Diskrete Iterierte Funktionensysteme (bibtex)
by Michael A. Neuhauser
Abstract:
Iterated function systems (IFS) are sets of contractive functions. They define a unique fractal attractor that can be represented as a binary image. They also define a unique invariant measure that can be represented as a greyvalue or color image. IFS are well suited for image coding, because even IFS with few functions are able to generate realistic images of natural objects. Firstly the mathematical basics of IFS will be presented in this work. Then the properties of affine transformation on euclidean spaces will be investigated. An convenient representation of affine transformations on the euclidian plane is developed. A discretization of the transformation is done next, this allows to present two efficient algorithms that compute a discrete attractor. By using image pyramids one of this algorithms is extended and enhanced. An efficient method for calculation of a discrete invariant measure is given. The discrete transformations introduce an error that results in a difference between the attractor, the invariant measure and their discrete counterparts. An error bound for this difference is shown. It follows from this error bound that it is always possible to achieve an accuracy that is sufficient for image coding purposes.
Reference:
Diskrete Iterierte Funktionensysteme (Michael A. Neuhauser), Technical report, PRIP, TU Wien, 1993.
Bibtex Entry:
@TechReport{TR017,
  author =	 "Michael A. Neuhauser",
  institution =	 "PRIP, TU Wien",
  month =	 feb,
  number =	 "PRIP-TR-017",
  title =	 "Diskrete {I}terierte {F}unktionensysteme",
  year =	 "1993",
  url =		 "ftp://ftp.prip.tuwien.ac.at/pub/publications/trs/tr17.ps.gz",
  abstract =	 "Iterated function systems (IFS) are sets of
                  contractive functions. They define a unique fractal
                  attractor that can be represented as a binary
                  image. They also define a unique invariant measure
                  that can be represented as a greyvalue or color
                  image. IFS are well suited for image coding, because
                  even IFS with few functions are able to generate
                  realistic images of natural objects. Firstly the
                  mathematical basics of IFS will be presented in this
                  work. Then the properties of affine transformation
                  on euclidean spaces will be investigated. An
                  convenient representation of affine transformations
                  on the euclidian plane is developed. A
                  discretization of the transformation is done next,
                  this allows to present two efficient algorithms that
                  compute a discrete attractor. By using image
                  pyramids one of this algorithms is extended and
                  enhanced. An efficient method for calculation of a
                  discrete invariant measure is given. The discrete
                  transformations introduce an error that results in a
                  difference between the attractor, the invariant
                  measure and their discrete counterparts. An error
                  bound for this difference is shown. It follows from
                  this error bound that it is always possible to
                  achieve an accuracy that is sufficient for image
                  coding purposes.",
}
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