Fractal Image Compression

The first fully automated fractal image compression algorithm was published by Arnaud Jacquin in 1989. Given an image f , the encoder finds a contractive affine image transformation (fractal transform) T such that the fixed point of T is close to f . The decoding is by iteration of the fractal transform starting from an arbitrary image. Due to the contraction mapping principle, the sequence of iterates converges to the fixed point of T . Several researchers have improved the original algorithm. This document describes the contribution of our group. 1. Reduction of the encoding complexity In the encoding step, the image is partitioned into disjoint blocks (range blocks). For each range block, another block (domain block) is selected from the same image. The goal is to approximate the pixel intensities of the range block with those of a domain block. Because good approximations are obtained when many domain blocks are allowed, searching the pool of domain blocks is time-consuming. 1. Nearest neighbor search [11, 13]: Range blocks and domain blocks are assigned d-dimensional feature vectors such that searching in the pool of domain blocks can be restricted to the domain blocks whose feature vectors are the nearest neighbors of the feature vector of the current range block. Thus, the sequential search in the domain pool is replaced by multi-dimensional nearest neighbor searching which can be run in logarithmic time. 2. Clustering methods [1, 4]: The domain blocks are classified by clustering their feature vectors in Voronoi cells whose centers are designed from the test image or from a set of training images. For each range block, matches are sought in the neighboring classes only. 3. Fast search via fast convolution [12, 15, 7]: This is a lossless method in a sense that the domain block that yields the minimal approximation error is found. A calculation of the inner products between the range blocks and codebook blocks (blocks formed from the domain blocks) dominates the computational costs in the encoding. The codebook blocks are typically defined by downfiltering the image to half its resolution. The inner products are the finite impulse response (FIR) of the downfiltered image with respect to the range blocks. Thus, the cross-correlation of the range block with the downfiltered image is required. This discrete two-dimensional convolution can be carried out more efficiently in the frequency domain when the range block is not too small. 4. Domain pool reduction [14]: An adjustable number of domains are excluded from the domain pool. We studied the effects on computation time, image fidelity and compression ratio. We showed that there is no need for keeping domains with low intensity variance in the pool. 2. Use of adaptive partitions We devised a fractal coder that finds the image partition by a split and merge process, yielding range blocks as unions of edge-connected small square blocks. This fractal coder has a better rate-distortion performance and subjective quality than the widely used quadtree-based fractal coders [6]. Figure 1 compares the subjective quality performance of our fractal coder with that of the state-of-the-art wavelet coder of Said and Pearlman (SPIHT). 3. Rate-distortion fractal coding Usually, the fractal transform is found in a heuristic way. We proposed rate-distortion based fractal coding [16] where the fractal transform is optimal in the sense that it guarantees the lowest approximation error over a large set of admissible transforms subject to a rate constraint. We begin with a fine scale hierarchical partition of the image and use a pruning strategy based on the generalized BFOS algorithm. We give results for rectangular partitions. 4. Computational complexity Standard fractal coding is a greedy suboptimal algorithm. We showed that the problem of finding the optimal fractal code is computationally intractable [10].