Toward a Unified Methodology for Fractal Extension of Various Shape Preserving Spline Interpolants
01 Jan 2015-pp 223-238
TL;DR: In this paper, a unified approach for the fractal generalization of various traditional nonrecursive polynomial and rational splines is presented, where the elements of the iterated function system are identified befittingly.
Abstract: Fractal interpolation, one in the long tradition of those involving the interpolatary theory of functions, is concerned with interpolation of a data set with a function whose graph is a fractal or a self-referential set. The novelty of fractal interpolants lies in their ability to model a data set with either a smooth or a nonsmooth function depending on the problem at hand. To broaden their horizons, some special class of fractal interpolants are introduced and their shape preserving aspects are investigated recently in the literature. In the current article, we provide a unified approach for the fractal generalization of various traditional nonrecursive polynomial and rational splines. To this end, first we shall view polynomial/rational FIFs as \(\alpha \)-fractal functions corresponding to the traditional nonrecursive splines. The elements of the iterated function system are identified befittingly so that the class of \(\alpha \)-fractal function \(f^\mathbf {\alpha }\) incorporates the geometric features such as positivity, monotonicity, and convexity in addition to the regularity inherent in the generating function f. This general theory in conjuction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions. Even though the results obtained in this article are generally enough, we wish to apply it on a specific rational cubic spline with two free shape parameters.
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TL;DR: The appearance of fractal interpolation function represents a revival of experimental mathematics, raised by computers and intensified by powerful evidence of its applications as mentioned in this paper, and it represents a new direction for experimental mathematics.
Abstract: The appearance of fractal interpolation function represents a revival of experimental mathematics, raised by computers and intensified by powerful evidence of its applications. This paper is devote...
8 citations
TL;DR: In this article, a new rational cubic spline is introduced with the help of the iterated function system (IFS) that contains rational functions, and the convergence analysis is established.
Abstract: In this article, a new
$$\alpha$$
-fractal rational cubic spline is introduced with the help of the iterated function system (IFS) that contains rational functions. The numerator of the rational function contains a cubic polynomial and the denominator of the rational function contains a quadratic polynomial with three shape parameters. The convergence analysis of the
$$\alpha$$
-fractal rational cubic spline is established. By restricting the scaling factors and the shape parameters, the
$$\alpha$$
-fractal rational cubic spline is constrained between two piecewise linear functions whenever interpolation data lies in between two piecewise linear functions. Also, positivity and monotonicity of the
$$\alpha$$
-fractal rational cubic spline are discussed. Numerical examples are provided to support the theoretical results.
6 citations
Posted Content•
TL;DR: This paper investigates some univariate and bivariate constrained interpolation problems using rational quartic fractal interpolation functions, which has been submitted long back in a reputed journal and revised as per the journal requirement.
Abstract: This paper investigates some univariate and bivariate constrained interpolation problems using rational quartic fractal interpolation functions, which has been submitted long back in a reputed journal and revised as per the journal requirement. This research is extension of the work [S. K. Katiyar and A. K. B. Chand, Shape Preserving Rational Quartic Fractal Functions, Fractal, in Press].
1 citations
Cites background from "Toward a Unified Methodology for Fr..."
...As a submissive contribution to this goal, Chand and coworkers have initiated the study on shape preserving fractal interpolation and approximation using various families of polynomial and rational IFSs (see, for instance, [12, 13, 14, 15, 16, 17, 18, 19])....
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01 Jan 2021
TL;DR: In this article, a general construction of fractal rational functions is introduced for the first time in the literature, which allows to insert shape parameters for positivity-preserving univariate interpolation.
Abstract: Coalescence hidden variable fractal interpolation function (CHFIF) proves more versatile than classical interpolant and fractal interpolation function (FIF). Using rational functions and CHFIF, a general construction of \(\mathbf {A}\)-fractal rational functions is introduced for the first time in the literature. This construction of \(\mathbf {A}\)-fractal rational function also allows us to insert shape parameters for positivity-preserving univariate interpolation. The convergence analysis of the proposed scheme is established. With suitably chosen numerical examples and graphs, the effectiveness of the positivity-preserving interpolation scheme is illustrated.
1 citations
TL;DR: In this article, rational cubic spline FIFs (RCSFIFs) with a quadratic denominator involving two shape parameters are constructed, and the elements of the iterated function system in each subinterval are identified so that the graph of the resulting fractal curve lies within a prescribed rectangle.
Abstract: This paper sets a theoretical foundation for applications of fractal interpolation functions (FIFs). We construct rational cubic spline FIFs (RCSFIFs) with a quadratic denominator involving two shape parameters. The elements of the iterated function system (IFS) in each subinterval are identified so that the graph of the resulting $$\mathcal{C}^1$$
-RCSFIF lies within a prescribed rectangle. These parameters include, in particular, conditions on the positivity of the
$$\mathcal{C}^1$$
-RCSFIF. The problem of visualization of constrained data is also addressed when a data set is lying above a straight line and the proposed fractal curve is required to lie on the same side of the line. We illustrate our interpolation scheme with some numerical examples.
1 citations
References
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TL;DR: In this article, a monotone piecewise bicubic interpolation algorithm was proposed for data on a rectangular mesh, where the first partial derivatives and first mixed partial derivatives are determined by the mesh points.
Abstract: In a 1980 paper [SIAM J. Numer. Anal., 17 (1980), pp. 238–246] the authors developed a univariate piecewise cubic interpolation algorithm which produces a monotone interpolant to monotone data. This paper is an extension of those results to monotone $\mathcal{C}^1 $ piecewise bicubic interpolation to data on a rectangular mesh. Such an interpolant is determined by the first partial derivatives and first mixed partial (twist) at the mesh points. Necessary and sufficient conditions on these derivatives are derived such that the resulting bicubic polynomial is monotone on a single rectangular element. These conditions are then simplified to a set of sufficient conditions for monotonicity. The latter are translated to a system of linear inequalities, which form the basis for a monotone piecewise bicubic interpolation algorithm.
2,174 citations
TL;DR: In this article, the authors introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data according tof(x��i)=y fixmei fori e {0,1,⋯,N}.
Abstract: Let a data set {(x
i,y
i) ∈I×R;i=0,1,⋯,N} be given, whereI=[x
0,x
N]⊂R. We introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data according tof(x
i)=y
i fori e {0,1,⋯,N}. Results are presented on the existence, coding theory, functional equations and moment theory for such fractal interpolation functions. Applications to the approximation of naturally wiggly functions, which may show some kind of geometrical self-similarity under magnification, such as profiles of cloud tops and mountain ranges, are envisaged.
736 citations
"Toward a Unified Methodology for Fr..." refers background or methods in this paper
...Proposition 1 (Barnsley [1]) The IFS {X;wi : i ∈ NN−1} defined above admits a unique attractor G, and G is the graph of a continuous function g : I → R which obeys g(xi ) = yi for i ∈ NN ....
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...This case is proposed by Barnsley [1] and Navascués [11] as generalization of any continuous function....
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...It is known [1] that there exists a metric on R2, equivalent to the Euclidean metric, with respect to which wi , i ∈ NN−1, are contractions....
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...For a detailed study, the reader may consult [1, 2, 15]....
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...To model such a data set, Barnsley [1] introduced the notion of Fractal Interpolation Function (FIF) based on the theory of Iterated Function System (IFS)....
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TL;DR: The calculus of deterministic fractal functions is introduced in this article, which can be explicitly indefinitely integrated any number of times, yielding a hierarchy of successively smoother interpolation functions which generalize splines and which are attractors for iterated function systems.
237 citations
176 citations
"Toward a Unified Methodology for Fr..." refers background in this paper
...This case is proposed by Barnsley [1] and Navascués [11] as generalization of any continuous function....
[...]
TL;DR: In this paper, a criterion for the positivity of a cubic polynomial on a given interval is derived, and a necessary and sufficient condition is given under which cubicC 1-spline interpolants are nonnegative.
Abstract: A criterion for the positivity of a cubic polynomial on a given interval is derived. By means of this result a necessary and sufficient condition is given under which cubicC
1-spline interpolants are nonnegative. Further, since such interpolants are not uniquely determined, for selecting one of them the geometric curvature is minimized. The arising optimization problem is solved numerically via dualization.
150 citations