Benoit Mandelbrot, the inventor of fractal geometry and the term "fractal," showed in his book, The Fractal Geometry of Nature, that realistic landscapes, clouds, galaxies, and so forth, can be synthesized as computer graphics by using fractal formulae. Artificial shapes generated by fractal geometry mimic nature's irregularities and oddities so well that they have been used as scenery on movie sets.
Euclidean geometry used in ordinary computer graphics does not deal effectively with natural forms such as clouds lightning bolts, snow flakes or mountains. The geometry of the real world is irregular or "fractal." Mandelbrot noted that snowflakes are geometric extensions of smaller versions of themselves.
Computer graphics has incorporated fractal geometry, resulting in a new look at the natural world. Fractals and computers can create fantastic visual images unsurpassed by anything else. Fractals are produced by starting with an initial shape and infinitely adding new shapes created by repeated transformations, such as shrink, move and rotate. Fractal concepts have become an important tool for the analysis of nonlinear processes and for describing biological and physical structures.
Fractal patterns exhibit the peculiar property of looking similar at whatever scale they are viewed. For example, fractal snowflakes are generated by repeatedly repositioning and shrinking a triangle. Such "snowflakes" have edges that contain miniature replicas of the larger snowflake pattern. Upon closer viewing, these small snowflakes have snowflake shapes on their edges, ad infinitum. At any one scale, the snowflakes get smaller until they become dots, but if the scale is changed and you and zoom in on them, they appear yet again.
How fractal compression works
In the key discovery related to fractal compression, Michael Barnsley
demonstrated that any image can be imitated by a set of fractal patterns similar to the infinitely shrinking snowflakes.
Since this breakthrough in the mid-1980s, called the Fractal Transform, Barnsley and his colleagues have developed and continued to improve computer algorithms that rapidly and automatically translate pictures into fractal formulae. The program analyzes a picture into collections of shapes that resemble each other, except for location, size and orientation. Each such collection of similar shapes can be precisely described as a fractal formula with certain parameters.
Measured in bytes, fractal formulas turn out to be a much more compact way to reproduce a picture than a raster bit map; hence, fractal encoding is fractal compression.
The redundancy that fractal compression depends on is called "affine
redundancy" (as in "affinity"), the surprising similarity of shapes that are apparently scattered throughout any image. An image that has low "affine redundancy" can be hard to compress, yielding a relatively large file. With fractal techniques, this can be compensated by allowing more time for the compression programs to run. The time it takes to decompress the image is not affected by this size-time tradeoff during the compression stage.
