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On the Inverse Problem of Fractal Compression

In: Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems

Author

Listed:
  • Hannes Hartenstein

    (NEC Europe Ltd., Computer & Communication Research Lab)

  • Matthias Ruhl

    (Massachusetts Institute of Technology, Laboratory of Computer Science)

  • Dietmar Saupe

    (Universität Leipzig, Institut für Informatik)

  • Edward R. Vrscay

    (University of Waterloo, Department of Applied Mathematics)

Abstract

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 codingstrategy 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 differentiate methods. The differentiability of the attractor as a function of its luminance parameters is established. We also comment on the approximating behavior of collage coding, state a lower bound for the optimal attractor error, and outline an annealing scheme for improved fractal coding.

Suggested Citation

  • Hannes Hartenstein & Matthias Ruhl & Dietmar Saupe & Edward R. Vrscay, 2001. "On the Inverse Problem of Fractal Compression," Springer Books, in: Bernold Fiedler (ed.), Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pages 617-647, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-56589-2_26
    DOI: 10.1007/978-3-642-56589-2_26
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