IFS imaging beyond compression Original Research Article

Historically, most research in fractal image coding focused on its compression capabilities, seeking to obtain the best domain–range block matching with the minimal amount of fractal code. In the course of these investigations, it was implicitly known, yet perhaps not fully investigated, that a number of domain image subblocks will approximate a given range image subblock almost as well as the optimal domain subblock selected by collage coding. Recently we have presented a simple mathematical model of image self-similarity and a series of computer experiments that examine the self-similarity of natural images under contractive affine greyscale maps [S. Alexander, E. Vrscay, S. Tsurumi, A simple, general model for the affine self-similarity of images, in: A. Campilho, M. Karnel (Eds.), ICIAR, in: Lecture Notes in Computer Science, vol. 5112, Springer-Verlag, 2008, pp. 192–203; S. Alexander, Multiscale methods in image modelling and image processing, Ph.D. thesis, Dept. of Applied Mathematics, University of Waterloo, 2005]. We demonstrated that typical natural images have large numbers of such maps that closely approximate all areas (i.e. range subblock) of the image. In the present work, some implications of this are explored. A novel IFS-type ‘fractal operator’ with overlapping maps (IFSOM) is presented. Application of this technique in a multiscale manner is demonstrated to have denoising performance comparable to current state-of-the-art methods. Extensions of this preliminary work are discussed.