Iterated Function Systems and Two-Dimensional Shape Representation

Abstract

An application of Iterated Function Systems to the problem of automatic generation of two-dimensional shape representation is presented. A definition of an Iterated Function System is given, and the properties that make it suitable for shape representation are described. Details of programs implementing shape coding and rendering are provided together with an assessment of their performances. A fundamental requirement of any vision system designed to perform recognition tasks is a library of representatio ns of objects it is likely to encounter. The classic approach to constructing such representations is to decompose the shape into simple primitive elements. However, for complex shapes it is usually necessary to use a large number of such primitives to achieve an accurate decomposition. The alternative is to define more complex, context-specific primitives, which would be of practical use for only a small set of shapes. The advantage of using an Iterated Function System (IFS) instead is that it enables us to construct a recursive definition of shape. This is achieved by using contractive affine transformations of the original shape as the primitives in its decomposition, and thus removing the need for shape primitives to be defined prior to encoding. As a first approximation an IFS coding can be thought of as simply a compact list of numbers corresponding to the transform coefficients that describe how a shape maps into itself - in effect how the shape can be covered by a collage of smaller versions of itself. These numbers can then be used by a very simple rendering algorithm to produce a reconstruction of the whole of the original shape. This reconstruction is known as the 'attractor' of the IFS. The rendering process is rapid, stable, and does not depend on any information external to the IFS to enable accurate reconstruction. The nature of the IFS code enables the rendering of a shape with the same speed and accuracy independent of the size and orientation required. Previous applications of IFS theory to shape encoding [1,2,3], have relied heavily on the interaction of the user with computer programs in order to make the necessary shape collages, and in the case of [3], coding was restricted to shapes that were self-similar and that lend themselves easily to the IFS coding technique. The methods outlined in our paper constitute an entirely automatic way of producing IFS codings.