Bivariate Thiele-Like Rational Interpolation Continued Fractions with Parameters Based on Virtual Points

Abstract

The interpolation of Thiele-type continued fractions is thought of as the traditional rational interpolation and plays a significant role in numerical analysis and image interpolation. Different to the classical method, a novel type of bivariate Thiele-like rational interpolation continued fractions with parameters is proposed to efficiently address the interpolation problem. Firstly, the multiplicity of the points is adjusted strategically. Secondly, bivariate Thiele-like rational interpolation continued fractions with parameters is developed. We also discuss the interpolant algorithm, theorem, and dual interpolation of the proposed interpolation method. Many interpolation functions can be gained through adjusting the parameter, which is flexible and convenient. We also demonstrate that the novel interpolation function can deal with the interpolation problems that inverse differences do not exist or that there are unattainable points appearing in classical Thiele-type continued fractions interpolation. Through the selection of proper parameters, the value of the interpolation function can be changed at any point in the interpolant region under unaltered interpolant data. Numerical examples are given to show that the developed methods achieve state-of-the-art performance.

Rational Bi-Quartic Spline With Six Parameters for Surface Interpolation With Application in Image Enlargement

“…In contrast, our proposed scheme is easy to be implemented via efficient algorithm. Therefore, the main finding in this study has more merit than the work of Zou et al [25] and Tang and Zhu [26] for the following reasons:…”

“…PSNR value is used as measurement to determine the effectiveness of the proposed scheme. However, Zou et al [25] also have discussed image interpolation using bivariate Thiele-Like rational continued fractional. They have tested their scheme to image interpolation of Lena's grayscale image.…”

“…Therefore, our proposed scheme is easier to be implemented than wavelets. (g) In Zhou et al [25] the bivariate Thiele-Like rational interpolation continued fractions is used for image interpolation. Meanwhile in Tang and Zhu [26] a new version of C 1 -continuous Coons patches over triangular domain is constructed for the application in image interpolation.…”

A Nonstationary Ternary 4-Point Shape-Preserving Subdivision Scheme

“…Ashraf et al analyzed the ternary four-point rational interpolating subdivision scheme's geometric properties [18,19]. More subdivision schemes are studied in [17,[20][21][22].…”

Analysis of Geometric Properties of Ternary Four-Point Rational Interpolating Subdivision Scheme

“…Zulkifli et al [18] discussed the application of new rational bi-cubic Ball function with six parameters in image interpolation, especially for the gray scale image. For more recent work on SS one may refer to References [19][20][21][22][23].…”