Application Of Recurrent Iterated Function Systems To Images

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Abstract
A new fractal technique for the analysis and compression of digital images is presented. It is shown that a family of contours extracted from an image can be modelled geometrically as a single entity, based on the theory of recurrent iterated function systems (RIFS). RIFS structures are a rich source for deterministic images, including curves which cannot be generated by standard techniques. Control and stability properties are investigated. We state a control theorem - the recurrent collage theorem - and show how to use it to constrain a recurrent IFS structure so that its attractor is close to a given family of contours. This closeness is not only perceptual; it is measured by means of a min-max distance, for which shape and geometry is important but slight shifts are not. It is therefore the right distance to use for geometric modeling. We show how a very intricate geometric structure, at all scales, is inherently encoded in a few parameters that describe entirely the recurrent structures. Very high data compression ratios can be obtained. The decoding phase is achieved through a simple and fast reconstruction algorithm. Finally, we suggest how higher dimensional structures could be designed to model a wide range of digital textures, thus leading our research towards a complete image compression system that will take its input from some low-level image segmenter.

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