Fractal image compression and recurrent iterated function systems

ractal geometry provides a basis for modF eling the infinite detail found in nature. Fractal methods are quite popular in computer graphics for modeling natural phenomena such as mountains, clouds, and many kinds of plants. Linear fractal models such as the iterated function system (IFS), recurrent iterated function system (RIFS), and Lindenmeyer systern (L-system) concisely describe complex objects using self-reference. These models hold much promise in computer graphics as geometric representations of detail. Fractal techniques have recently found application in the field of image compression. The use of fractals for compression has grown into a well-established area of signal processing, but this use sacrifices its "fractal" origins in the search for optimal coding gain. The siidebar (p. 26) identifies the differences between the fields of fractal image compression and fractal geometry. This article rebuilds the relationship between them to better facilitate the sharing of new results. Many geometric representations exist for smooth shapes, and each has certain benefits and drawbacks. Computer-aided geometric design has produced many algorithms to convert a given curve or surface description into the most appropriate geometric representation for a given task. Likewise, there are several models for linear fractal shapes and several methods for converting between the representations, such as from L-system to RIFS,' from RIFS to L-system,' and from L-system and RIFS to constructive solid g e ~ m e t r y . ~ The representation used by fractal image compression has been called partilioned IFS4 or, synonymously, local IFS.' This article describes a method for converting fractal image compression's partitioned/local IFS to fractal geometry's RIFS. This conversion algorithm allows fractal image compression to represent any input shape as a linear fractal and permits algorithms developed for linear fractals to be applied to a wider variety of shapes.