Hierarchical Interpretation of Fractal Image Coding and Its Application to Fast Decoding

The basics of a block oriented fractal image coder, are described. The output of the coder is an IFS (Iterated Function System) code, which describes the image as a xed-point of a contractive transformation. A new hierarchical interpretation of the IFS code, which relates diierent scales of the xed-point to the code, is presented and proved. The proof is based on nding a function of a continuous variable, from which diierent scales of the signal can be derived. Its application to a fast decoding algorithm is then described, leading typically to an order of magnitude reduction of computation time. The use of fractal shapes to describe real world scenes has been shown to result in very realistic images 1]. This is due to the self-similarity property of fractal shapes, a property which is frequently encountered in real world scenes 2]. One way of creating a fractal shape is by considering it as a xed-point of a contractive Iterated Function System (IFS) 3]. The result is that a complex image can be described by a rather small number of IFS parameters. Thus, the IFS provides an eecient representation of a fractal image. Two issues are involved in coding an image by an IFS : encoding-Given an image, nd a contractive IFS such that its xed point is suuciently 'close' to the image. The parameters of this IFS are then called the IFS code of the image. Throughout, the terms 'IFS' and 'IFS code' will be used interchangeably. This problem is known as the 'inverse problem'. decoding-Given a contractive IFS, nd its xed point. The problem of encoding has been widely addressed 3]-6]. Since general images are not necessarily self similar, a way to nd the IFS code is to partition the image into blocks, and to restrict the functions forming the IFS to a set of transformations on these blocks. Given the set of allowed transformations and the image partition, the task is to nd an IFS such that its xed point is as 'close' to the original image as possible. The problem of decoding can be solved by applying the contractive-mapping theorem 7]. Namely, iterate repeatedly the IFS on any starting image, and the resulting image will converge to the desired xed-point. The main drawback with this decoding scheme is its computational cost. This cost is high because the iterations are done on a full size image. In this paper …