Fractal Interpolation Functions: A Short Survey

Navascués, M. A.; Chand, A. K. B.; Viswanathan, P.; Sebastián, María Victoria

doi:10.4236/am.2014.512176

Cited by 20 publications

(9 citation statements)

References 22 publications

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“…The calculation of this quantifier can be simplified, since it is not an excessively restrictive parameter in terms of signal stationarity hypotheses, and it does not need too long records. Our team has developed several low-cost computational algorithms for calculating the fractal dimension, for example, from the reconstruction of the EEG signal using fractal interpolation functions built with systems of iterated functions [18,19]. In this work, a procedure has been implemented to calculate the fractal dimension D of the trace of each EEG channel as a curve of the real plane, using the existing relationship with another quantization parameter, the Hurst exponent, denoted by H:…”

Section: Methodsmentioning

confidence: 99%

Fractal Dimension as Quantifier of EEG Activity in Driving Simulation

et al. 2021

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Dynamical systems and fractal theory methodologies have been proved useful for the modeling and analysis of experimental datasets and, in particular, for electroencephalographic signals. The computation of the fractal dimension of approximation curves in the plane enables the assignment of numerical values to bioelectric recordings in order to discriminate between different states of the observed system. The procedure does not require the stationarity of the signals nor extremely long segments of data. In previous works, we checked that this parameter is a good index for brain activity. In this paper, we consider this measurement in order to quantify the geometric complexity of the brain waves in states of rest and during vehicle driving simulation in different scenarios. This work presents evidence that the fractal dimension allows the detection of the brain bioelectric changes produced in the areas that carry out the different driving simulation tasks, increasing with their complexity.

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Section: Methodsmentioning

confidence: 99%

Fractal Dimension as Quantifier of EEG Activity in Driving Simulation

et al. 2021

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“…Barnsley introduced fractal interpolation functions (FIF) using iterated function system (IFS) theory which is an important part of fractals (see [1], [2], [11] for further information).…”

Section: Introductionmentioning

confidence: 99%

“…Barnsley introduced fractal interpolation functions (FIF) using the iterated function system (IFS) theory, which is an important part of fractals (see [1,2,13] for further information). He showed that there exists a function interpolating a given data set such that the graph of this function is the attractor of an IFS.…”

Section: Introductionmentioning

confidence: 99%

Graph-directed fractal interpolation functions

Deniz¹,

Özdemir²

2017

Turk J Math

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It is known that there exists a function interpolating a given data set such that the graph of the function is the attractor of an iterated function system, which is called a fractal interpolation function. We generalize the notion of the fractal interpolation function to the graph-directed case and prove that for a finite number of data sets there exist interpolation functions each of which interpolates a corresponding data set in R 2 such that the graphs of the interpolation functions are attractors of a graph-directed iterated function system.

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“…Neverthless, to ease the acess for the nonexpert the essentials of fractal interpolation that form background for our study have been collected in this section. For a detailed study, the reader may consult [1,2,15].…”

Section: Basic Factsmentioning

confidence: 99%

Toward a Unified Methodology for Fractal Extension of Various Shape Preserving Spline Interpolants

Katiyar

Chand

2015

Springer Proceedings in Mathematics &Amp; Statistics

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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 α-fractal functions corresponding to the traditional nonrecursive splines. The elements of the iterated function system are identified befittingly so that the class of α-fractal function f α 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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Fractal Interpolation Functions: A Short Survey

Cited by 20 publications

References 22 publications

Fractal Dimension as Quantifier of EEG Activity in Driving Simulation

Fractal Dimension as Quantifier of EEG Activity in Driving Simulation

Graph-directed fractal interpolation functions

Toward a Unified Methodology for Fractal Extension of Various Shape Preserving Spline Interpolants

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