On the Parameter Identification Problem in the Plane and the Polar Fractal Interpolation Functions

Dalla, Leoni; Drakopoulos, Vasileios

doi:10.1006/jath.1999.3380

Cited by 45 publications

(13 citation statements)

References 13 publications

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“…We leave this direction open for the future work. However, it seems to be not devoid of interest to point out that the problem of getting a necessary and sufficient condition based on the parameters involved in the IFS so as to have the corresponding FIF within a prescribed rectangle is unsolved even for the more tractable case, namely, the case of an affine FIF (see [8,29]). …”

Section: Remark 12mentioning

confidence: 97%

“…However, there is no indication about the conditions that the IFS parameters must obey, so that the attractor remains within the rectangle K. One possible explanation of this is that C can be taken as a sufficiently large compact set. Dalla and Drakopoulos [8] (x 1 , c), (x 1 , d), (x n , c), (x n , d). However, the non-linearity of the present v i (x, f ) demands a different approach.…”

Section: Rational Quadratic Fif Within a Prescribed Axis-aligned Rectmentioning

confidence: 99%

“…This was motivated by and extends the parameter identification problem studied earlier by Dalla and Drakopoulos [8] for an affine FIF. It is felt that these types of parameter identification problems discussed here can find applications in fields such as image compression, data visualization, modern data analysis, and CAD/CAM.…”

Section: Introductionmentioning

confidence: 98%

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A rational iterated function system for resolution of univariate constrained interpolation

Viswanathan

Chand

Navascués

2014

RACSAM

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Iterated Function Systems (IFSs) provide a standard framework for generating Fractal Interpolation Functions (FIFs) that yield smooth or non-smooth approximants. Nevertheless, the most widely studied FIFs so far in the literature that are obtained through polynomial IFSs are, in general, incapable of reproducing important shape properties inherent in a given data set. Abandoning the polynomiality of the functions defining the IFS, we introduce a new class of rational IFS that generates fractal functions (self-referential functions) for solving constrained interpolation problems. Suitable values of the rational IFS parameters are identified so that: (i) the corresponding FIF inherits positivity and/or monotonicity properties present in the data set, and (ii) the attractor of the IFS lies within an axis-aligned rectangle. The proposed IFS schemes for the shape preserving interpolation generalize some of the classical non-recursive interpolation methods, and expand the interpolation/approximation, including approximants for which functions themselves or the first derivatives can even be non-differentiable in a dense set of points of the domain. For appropriate values of the IFS parameters, the resulting rational quadratic FIF converges uniformly to the original function Φ ∈ C 3 [x 1 , x n ] with h 3 order of convergence, where h denotes the norm of the partition. We also provide a number of examples intended to demonstrate the proposed schemes and to suggest how these schemes outperform their classical counterparts.

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Section: Remark 12mentioning

confidence: 97%

Section: Rational Quadratic Fif Within a Prescribed Axis-aligned Rectmentioning

confidence: 99%

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A rational iterated function system for resolution of univariate constrained interpolation

Viswanathan

Chand

Navascués

2014

RACSAM

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“…Upper and lower bounds of the vertical scaling factors that constrain an affine fractal interpolation function within an axis-aligned rectangle are determined in [13]. Recently, this type of containment problem is solved with a general method that does not rely on the affinity of the IFS and successively employed by taking cubic FIF as an example [14].…”

Section: Fractal Interpolation Theory: An Overviewmentioning

confidence: 99%

Fractal Interpolation Functions: A Short Survey

Navascués¹,

Chand²,

Viswanathan³

et al. 2014

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The object of this short survey is to revive interest in the technique of fractal interpolation. In order to attract the attention of numerical analysts, or rather scientific community of researchers applying various approximation techniques, the article is interspersed with comparison of fractal interpolation functions and diverse conventional interpolation schemes. There are multitudes of interpolation methods using several families of functions: polynomial, exponential, rational, trigonometric and splines to name a few. But it should be noted that all these conventional nonrecursive methods produce interpolants that are differentiable a number of times except possibly at a finite set of points. One of the goals of the paper is the definition of interpolants which are not smooth, and likely they are nowhere differentiable. They are defined by means of an appropriate iterated function system. Their appearance fills the gap of non-smooth methods in the field of approximation. Another interesting topic is that, if one chooses the elements of the iterated function system suitably, the resulting fractal curve may be close to classical mathematical functions like polynomials, exponentials, etc. The authors review many results obtained in this field so far, although the article does not claim any completeness. Theory as well as applications concerning this new topic published in the last decade are discussed. The one dimensional case is only considered.

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“…In view of their diverse applications, there has been steadily increasing interest in the theory of fractal functions, and it still continues to be a hot topic of research. Following the publication of Fractals Everywhere [2], a beautiful exposition of IFS theory, fractal functions and their applications, various related issues such as calculus, Holder continuity, convergence, stability, smoothness, determination of scaling parameters, and perturbation error have been investigated in the literature [3][4][5][6][7][8][9][10][11][12][13]. The concept of smooth FIFs has been used to generalize the traditional splines [14][15][16][17][18] and to demonstrate that the interaction of classical numerical methods with fractal theory provides new interpolation schemes that supplement the existing ones.…”

Section: Prologuementioning

confidence: 99%

Fractal Functions in Interpolation and Approximation: A Bird's-Eye View

AKB¹

2014

J Appl Computat Math

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This article is targeted to provide a brief and coarse discussion on theory of fractal functions and their recent developments including some of the researches of the authors. Materials on fractal functions provided by this paper is not claimed to be exhaustive, but is intended to be read before or in parallel with technical papers available in the literature on this subject. We have made an earnest attempt to supply an overall flavor of the subject of univarite fractal approximation and useful source of links to assist a novice and perhaps to an expert in fractal functions too.

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On the Parameter Identification Problem in the Plane and the Polar Fractal Interpolation Functions

Cited by 45 publications

References 13 publications

A rational iterated function system for resolution of univariate constrained interpolation

A rational iterated function system for resolution of univariate constrained interpolation

Fractal Interpolation Functions: A Short Survey

Fractal Functions in Interpolation and Approximation: A Bird's-Eye View

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