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.

SOLVING THE INVERSE PROBLEM FOR FUNCTION/IMAGE APPROXIMATION USING ITERATED FUNCTION SYSTEMS I: THEORETICAL BASIS

Fractals ◽  
1994 ◽  
Vol 02 (03) ◽  
pp. 325-334 ◽  
Author(s):  
BRUNO FORTE ◽  
EDWARD R. VRSCAY
Keyword(s):  
Inverse Problem ◽  
Function Approximation ◽  
Image Representation ◽  
Iterated Function Systems ◽  
Rigorous Solution ◽  
Compact Metric Space ◽  
Image Approximation ◽  
Iterated Function ◽  
Contractive Operator ◽  
Contraction Maps

We are concerned with function approximation and image representation using Iterated Function Systems (IFS) over ℒp (X, µ): An N-map IFS with grey level maps (IFSM), to be denoted as (w, Φ), is a set w of N contraction maps wi: X → X over a compact metric space (X, d) (the "base space") with an associated set Φ of maps ϕi: R → R. Associated with each IFSM is a contractive operator T with fixed point [Formula: see text]. We provide a rigorous solution to the following inverse problem: Given a target υ ∈ ℒp(X, µ) and an ∊ > 0, find an IFSM whose attractor satisfies [Formula: see text].

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.

scholarly journals Box dimension for graphs of fractal functions

1997 ◽  
Vol 40 (2) ◽  
pp. 331-344
Author(s):  
Gavin Brown ◽  
Qinghe Yin
Keyword(s):  
Iterated Function Systems ◽  
Box Dimension ◽  
Affine Maps ◽  
Iterated Function ◽  
Fractal Functions

We calculate the box-dimension for a class of nowhere differentiable curves defined by Markov attractors of certain iterated function systems of affine maps.

scholarly journals On the existence of the Hutchinson measure for generalized iterated function systems

2020 ◽  
Vol 19 (3) ◽  
Author(s):  
Filip Strobin
Keyword(s):  
Iterated Function Systems ◽  
Iterated Function ◽  
Contractive Maps

AbstractWe prove that each generalized (in the sense of Miculescu and Mihail) IFS consisting of contractive maps generates the unique generalized Hutchinson measure. This result extends the earlier result due to Miculescu in which the assertion is proved under certain additional contractive assumptions.

A new class of markov processes for image encoding

1988 ◽  
Vol 20 (01) ◽  
pp. 14-32 ◽  
Author(s):  
Michael F. Barnsley ◽  
John H. Elton
Keyword(s):  
Invariant Measures ◽  
Metric Spaces ◽  
Iterated Function Systems ◽  
Interesting Case ◽  
Image Encoding ◽  
New Class ◽  
Affine Maps ◽  
Iterated Maps ◽  
Iterated Function ◽  
Infinite Extent

A new class of iterated function systems is introduced, which allows for the computation of non-compactly supported invariant measures, which may represent, for example, greytone images of infinite extent. Conditions for the existence and attractiveness of invariant measures for this new class of randomly iterated maps, which are not necessarily contractions, in metric spaces such as , are established. Estimates for moments of these measures are obtained. Special conditions are given for existence of the invariant measure in the interesting case of affine maps on . For non-singular affine maps on , the support of the measure is shown to be an infinite interval, but Fourier transform analysis shows that the measure can be purely singular even though its distribution function is strictly increasing.

Sign in / Sign up

Export Citation Format

Share Document