Barnsley's Fractal Compression

1985

The Barnsley fern assembling under iterated function system rules. Barnsley's 1985 papers proposed compressing the equation instead of the pixels. Place: Georgia Institute of Technology, Atlanta; later Iterated Systems Inc., Norcross, Georgia Figures: Michael Barnsley, mathematician · Alan Sloan, co-founder of Iterated Systems · the IFS (iterated function system) research community Michael Barnsley publishes the Collage Theorem in 1985. The idea: any image can be approximated as the fixed point of a contractive iterated function system, a small set of affine transformations that, applied recursively, converge to the original picture. Store the transformations, throw away the pixels. The compression ratios on suitable imagery are spectacular, sometimes 100-to-1 or better. The Sierpinski triangle compresses to four numbers. A fern compresses to twenty-eight. Barnsley and Alan Sloan found Iterated Systems in 1987 to commercialize the idea. The encoder, which has to FIND the contractive transformations, is wildly expensive (hours per image on 1980s hardware). The decoder is fast and resolution-independent: the same compressed file renders at any output size. It is, on paper, the ideal format for the bandwidth-constrained future. Video: The chaos game and Barnsley fern, the iterated function systems behind Michael Barnsley's 1985 compression idea.