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Fractal approximation of curves and surfaces

Abstract : Approximation of natural objects (curves, surfaces, or images) with fractal models is an important center of interest for research. The general inverse problem paradigm concerns many application fields, including information representation for image transmission or compression, 3D reconstruction for visualization or CAGD. A large variety of studies, using specific models or methods, have been proposed to address this inverse fractal problem. The most known of them is the fractal image compression method introduced by Jacquin. Generally speaking, these techniques lack of flexibility in term of control over the approximated shape. Furthermore, iteration space used is the visualisation space, R². Previous work achieved a general framework for fractal modeling: fractal free forms. This model allows user to define self-similar objects in a space of a higher dimension. We propose a resolution of the inverse problem base on this model and a non-linear regression algorithm. This versatile method allows the approximation of curves and surfaces, even rough or smooth, and also grey-level images. A hierachical extension of this model is introduced for modeling heterogeneous objects, for which characteristics are varying in space. Two algorithms are proposed for the associated approximation problem. The first one computes, given a grey-level image and a distortion criteria, the model that gives a uniform repartition of this distortion. The second one is a complete optimisation of the rate/distortion ratio, given a rate budget. Results show that this type of model is interesting for low rate compression.
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Contributor : Eric Guérin <>
Submitted on : Friday, December 5, 2003 - 1:06:47 PM
Last modification on : Wednesday, November 20, 2019 - 3:07:09 AM
Long-term archiving on: : Friday, April 2, 2010 - 7:08:16 PM


  • HAL Id : tel-00003908, version 1



Eric Guérin. Fractal approximation of curves and surfaces. Graphics [cs.GR]. Université Claude Bernard - Lyon I, 2002. ⟨tel-00003908⟩



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