366 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 9, NO. 3, MARCH 2000 A Graph-Theoretic Approach for Studying the Convergence of Fractal Encoding Algorithm Jayanta Mukherjee, Pramod Kumar, and S. K. Ghosh Abstract—In this paper, we present a graph-theoretic interpre- tation of convergence of fractal encoding based on partial iterated function system (PIFS). First we have considered a special circum- stance, where no spatial contraction has been allowed in the en- coding process. The concept leads to the development of a linear time fast decoding algorithm from the compressed image. This con- cept is extended for the general scheme of fractal compression al- lowing spatial contraction (on averaging) from larger domains to smaller ranges. A linear time fast decoding algorithm is also pro- posed in this situation, which produces a decoded image very close to the result obtained by an ordinary iterative decompression al- gorithm. Index Terms—Attractor, contractive transform, fixed point, fractal compression, partial iterated function system (PIFS). I. INTRODUCTION F OR THE last ten years, fractal encoding schemes based on iterated function systems (IFS) [1] have drawn significant attention from researchers. One of the important issues for fractal encoding scheme is the convergence of the decoding algorithm. In the outset of Jacquin’s proposed algorithm [3] based on partial iterated function system (PIFS), there are various approaches in [4], [5], [7] [13], and [15]–[17] to prove the convergence of the algorithm and find out the tighter bounds on the convergence criterion. In this paper, we will first revisit these criterion (with the underlying assumptions) and present a graph theoretic interpretation of convergence of fractal encoding. First, we will consider a special circumstance, where no spatial contraction has been allowed in the encoding process. The concept leads to the development of a linear time fast decoding algorithm from the compressed image. This concept is extended for the general scheme of fractal compression allowing spatial contraction (on averaging) from larger domains to a smaller ranges. A linear time fast decoding algorithm is also proposed in this situation, which produces a decoded image very close to the result obtained by an ordinary iterative decompression algorithm. It is interesting to note that earlier, Oien and Lepsoy [14] pro- posed a noniterative method to decode an encoded image. They considered averaging operation while shrinking a larger domain to smaller range and also put the constraint on the domain size Manuscript received January 4, 1999; revised August 18, 1999. The associate editor coordinating the review of this manuscript and approving it for publica- tion was Dr. Yoshitaka Hashimoto. The authors are with the Department of Computer Science and En- gineering, Indian Institute of Technology, Kharagpur, India 721 302 (e-mail: jay@cse.iitkgp.ernet.in; pkumar@cse.iitkgp.ernet.in; som@cse. iitkgp.ernet.in). Publisher Item Identifier S 1057-7149(00)01503-7. which should be of ( is an integer and ) times of the size of a range block. In their encoding assumption, the range partition and domain construction is such that every domain block in the image is made up of an integral number of range blocks. In our method, there is no such constraint on the domain construction and the domain size may also be equal to a range size. There are also other efforts for speeding up the decoding operation. Hamzaoui [7], [9] proposed a fast decoding algorithm using the updated pixels under same iteration. Hamzaoui [8] also proposed to use a suitable order of decoding range blocks for fast convergence. The ordering is based on the frequency with which a pixel is used in the fractal code. By using multires- olution fractal decoding from low resolution to high resolution, Baharav et al. [2] also achieved speed-up in decoding encoded images. The speed of their decoding algorithm is approximately twice of the conventional one. There are a few other important observations drawn from this study. While compressing without spatial contraction only a few transformations are required to be contractive (magnitude of the scaling factor 1). Another important observation is that it is not true that larger domain pool will always give better quality of decompressed image (considering other parameters remain same). In fact we have observed that small number of domain pools are also giving significantly better quality of picture after decoding. We have identified the presence of redundancy in the encoding process. This also plays a role in determining the quality of the decompressed image. In Section II, we briefly present the conventional encoding and decoding schemes. First, the basic fractal coding algorithm in the line of Jacquin’s proposed scheme has been presented and then follows a study of its convergence criterion under different conditions. In Section III, we have presented our graph theo- retic approach for studying the convergence of the fractal en- coding scheme. Subsequently, the fast decoding algorithms and our analysis on the characterization of fractal encoding schemes are presented. II. CONVENTIONAL SCHEMES A. Fractal Encoding Scheme Based on PIFS An image is defined as the mapping of points in discrete two-dimensional (2-D) space to grey level values be- longing to real set such that . The input image is partitioned into nonoverlapping range blocks, each of size and overlapping domain blocks of size . For every range block belonging to , we find a suitable domain block belonging to and an associ- ated affine transformation such that can be reconstructed 1057–7149/00$10.00 © 2000 IEEE