. The inverse problem of fractal compression amounts to determining a contractive operator such that the corresponding xed point approximates a given target function. The standard method based on the collage coding strategy is known to represent a suboptimal method. Why does one not search for optimal fractal codes? We will prove that optimal fractal coding, when considered as a discrete optimization problem, constitutes an NP-hard problem, i.e., it cannot be solved in a practical amount of time. Nevertheless, when the fractal code parameters are allowed to vary continuously, we show that one is able to improve on collage coding by ne-tuning some of the fractal code parameters with the help of dierentiable methods. The dierentiability of the attractor as a function of its luminance parameters is established. We also comment on the approximating behavior of collage coding, state a lower bound for the optimal attractor error, and outline an annealing scheme for improved fractal codin...

Standard graphics systems encode pictures by assigning an address and colour attribute for each point of the object resulting in a long list of addresses and attributes. Fractal geometry enables a newer class of geometrical shapes to be used to encode whole objects, thus image compression is achieved. Compression ratios of 10,000:1 have been claimed by researchers 1 in this field. The fractal equations describing these shapes are very simple equations. Specifically, iterated function system (IFS) codes are investigated. The difficult inverse problem of finding a suitable IFS code whose fractal image is to represent the real image and hence achieve compression is investigated through the use of: a) a library of IFS codes and complex moments, b) the method of simulated annealing, for solving non-linear equations of many parameters. Image Compression Image compression is reducing the number of bits required to represent an image in such a way that either an exact replica of the image (lossless compression) or an approximate replica (lossy compression) of the image can be retrieved. 1 M.F. BARNSLEY, A.D. SLOAN, "A better way to compress images", BYTE, Jan 1988, p.215-223. Proc. 1 st Seminar on Information Technology and its Applications (ITA `91), Markfield Conf. Centre, Leicester, U.K., 29 Sept., 1991. 1 Canonical Representation of Digital Images A digital picture consists of an n m array of integer numbers or picture elements (pels), see Fig.1. n m pixels pixels Fig.1 Canonical Representation of Digital Images. If it takes B bits to encode each pel, then: n m B bits are required to represent the picture digitally. Thus for a 512 512 raster with 8 bits/pel: 512 512 8 = 2,097,152 bits. (A large number!) Reasons for Compressing Images 1) To reduce the speed...