# Colloquium of the Department and the PhD Program

## title

*Recent studies in fractal image coding*

## speaker

## date & place

Tuesday, 28.06.2005, 14:15 h

Room Z714

Room Z714

## abstract

Part ILet T be a contraction mapping on an appropriate space of functions

*B(X)*. Then the evolution equation

*y*can be used to produce a continuous evolution

_{t}= Ty - y*y(x,t)*from an arbitrary initial condition

*y*to the fixed point

_{0}*y*. This simple observation is applied in the context of iterated function systems, in particular, fractal image coding. Such an evolution equation technique can also be applied to complex analytic mappings which are not strictly contractive but which possess invariant attractor sets, e.g. Newton's method in the complex plane. (Work with J. Bona.)

^{'}∈ B(X)Part II

Historically, most research in fractal image coding focussed 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 good number of domain image subblocks approximate a given range block almost as well as the optimal domain subblock selected by collage coding. In this part of the talk, we report on a series of computer experiments that examine how well or poorly

*n x n*-pixel range blocks are approximated by

*2n x 2n*domain blocks in an exhaustive pool using affine greyscale maps. We examine some distributions of collage distances within an image and in an ensemble of images. We also examine the effects of noise on such distributions. In the limit of zero signal-to-noise ratio, these distributions are observed to approach a distribution that can be derived analytically. (Work with S. Alexander and S. Tsurumi.)