Iterierte Funktionensysteme - Das eindimensionale inverse Problem (bibtex)
by Irene J. Leitgeb
Abstract:
Since images of natural objects often have fractal features, Iterated Function Systems (IFS) seem quite suited for their representation in computers. A further reason to use IFS is their low requirements of memory. An IFS is a set of contractive functions that defines a fractal attractor; this attractor can be interpreted as a binary image. First of all, an explanation of IFS and the inverse problem will be given in this work, followed by an outline of existing approaches to the solution of the inverse problem. A brief introduction to the mathematical basis of IFS will then be given. Then an algorithm is presented that is based upon existing discretisation of transformations of an IFS and algorithms that find a discrete attractor of an IFS. The algorithm solves the inverse problem in the 1D case. It uses the ratio of the length of the black and the white connected components of an attractor which is invariant under the transformations of an IFS. In the discrete case an interval can be defined for this ratio. This information is used to find possible transformations. In consideration of sampling effects, the algorithm tries then to find the parameters for each possible transformation if they exist.
Reference:
Iterierte Funktionensysteme - Das eindimensionale inverse Problem (Irene J. Leitgeb), Technical report, PRIP, TU WIEN, 1993.
Bibtex Entry:
@TechReport{PTR-Leitgeb93a,
  author =	 "Irene J. Leitgeb",
  institution =	 "PRIP, TU WIEN",
  number =	 "PRIP-TR-023",
  title =	 "Iterierte {F}unktionensysteme - {D}as
                  eindimensionale inverse {P}roblem",
  year =	 "1993",
  url =		 "ftp://ftp.prip.tuwien.ac.at/pub/publications/trs/tr23.ps.gz",
  abstract =	 "Since images of natural objects often have fractal
                  features, Iterated Function Systems (IFS) seem quite
                  suited for their representation in computers. A
                  further reason to use IFS is their low requirements
                  of memory. An IFS is a set of contractive functions
                  that defines a fractal attractor; this attractor can
                  be interpreted as a binary image. First of all, an
                  explanation of IFS and the inverse problem will be
                  given in this work, followed by an outline of
                  existing approaches to the solution of the inverse
                  problem. A brief introduction to the mathematical
                  basis of IFS will then be given. Then an algorithm
                  is presented that is based upon existing
                  discretisation of transformations of an IFS and
                  algorithms that find a discrete attractor of an
                  IFS. The algorithm solves the inverse problem in the
                  1D case. It uses the ratio of the length of the
                  black and the white connected components of an
                  attractor which is invariant under the
                  transformations of an IFS. In the discrete case an
                  interval can be defined for this ratio. This
                  information is used to find possible
                  transformations. In consideration of sampling
                  effects, the algorithm tries then to find the
                  parameters for each possible transformation if they
                  exist.",
}
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