Curve Fitting by Fractal Interpolation

Manousopoulos, Polychronis; Drakopoulos, Vasileios; Theoharis, Theoharis

doi:10.1007/978-3-540-79299-4_4

Cited by 23 publications

(15 citation statements)

References 10 publications

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“…Furthermore, we suppose that the idea of mapping on the basis of FJI can be extended to IFS as they are iterative. Consequently, the results of the present research can be applied to various applications associated with fractal interpolation functions such as signal processing and modeling coastlines and shapes [130,131,132,133,134].…”

Section: Discussionmentioning

confidence: 95%

Dynamics with Chaos and Fractals

Alejaily

Fen

Akhmet³

2020

Nonlinear Systems and Complexity

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, 98 pages In this thesis, we study how to construct and analyze dynamics for chaos and fractals. After the introductory chapter, we discuss in the second chapter the chaotic behavior of hydrosphere parameters and their influence on global weather and climate. For this purpose, we investigate the nature and source of unpredictability in the dynamics of sea surface temperature. The impact of sea surface temperature variability on the global climate is clear during some global climate patterns like the El Niño-Southern Oscillation. The interactions between these types of global climate patterns may transmit chaos. We discuss the unpredictability as a global phenomenon through extension of chaos horizontally and vertically by using theoretical as well as numerical analyses. In the third chapter, we introduce a technique concerning the construction of dynamics for fractals. Deterministic fractals like Julia sets are crucial for understanding the fractal phenomenon. These structures are deterministically arising from simple dynamics of iteration of analytic functions. On the basis of the dynamics, we develop a scheme to map fractals through iterations. This allows to involve fractals as states of dynamical systems as well as to introduce dynamics in fractals through differential and discrete equations. Creation of effective frameworks v for applications of fractals and assisting to explore the relationship between fractals and chaos by the involvement of dynamics in fractals are important aspects in this direction.

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Section: Discussionmentioning

confidence: 95%

Dynamics with Chaos and Fractals

Alejaily

Fen

Akhmet³

2020

Nonlinear Systems and Complexity

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“…Similar to Case 1, we first consider the constrained rational fractal interpolation below g(x). The constraint for the vertical scaling factor is defined by (16),…”

Section: Visualization Of Constrained 2d Datamentioning

confidence: 99%

“…In [13], authors prove the stability of the fractal interpolants. In [16], an A c c e p t e d M a n u s c r i p t economical new method for curve fitting using fractal interpolation was introduced. In [6], Barnsley and Harrington prove that the integral of a FIF can yield a smooth fractal interpolation function which generalizes spline and is an attractor for iterated function systems, it shows existence of differentiable fractal interpolation functions.…”

Section: Introductionmentioning

confidence: 99%

Visualization of constrained data by smooth rational fractal interpolation

Liu

Bao

2015

International Journal of Computer Mathematics

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A family of smooth rational spline fractal interpolation function (SRFIF) is presented with the help of classical rational cubic spline. Each SRFIF of the family is identified uniquely by the values of vertical scaling factor s i and shape parameters α i and β i , and the shape of the fractal interpolating curves can be constrained and modified by selecting suitable parameters and vertical scaling factor. In order to meet the needs of practical design, a new method is developed to visualize constrained data in the view of constrained curves. In the special case, the methods of linear constraint and quadratic constraint are discussed. Also, convergence analysis shows that SRFIF gives a good approximation to the original function.

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“…Fractal interpolation curve for discrete data points, and best approximations to the FIF were depicted through continuous functions (Kang et al 2014; Li and Gao 2014). Manousopoulos et al (2008Manousopoulos et al ( , 2011 proposed a novel approach to the curve fitting by FIF and RFIF using contraction factor (i.e., vertical scaling factor), and compared results using geographic data against the existing procedures in literature. Quadratic functions were used in representation of fractal interpolation in Shen and Feng (2017), and it was observed that by taking constant scale factor the interpolated curve was essentially parabolic.…”

Section: Introductionmentioning

confidence: 99%

Simulation and modeling of diffraction pattern of crystallography: a fractal approach

2020

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Modeling of diffraction pattern is prerequisite to analyze and explore hidden characteristic of crystals, minerals, proteins and waves. Demonstration of diffraction pattern is a challenging task as it exhibits non-linear structure that makes their representation complex. The main objective, here, is to present simple and efficient scheme to draw heterogeneous diffractive pattern. The proposed approach yields fast and reliable description of diffraction pattern. This approach has been applied to two different materials namely nickel ferrite (NiFe 2 O 4) and iron oxide (Fe 3 O 4). The recurrent fractal interpolation scheme with variable scaling factors projects and simulates the diffraction pattern more efficiently. The presented work shows that the proposed algorithm is reliable and competitive in simulation and modeling of diffraction pattern.

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Curve Fitting by Fractal Interpolation

Cited by 23 publications

References 10 publications

Dynamics with Chaos and Fractals

Dynamics with Chaos and Fractals

Visualization of constrained data by smooth rational fractal interpolation

Simulation and modeling of diffraction pattern of crystallography: a fractal approach

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