Positivity and Stability of Rational Cubic Fractal Interpolation Surfaces

Fractal interpolation has become popular in 2D and 3D data visualization to model a wide variety of physical phenomena. The purpose of this paper is to present a promising new class of positivity preserving C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}^1$$\end{document}-rational cubic spline fractal interpolation surfaces (FISs) to stitch surface data arranged over a rectangular grid. We develop the rational cubic FIS as a combination of blending functions and the rational fractal boundary curves that are constructed along the grid lines of the interpolation domain. We investigate the stability results of the rational cubic FIS with respect to its independent and dependent variables, and the corresponding first order partial derivatives at knots. We identify the scaling factors and shape parameters so that the positivity feature of given data along the grid lines is translated to the corresponding rational fractal boundary curves. In particular, our rational cubic FIS is positive whenever the corresponding fractal boundary curves are positive. Numerical examples are provided to illustrate: (i) the construction of positive rational cubic FISs for a given positive surface data, (ii) the effects on the shape of the positive rational cubic FIS due to the changes in the scaling factors or/and shape parameters, and (iii) the stability results of the proposed rational cubic FIS.