Pattern Matching Image Compression

We propose a non-transform image compression scheme based on approximate pattern matching, that we name Pattern Matching Image Compression (PMIC). The main idea behind it is a lossy extension of the Lempel-Ziv data compression scheme in which one searches for the longest prefix of an uncompressed image that approzimately occurs in the already processed image. We consider both the Hamming distance and the square error distortion. The theoretical basis for such a scheme was laid out by Luczak and Szpankowski [2]. A straightforward implementation of the basic scheme described in [2] on real images (structured data) seems not to be attractive from a practical point of view. The main algorithm is therefore enhanced with several new features such as searching for reverse approximate matching, recognizing substrings in images that are additively shifted versions of each other, introducing a variable and adaptive maximum distortion level D, and so forth. These enhancements are crucial to the overall quality of our scheme. The actual implementation (cf. [l]) suggests: l PMIC is slower in compression but the fastest possible in decompression, l Overall quality of compression is very good and competitive with JPEG and wavelet compression (especially when the image contains a substantial amount of high frequency components), . PMIC is systematically better than fractal compression. A unique feature of the proposed algorithm is that asymptotic performance of the scheme can be theoretically established. More precisely, under stationary mixing probabilistic model of an image and fixed maximum distortion level D, it is shown in [2] that the compression ratio is asymptotically equal to the so called generalited Re’nyi entropy Q(D). This entropy is in general smaller than the optimal rate distortion function R(D), but there is numerical evidence that these two quantities do not differ too much for small and medium values of D. The main algorithmic challenge consists of finding efficient algorithms for constructing the longest prefix that approximately occurs in the training sequence, both for Hamming distance and square error distortion. We give efficient algorithms for this problem that, when used as subroutines in the compression of an N x N image, result in 0( N2 log N) t ime if the database used is of size O(log N), and in O(N3) time if the database used is of size O(N). The next stage of this research will extend it to video and audio. The former extension is conceptually similar to what we already did, while the latter is substantially different and will be reported elsewhere.