix in the box spline space compared to the corresponding tensor-product B-spline space. The improvements in accuracy are quantified numerically and visualized in our experiments with synthetic as well as real biomedical datasets.

关键词： Box splines interpolation and approximation irregular sampling volumetric reconstruction

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A data-adaptive design of a spherical basis function network for gravity field modelling

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IAG Symposium on Dynamic Planet

作者： Klees, R. Wittwer, T. Delft Univ Technol DEOS Kluyverweg 1 NL-2629 HS Delft Netherlands

ISBN: (纸本)9783540493495

A data-adaptive strategy for the design of spherical basis function (SBF) networks is presented. The strategy comprises a data-driven selection of the SBFs prior to parameter estimation and an automatic selection of the bandwidth of the SBFs. The selection procedure takes the data distribution and the data noise into account. The strategy can be applied to all interpolation and approximation problems that use SBFs. It is particularly suited for the local inversion of magnetic and gravity data acquired by terrestrial, airborne or space-borne sensors. It is shown that for inhomogeneous and scattered data distribution, the method requires significantly less basis functions than commonly used strategies, which range from template networks of SBFs (e.g. from hierarchical subdivision schemes or equal angular grids) to SBFs located below each data point or below a subset of the data points. Moreover, the system of normal equations is much better conditioned, which often make regularization superfluous. Finally, it is shown that the approximation errors are smaller than when using standard SBF networks.

关键词： spherical basis functions network design gravity field modelling interpolation and approximation

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Regular algebraic curve segments (II) - interpolation and approximation

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COMPUTER AIDED GEOMETRIC DESIGN 2000年第6期17卷 503-519页

作者： Xu, GL Bajaj, CL Chu, CI Univ Texas Dept Comp Sci Austin TX 78712 USA Chinese Acad Sci ICMSEC State Key Lab Sci & Engn Beijing Peoples R China Hong Kong Baptist Univ Dept Math Kowloon Hong Kong Peoples R China

In this paper (part two of the trilogy) we introduce three classes of reduced form D-regular algebraic curve splines and use them for interpolation and approximation of various algebraic curves. Explicit formulas for interpolation and approximation are also given in some low degree cases. (C) 2000 Elsevier Science B.V. All rights reserved.

关键词： algebraic curve discriminating family transversal family BB-form reduced form interpolation and approximation

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A-splines:: local interpolation and approximation using G^k-continuous piecewise real algebraic curves

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COMPUTER AIDED GEOMETRIC DESIGN 1999年第6期16卷 557-578页

作者： Bajaj, CL Xu, GL Univ Texas Dept Comp Sci Austin TX 78712 USA Chinese Acad Sci ICMSEC State Key Lab Sci & Engn Comp Beijing Peoples R China

We provide sufficient conditions for the Bernstein-Bezier (BB) form of an implicitly defined bivariate polynomial over a triangle, such that the zero contour of the polynomial defines a smooth and single sheeted real algebraic curve segment. We call a piecewise G(k)-continuous chain of such real algebraic curve segments in BE-form as an A-spline (short for algebraic spline). We prove that the degree n A-splines can achieve in general G(2n-3) continuity by local fitting and still have degrees of freedom to achieve local data approximation. As examples, we show how to construct locally convex cubic A-splines to interpolate and/or approximate the vertices of an arbitrary planar polygon with up to G(4) continuity, to fit discrete points and derivatives data, and approximate high degree parametric and implicitly defined curves. Additionally, we provide computable error bounds. (C) 1999 Elsevier Science B.V. All rights reserved.

关键词： algebraic curve G(k)-continuous interpolation and approximation BB-form on triangle

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Fast curve and surface generation for interactive shape design

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Computers in Industry 1982年第1-2期3卷 105-111页

作者： Strasser, Wolfgang Fachbereich Informatik Fachgebiet Graphisch-Interaktive Systeme Technishe Hochschule Darmstadt D-6100 Darmstadt Steubenplatz 12 Germany

Interactive curve and surface design using interactive graphics systems is very attractive. However the designer's expectations can be satisfied only if easily understood definition methods are available and if the system responds without recognizable delay. These requirements can be met by careful selection of curve and surface definition methods. A subclass of B-spline is found to fulfill all requirements and can easily be implemented in fast special purpose hardware for surface generation and curve interpolation. © 1982.

关键词： B-splines curve and surface generators interactive curve design interpolation and approximation

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