Evolutionary Fractal Image

This paper introduces evolutionary computing to frac-tal image compression. In fractal image compression 1] a partitioning of the image into ranges is required. We propose to use evolutionary computing to nd good partitionings. Here ranges are connected sets of small square image blocks. Populations consist of N p conng-urations, each of which is a partitioning with a fractal code. In the evolution each connguration produces children who inherit their parent partitionings except for two random neighboring ranges which are merged. From the oospring the best ones are selected for the next generation population based on a tness criterion (collage error). We show that a far better rate-distortion curve can be obtained with this approach as compared to traditional quad-tree partitionings. Finding the optimal partitioning at a given bit-rate is an unsolved problem in fractal image compression. The space of all partitionings with a given number of ranges is simply too huge. Traditionally, deterministically derived quad-tree 2, 3], rectangular 4], triangular 5, 6], and other polygonal 7] partitionings are used. We follow 8] and deene ranges as unions of edge-connected small square image blocks. The type of fractal image encoding chosen is the standard one: For a range block R we consider a pool of domain blocks twice the linear size. The domain blocks are shrunken by pixel averaging to match the range block size. This pool of codebook blocks is enlarged by including all 8 isomet-ric versions (rotations and ips) of a block. This gives a pool of codebook blocks D 1 ; : : : ; D ND. For range R and codebook block D we let (s; o) = arg min s;o2R kR ? (sD + o1)k 2 where 1 is the at block with maximal intensity at every pixel. The coeecient s is clamped to ?1; 1] to ensure convergence in the decoding and then both s and o are uniformly quantized yielding s and o. The collage error for range R is E(D; R) = kR?(sD+o1)k 2. Sorting the codebook blocks D k with respect to increasing collage error E(D k ; R) yields indices k 1 ; : : : ; k ND. The fractal code for range R consists of the optimal index k 1 and the corresponding quantized scaling and ooset parameters s and o. Initially we subdivide the image to be encoded into atomic blocks of the same size, e.g., of size …