Graph-Directed Coalescence Hidden Variable Fractal Interpolation Functions

Abstract

Fractal interpolation function (FIF) is a special type of continuous function which interpolates certain data set and the attractor of the Iterated Function System (IFS) corresponding to a data set is the graph of the FIF. Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is both self-affine and non self-affine in nature depending on the free variables and constrained free variables for a generalized IFS. In this article, graph directed iterated function system for a finite number of generalized data sets is considered and it is shown that the projection of the attractors on  2 is the graph of the CHFIFs interpolating the corresponding data sets.

Genetic Estimation of Iterated Function Systems for Accurate Fractal Modeling in Pattern Recognition Tools

“…We first implemented the linear self-affine fractal interpolation with genetic optimization of the interpolation nodes (called method 1 in the following). For comparison purposes, moreover, the suboptimal procedure described in [4] and called the M azel/Hayes method has been implemented. In summary, we implemented the following versions of the piecewise self-affine fractal interpolation:…”

“…However, the types of data an IFS can represent well are limited. Thus different IFS models, such as the piecewise self-affine [3] or the hidden-variable [4] models, has been developed in order to represent signals that are neither self-affine nor self-similar. In this paper, we use self-affine and piecewise self-affine IFS fractal models.…”

“…Alternatively, the inverse problem can be formulated through least squares [6], and therefore a function minimization algorithm can be used. In [4], an optimization approach for the determination of the parameters of IFS models is used. However, the optimization problem turns out to be quite complex, because the function to be minimized is nonlinear with many local minima and because many parameters are involved.…”