On the inverse problem of fractal compression

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2001
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Hartenstein, Hannes
Ruhl, Matthias
Vrscay, Edward R.
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FIEDLER, Bernold, ed.. Ergodic theory, analysis and efficient simulation of dynamical Systems. Berlin [u.a.]: Springer, 2001, pp. 617-647. ISBN 3-540-41290-5
Zusammenfassung

The inverse problem of fractal compression amounts to determining a contractive operator such that the corresponding fixed point approximates a given target function. The standard method based on the collage coding strategy is known to represent a suboptimal method. Why does one not search for optimal fractal codes? We will prove that optimal fractal coding, when considered as a discrete optimization problem, constitutes an NP-hard problem, i.e., it cannot be solved in a practical amount of time. Nevertheless, when the fractal code parameters are allowed to vary continuously, we show that one is able to improve on collage coding by fine-tuning some of the fractal code parameters with the help of differential methods. The differentiability of the attractor as a function of its luminance parameters is established. We also comment on the approximating behaviour of collage coding, state a lower bound for the optimal attractor error, and outline an annealing scheme for improved fractal coding.

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ISO 690HARTENSTEIN, Hannes, Matthias RUHL, Dietmar SAUPE, Edward R. VRSCAY, 2001. On the inverse problem of fractal compression. In: FIEDLER, Bernold, ed.. Ergodic theory, analysis and efficient simulation of dynamical Systems. Berlin [u.a.]: Springer, 2001, pp. 617-647. ISBN 3-540-41290-5
BibTex
@incollection{Hartenstein2001inver-23256,
  year={2001},
  title={On the inverse problem of fractal compression},
  isbn={3-540-41290-5},
  publisher={Springer},
  address={Berlin [u.a.]},
  booktitle={Ergodic theory, analysis and efficient simulation of dynamical Systems},
  pages={617--647},
  editor={Fiedler, Bernold},
  author={Hartenstein, Hannes and Ruhl, Matthias and Saupe, Dietmar and Vrscay, Edward R.}
}
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    <dcterms:abstract xml:lang="eng">The inverse problem of fractal compression amounts to determining a contractive operator such that the corresponding fixed point approximates a given target function. The standard method based on the collage coding strategy is known to represent a suboptimal method. Why does one not search for optimal fractal codes? We will prove that optimal fractal coding, when considered as a discrete optimization problem, constitutes an NP-hard problem, i.e., it cannot be solved in a practical amount of time. Nevertheless, when the fractal code parameters are allowed to vary continuously, we show that one is able to improve on collage coding by fine-tuning some of the fractal code parameters with the help of differential methods. The differentiability of the attractor as a function of its luminance parameters is established. We also comment on the approximating behaviour of collage coding, state a lower bound for the optimal attractor error, and outline an annealing scheme for improved fractal coding.</dcterms:abstract>
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