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Inverse and approximation problem for two-dimensional fractal sets.

R Rinaldo1, A Zakhor

  • 1Dept. of Electr. Eng., California Univ., Berkeley, CA.

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|January 1, 1994
PubMed
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This study introduces methods for finding fractal parameters from images using wavelet transforms and image moments. These techniques enable fractal image representation and approximation for modeling natural phenomena.

Area of Science:

  • Fractal Geometry
  • Image Analysis
  • Computational Mathematics

Background:

  • Fractals are extensively used to model natural phenomena and images due to their complex geometry.
  • Iterated Function Systems (IFS) provide a recursive definition for describing and classifying deterministic fractals.
  • Developing image representation schemes based on IFS parameters is conceivable for fractal images.

Purpose of the Study:

  • To address two distinct problems: an inverse problem and an approximation problem related to fractal image representation.
  • To find Iterated Function System (IFS) parameters for signals exactly generated by an IFS.
  • To approximate arbitrary images using fractal IFS-generated images with matching moments.

Main Methods:

  • Utilizing wavelet transform and image moments to solve the inverse problem of finding IFS parameters.

Related Experiment Videos

  • Developing an approximation problem solution by finding fractal IFS-generated images whose moments match the original image moments.
  • Employing a special form of IFS model for generating approximating measures for arbitrary images.

    • Main Results:

    • Successfully demonstrated the inverse problem solution using wavelet transform and image moments.
    • Presented an approximation method for generating fractal images that match moments of original images.
    • Experimental results verified the efficacy of the proposed fractal image representation and approximation approaches.

    Conclusions:

    • The study successfully established methods for both exact reconstruction (inverse problem) and approximation of fractal images using IFS.
    • The developed techniques offer a general basis for approximating arbitrary images using fractal models.
    • The findings contribute to advanced image representation and analysis techniques leveraging fractal geometry.