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**1 - 2**of**2**### On the Inverse Problem of Fractal Compression

"... . 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 optim ..."

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. 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...

### Fractal Image Compression

- Proc. 1st Seminar on Information Technology and its Applications (ITA'91
, 1991

"... 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 achieve ..."

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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...