Festbildcodierung bei niedrigen Bitraten unter Verwendung fraktaler Methoden im Orts- und Frequenzbereich

In January 1988, Michael Barnsley and Alan Sloan published an article titled "A Better Way to Compress Images" in BYTE magazine, introducing a new method for image compression based on fractal geometry. They claimed that this method could achieve compression ratios as high as 10,000:1. However, their article did not disclose the specific process for determining these highly compressive fractal codes. This fueled speculation that their method could vastly outperform existing techniques, which achieved compression ratios around 50:1.

The first major assessment of these claims came in August 1989 when Arnaud Jacquin, under Barnsley’s supervision, presented a block-based automated method for fractal encoding of grayscale images in his doctoral dissertation. Although Jacquin's method only achieved compression ratios of approximately 16:1, far below the initial claims, it sparked significant interest and research activity, including the present work.

This dissertation contributes to the advancement and clarification of fractal image coding by conducting a comprehensive study of block-based fractal coding. It analyzes fundamental issues of fractal image coding and proposes solutions to improve the quality of encoded images, reduce encoding times, and mitigate error propagation during decoding.

Given the complexity and wide range of parameters involved in theoretical fractal image coding, this work dissects the fractal coder into fundamental components for individual analysis and optimization. Special attention is given to bitrate-efficient transmission and the effects of quantizing fractal transformation parameters.

Two key insights emerged from the analysis:

1. Despite its seemingly unique approach, fractal coding is closely related to existing coding techniques such as predictive coding, vector quantization, and wavelet coding, which can be seen as special cases of fractal coding.
2. The similarity in coding principles suggests a high potential for integrating fractal coding with other techniques.

This dissertation introduces new purely fractal and hybrid coders, which demonstrate significant improvements over Jacquin’s classical fractal coding and non-fractal methods like the JPEG standard. Notably, for extremely high compression ratios, with bitrates ranging from 0.2 to 0.05 bits per pixel, these new coders deliver compression results with good objective and subjective visual quality, particularly important at such low bitrates.

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