Fractal image compression and recurrent iterated function systems

Linear fractal models such as the iterated function system (IFS), recurrent IFS (RIFS), and Lindenmeyer system (L-system) concisely describe complex objects using self-reference. These models hold much promise in computer graphics as geometric representations of detail. The use of fractals for image compression sacrifices its fractal origins in the search for optimal coding gain. This article rebuilds the relationship between the fields of fractal image compression (FIC) and fractal geometry 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. The representation used by FIC has been called partitioned IFS or local IFS. This article describes a method for converting FIC's partitioned/local IFS to fractal geometry's RIFS. This conversion algorithm allows FIC 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. When every domain element can be expressed as the union of range elements, FIC produces a structure that is equivalent to a RIFS, enabling it to automatically model arbitrary shapes.