Irregular Pyramids (bibtex)
by Walter G. Kropatsch, Annick Montanvert
Abstract:
A pyramid is a stack of images with exponentially decreasing resolutions. Many image processing algorithms run on this hierarchical structure in $O(łog n)$ parallel processing steps where $n$ is a side of the input image. Perturbations in the structure may disturb the originally regular neighborhood relations and also the stability of the results. On the other hand, biological vision is based on piecewise regular patches in the retina, e.g. of a monkey or a human. P. Meer's stochastic pyramid is such an irregular structure. The parallel generation of the structure is governed by two ``decimation rules'' that also characterize a maximal independent set on the neighborhood graph of the image pixels. In general, the number of neighbors in the decimated graph may increase. It is shown that the decimation $G'(V',E')$ of any neighborhood-graph $G(V,E)$ preserves the degrees in the corresponding dual graphs. However the dual of $G'$ is not always a ``good'' decimation of the dual of $G$. Investigating in parallel dual decimations of regular graphs, one finds unique solutions that have interesting properties for image pyramids. Besides the above theoretical motivation for irregular structures, we can find similar strutures in the retinas of monkeys (and also of humans).
Reference:
Irregular Pyramids (Walter G. Kropatsch, Annick Montanvert), Technical report, PRIP, TU Wien, 1992.
Bibtex Entry:
@TechReport{TR005,
  author =	 "Walter G. Kropatsch and Annick Montanvert",
  institution =	 "PRIP, TU Wien",
  number =	 "PRIP-TR-005",
  title =	 "Irregular {P}yramids",
  year =	 "1992",
  url =		 "ftp://ftp.prip.tuwien.ac.at/pub/publications/trs/tr5.ps.gz",
  abstract =	 "A pyramid is a stack of images with exponentially
                  decreasing resolutions. Many image processing
                  algorithms run on this hierarchical structure in
                  $O(\log n)$ parallel processing steps where $n$ is a
                  side of the input image. Perturbations in the
                  structure may disturb the originally regular
                  neighborhood relations and also the stability of the
                  results. On the other hand, biological vision is
                  based on piecewise regular patches in the retina,
                  e.g. of a monkey or a human. P. Meer's stochastic
                  pyramid is such an irregular structure. The parallel
                  generation of the structure is governed by two
                  ``decimation rules'' that also characterize a
                  maximal independent set on the neighborhood graph of
                  the image pixels. In general, the number of
                  neighbors in the decimated graph may increase. It is
                  shown that the decimation $G'(V',E')$ of any
                  neighborhood-graph $G(V,E)$ preserves the degrees in
                  the corresponding dual graphs. However the dual of
                  $G'$ is not always a ``good'' decimation of the dual
                  of $G$. Investigating in parallel dual decimations
                  of regular graphs, one finds unique solutions that
                  have interesting properties for image
                  pyramids. Besides the above theoretical motivation
                  for irregular structures, we can find similar
                  strutures in the retinas of monkeys (and also of
                  humans).",
}
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