Correspondence Between Fractal-Wavelet Transforms and Iterated Function Systems With Grey Level Maps

1997 ◽  
pp. 54-64 ◽  
Author(s):  
F. Mendivil ◽  
E. R. Vrscay
Keyword(s):  
Wavelet Transforms ◽  
Iterated Function Systems ◽  
Grey Level ◽  
Iterated Function

scholarly journals Continuity properties of attractors for iterated fuzzy set systems

1994 ◽  
Vol 36 (2) ◽  
pp. 175-193 ◽  
Author(s):  
B. Forte ◽  
M. Lo Schiavo ◽  
E. R. Vrscay
Keyword(s):  
Fuzzy Set ◽  
Iterated Function Systems ◽  
Restricted Domain ◽  
Grey Level ◽  
Compact Metric Space ◽  
Set Systems ◽  
Continuity Properties ◽  
Iterated Function ◽  
Composition Of Functions ◽  
Contraction Maps

AbstractAn N-map Iterated Fuzzy Set System (IFZS), introduced in [4] and to be denoted as (w, Φ), is a system of N contraction maps wi: X → X over a compact metric space (X, d), with associated “grey level” maps øi: [0, 1] → [0, 1]. Associated with an IFZS (w, Φ) is a fixed point u ∈ f*(X), the class of normalized fuzzy sets on X, u: X → [0, 1]. We are concerned with the continuity properties of u with respect to changes in the wi, and the φi. Establishing continuity for the fixed points of IFZS is more complicated than for traditional Iterated Function Systems (IFS) with probabilities since a composition of functions is involved. Continuity at each specific attractor u may be established over a suitably restricted domain of φi maps. Two applications are (i) animation of images and (ii) the inverse problem of fractal construction.

"Chaos Games" for Iterated Function Systems with Grey Level Maps

1998 ◽  
Vol 29 (4) ◽  
pp. 878-890 ◽  
Author(s):  
B. Forte ◽  
F. Mendivil ◽  
E. R. Vrscay
Keyword(s):  
Iterated Function Systems ◽  
Grey Level ◽  
Iterated Function

SOLVING THE INVERSE PROBLEM FOR FUNCTION/IMAGE APPROXIMATION USING ITERATED FUNCTION SYSTEMS II: ALGORITHM AND COMPUTATIONS

Fractals ◽  
1994 ◽  
Vol 02 (03) ◽  
pp. 335-346 ◽  
Author(s):  
BRUNO FORTE ◽  
EDWARD R. VRSCAY
Keyword(s):  
Inverse Problem ◽  
Quadratic Programming ◽  
Iterated Function Systems ◽  
Grey Level ◽  
Image Approximation ◽  
Affine Maps ◽  
Iterated Function ◽  
Contractive Maps ◽  
Target Set ◽  
Infinite Set

In this paper, we provide an algorithm for the construction of IFSM approximations to a target set [Formula: see text], where X ⊂ RD and µ = m(D) (Lebesgue measure). The algorithm minimizes the squared "collage distance" [Formula: see text]. We work with an infinite set of fixed affine IFS maps wi: X → X satisfying a certain density and nonoverlapping condition. As such, only an optimization over the grey level maps ϕi: R+ → R+ is required. If affine maps are assumed, i.e. ϕi = αit + βi, then the algorithm becomes a quadratic programming (QP) problem in the αi and βi. We can also define a "local IFSM" (LIFSM) which considers the actions of contractive maps wi on subsets of X to produce smaller subsets. Again, affine ϕi maps are used, resulting in a QP problem. Some approximations of functions on [0,1] and images in [0, 1]2 are presented.

scholarly journals Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

2020 ◽  
pp. 1-31
Author(s):  
Balázs Bárány ◽  
Károly Simon ◽  
István Kolossváry ◽  
Michał Rams
Keyword(s):  
Hausdorff Measure ◽  
Similar Case ◽  
Iterated Function Systems ◽  
The Self ◽  
Dimensional Hausdorff Measure ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Topological Sense

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

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