Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1729287, 10 pages

https://doi.org/10.1155/2017/1729287

## Accurate and Efficient Evaluation of Chebyshev Tensor Product Surface

^{1}School of Science, National University of Defense Technology, Changsha, China^{2}College of Computer, National University of Defense Technology, Changsha, China^{3}State Key Laboratory of Astronautic Dynamics, Xi’an, China

Correspondence should be addressed to Hao Jiang; nc.ude.tdun@gnaijoah

Received 10 March 2017; Accepted 3 July 2017; Published 27 September 2017

Academic Editor: Elisa Francomano

Copyright © 2017 Keshan He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A Chebyshev tensor product surface is widely used in image analysis and numerical approximation. This article illustrates an accurate evaluation for the surface in form of Chebyshev tensor product. This algorithm is based on the application of error-free transformations to improve the traditional Clenshaw Chebyshev tensor product algorithm. Our error analysis shows that the error bound is in contrast to classic scheme , where is working precision and is a condition number of bivariate polynomial , which means that the accuracy of the computed result is similar to that produced by classical approach with twice working precision. Numerical experiments verify that the proposed algorithm is stable and efficient.

#### 1. Introduction

Chebyshev polynomials have been extended to almost all mathematical and physical discipline, including spectral methods, approximation theory, and representation of potentials [1–4]. Bivariate Chebyshev polynomials have gained attention of the computer vision researchers [5, 6]. Over the years, researchers have focused on the implementation of Chebyshev tensor product series in image analysis [5–7]. The Chebyshev tensor product series can be used to approximate an image, which is essentially regarded as a two-dimensional spatial function [8]. Two separable univariate Chebyshev polynomials that are discrete and orthogonal can approximate two-dimensional signal. Mukundan et al. [5] introduce a new discrete Chebyshev tensor product based on Chebyshev polynomials. This discrete Chebeshev tensor product functions show the effectiveness as feature descriptors. Rahmalan et al. [6] propose a novel approach based on discrete orthogonal Chebyshev tensor product for an efficient image compression. Recently, Omar et al. [7] propose a novel method for fusing images using Chebyshev tensor product series. All above need an image reconstruction from a finite Chebyshev tensor product surface. Thus, developing fast and reliable algorithms to evaluate the Chebyshev tensor product series are of challenging interest [9]. The Clenshaw tensor product algorithm () [10] is one of algorithms that are used to evaluate Chebyshev tensor product series.

In order to get a high-precision approximation of an image, it is essential to evaluate the series accurately. Particularly, we require higher level of accurate numeric results for ill-conditioned cases. Li et al.’s double-double [11] (double-double numbers are represented as an unevaluated sum of a leading double and a trailing double) is a library used to improve the accuracy of numerical computation. However, the algorithm is time-consuming when an input image becomes larger.

Error-free transformation studied by Ogita et al. [12] is another direct possible method to improve the accuracy apart from increasing the working precision. Compensated algorithms to evaluate the univariate polynomials in different basis have been proposed in [12–15]. Inspired by their work, we extend the univariate compensated algorithm to tensor product case using the compensated Clenshaw algorithm [16] for evaluation of Chebyshev series [15]. We perform a compensated Clenshaw tensor product algorithm (for simplicity we denote it by CompCTP algorithm) to evaluate the polynomials expressed in Chebyshev tensor product form. The proposed algorithm produces the same accuracy as using twice working precision.

Since the image is fundamentally treated as two-dimensional spatial function, we use general two-dimensional function to illustrate our algorithm in the sequel. This paper has the following layout. In Section 2, we introduce some preliminaries and basic algorithms underlying our algorithm. In Section 3, we propose the compensated algorithm to compute surface in form of Chebyshev tensor product. In Section 4, we analyze forward error bound of the algorithm. In Section 5, a series of numerical experiments illustrate the accuracy and efficiency of the proposed algorithm.

#### 2. Preliminaries and Error-Free Transformations

##### 2.1. Basic Notations

At the present time, IEEE 64-bit floating arithmetic standard is implemented, which is sufficiently accurate for most scientific applications. Throughout the paper, we presume that all the computations are performed using IEEE-754 [17] standard in double precision so that neither overflow nor underflow occurs.

We assume that the computations are produced in a floating-point arithmetic which yields the modelswhere and is the working precision. For brevity we denote , . Besides, we denote [18] and use and as the computed element of .

Finally, we recall the Chebyshev polynomials and analogous Chebyshev polynomials [15, 19]. The forms of Chebyshev polynomials and analogous Chebyshev polynomials definite in the interval with three term recurrence are shown in (2) and (3), respectively.

##### 2.2. Error-Free Transformations

Rounding errors are an unavoidable consequence of working in finite precision arithmetic [18]. Error-free transformations (EFTs) are a technology of the floating-point operation , which transforms any pair of floating-point numbers into a new pair with and to obtain an accurate result. Two algorithms of EFTs are Donald et al.’s [20] (compensated summation of two floating-point numbers) and Dekker’s algorithm [21] (compensated product of two floating-point numbers).

The Clenshaw algorithm [16] is a recursive method to compute a linear combination of Chebyshev series . Reviewing work [15], the forward error bound of Clenshaw algorithm satisfies (4).

Theorem 1 (see [15]). *Let be a polynomial at point and Clenshaw denote the numerical result of Clenshaw algorithm; then*

Combining EFTs with Clenshaw algorithm, [15] proposes a compensated Clenshaw algorithm ( in Algorithms 3–7) to evaluate univariate finite Chebyshev series. The algorithm shows a smaller forward error bound than Clenshaw algorithm.

Theorem 2 (see [15]). *Let be a finite Chebyshev series. The forward error bound of compensated Clenshaw algorithm ()verifies*

Since the element is , comparing to (4), is more stable to get an accurate result.

#### 3. Accurate Algorithm to Evaluate Chebyshev Tensor Product Surface

In this section, we perform a compensated algorithm to evaluate Chebyshev tensor product series based on EFTs. The technology is to extend compensated Clenshaw algorithm to polynomials expressed in Chebyshev tensor product. In order to extend Clenshaw algorithm to tensor product case, we express the series asTherefore, we write Clenshaw tensor product algorithm to evaluate the Chebyshev tensor product series with a nested Clenshaw algorithm (Algorithm 1).