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A generalized fractal transform for measure-valued images


  • Davide LA TORRE


  • Edward R. VRSCAY



Fractal image coding generally seeks to express an image as a union of spatially contracted and greyscale modified copies of subsets of itself. Generally,images are represented as functions u(x) and the fractal coding method is conducted in the framework of L^2 or L^1. Here we formulate a method of fractal image coding on measure-valued images: At each point \mu(x) is a probability measure overthe range of allowed greyscale values. We construct a complete metric space (Y,d_Y )of measure-valued images, \mu : X -> M(Rg), where X is the base or pixel space and M(Rg) is the set of probability measures supported on the greyscale range Rg. A method of fractal transforms is formulated over the metric space (Y,d_Y ). Under suitable conditions, a transform M : Y -> Y is contractive, implying the existence of a unique fixed point measure-valued function \mu^*= M\mu^*. We also show that the pointwise moments of this measure satisfy a set of recursion relations that are generalizations of those satisfied by moments of invariant measures of Iterated Function Systems with Probabilities.

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  • Davide LA TORRE & Edward R. VRSCAY, 2008. "A generalized fractal transform for measure-valued images," Departmental Working Papers 2008-38, Department of Economics, Management and Quantitative Methods at Università degli Studi di Milano.
  • Handle: RePEc:mil:wpdepa:2008-38

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    1. Herb E. KUNZE & Davide LA TORRE & Edward R. VRSCAY, 2008. "From iterated function systems to iterated multifunction systems," Departmental Working Papers 2008-39, Department of Economics, Management and Quantitative Methods at Università degli Studi di Milano.
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    Cited by:

    1. La Torre, Davide & Marsiglio, Simone & Privileggi, Fabio, 2011. "Fractals and Self-Similarity in Economics: the Case of a Stochastic Two-Sector Growth Model," POLIS Working Papers 157, Institute of Public Policy and Public Choice - POLIS.
    2. Davide La Torre & Simone, Marsiglio & Mendivil, Franklin & Privileggi, Fabio, 2015. "Self-Similar Measures in Multi-Sector Endogenous Growth Models," Department of Economics and Statistics Cognetti de Martiis. Working Papers 201509, University of Turin.

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    Measure-valued images; multifunctions; self-similarity; fractal;

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