[Paper Review] Shape Preserving Rational Cubic Spline Fractal Interpolation

A. K. B. Chand, P. Viswanathan|arXiv (Cornell University)|Mar 9, 2015

Mathematical Dynamics and Fractals55 references3 citations

TL;DR

This paper introduces a new class of rational cubic spline fractal interpolation functions (FIFs) that combine shape-preserving rational splines with iterated function systems (IFS) to enable monotonicity and convexity preservation in irregular or self-similar data. By incorporating tension parameters and scaling factors, the method ensures C¹ continuity, guarantees shape preservation through solvable parameter constraints, and achieves O(h⁴) convergence for C⁴ functions, generalizing classical rational interpolation while offering enhanced flexibility for complex data.

ABSTRACT

Fractal interpolation functions (FIFs) developed through iterated function systems (IFSs) prove more versatile than classical interpolants. However, the applications of FIFs in the domain of `shape preserving interpolation' are not fully addressed so far. Among various techniques available in the classical numerical analysis, rational interpolation schemes are well suited for the shape preservation problems and shape modification analysis. In this paper, the capability of FIFs to generalize smooth classical interpolants, and the effectiveness of rational function models in shape preservation are intertwingly exploited to provide a new solution to the shape preserving interpolation problem in fractal perspective. As a common platform for these two techniques to work together, we introduce rational cubic spline FIFs involving tension parameters for the first time in literature. Suitable conditions on parameters of the associated IFS are developed so that the rational fractal interpolant inherits fundamental shape properties such as monotonicity, convexity, and positivity present in the given data. With some suitable hypotheses on the original function, the convergence analysis of the $\mathcal{C}^1$-rational cubic spline FIF is carried out. Due to the presence of the scaling factors in the rational cubic spline fractal interpolant, our approach generalizes the classical results on the shape preserving rational interpolation by Delbourgo and Gregory [SIAM J. Sci. Stat. Comput., 6 (1985), pp. 967-976]. The effectiveness of the shape preserving interpolation schemes are illustrated with suitably chosen numerical examples and graphs, which support the practical utility of our methods.

Motivation & Objective

• To address the gap in fractal interpolation methods for preserving shape properties like monotonicity and convexity in irregular or self-similar data.
• To develop a C¹-continuous rational cubic spline FIF that integrates shape parameters and scaling factors for enhanced control and computational efficiency.
• To generalize classical rational interpolation results by Delbourgo and Gregory through the inclusion of fractal parameters while maintaining convergence and shape fidelity.
• To provide a robust, flexible framework for interpolating data with varying derivative irregularity, especially in physical and engineering applications.

Proposed method

• Constructs a rational cubic spline FIF using an iterated function system (IFS) with affine transformations that include scaling factors and rational shape parameters.
• Employs a functional equation (3.3) to recursively generate the fractal interpolant, ensuring C¹ continuity across intervals.
• Incorporates tension parameters (r₁, r₂) and scaling factors (α₁, α₂) to control both the fractal dimension and the shape of the interpolant.
• Applies piecewise construction on subintervals, with adaptive node insertion where necessary to preserve monotonicity in regions with complex behavior.
• Derives sufficient conditions on scaling factors and shape parameters to ensure the interpolant inherits monotonicity and convexity from the data.
• Performs convergence analysis showing O(h⁴) error bounds for functions in C⁴(I), under appropriate parameter selection.

Experimental results

Research questions

• RQ1 Can a rational cubic spline FIF be constructed such that it preserves monotonicity and convexity of irregular or self-similar data?
• RQ2 How do scaling factors and shape parameters influence the shape preservation and convergence behavior of the fractal interpolant?
• RQ3 To what extent does the proposed FIF generalize classical rational interpolation schemes in terms of shape control and approximation accuracy?
• RQ4 What constraints on parameters ensure the existence of a range-restricted, shape-preserving fractal interpolant?
• RQ5 How does the convergence rate of the rational cubic spline FIF compare to classical methods under similar smoothness assumptions?

Key findings

• The proposed rational cubic spline FIF ensures C¹ continuity and shape preservation (monotonicity and convexity) by satisfying a finite set of solvable inequalities derived from parameter constraints.
• Sufficient conditions on scaling factors (below explicit bounds) and shape parameters (above explicit bounds) guarantee the existence of a shape-preserving interpolant.
• The method achieves O(h⁴) convergence rate for functions in C⁴(I), matching the best classical rational spline schemes.
• The inclusion of scaling factors allows the method to generalize classical rational interpolation results by Delbourgo and Gregory, extending their framework to fractal and self-similar data.
• For data with varying derivative irregularity, the proposed FIF outperforms classical counterparts in preserving shape and ensuring smoothness.
• Numerical examples demonstrate the method's effectiveness in constructing co-monotone and mixed-shape-preserving interpolants for complex, real-world data sets.