The application of logical transforms to lossless image compression using Boolean minimization

A rapidly advancing field, image compression has seen many recent developments. A number of well-known compression algorithms dominate the world of lossy compression, but in the realm of lossless compression, fewer techniques have been explored. The most successful lossy compression methods are based on transform methodologies, commonly involving the KLT, cosine, and wavelet transforms. However, well-known lossless compression techniques such as Huffman coding, arithmetic coding, the Lempel-Ziv algorithm (LZW), and runlength coding do not utilize transforms. In this paper, we investigate the application of transform-based algorithms to the domain of lossless compression. As a first step in this research, we have applied logical transforms to lossless compression. Logical transforms are matrix-based transforms that operate on vectors whose elements represent either a binary ‘0’ or ‘1’. The central idea of the logical transform method is, indeed, similar to that of the lossy case: the image at hand is divided into blocks, and each block is transformed. However, logical transforms are applied to blocks of binary data in bit planes of the grayscale image to be compressed. Using a sequence of logical transforms designed to perform the process of Boolean minimization, we were able to successfully compress images and achieve ratios comparable to that of alternative lossless techniques. Existing techniques for image compression using the minimization of Boolean “switching” functions rely chiefly on time-inefficient algorithms for minimization using Karnaughmaps (K-maps) or other approaches. Using K-maps presents several drawbacks. First, the complexity of minimizing a given Boolean function increases rapidly as the size of its output truth table increases. Minimizing a 6-variable (length 64) Boolean function, for example, is a relatively complex challenge compared to the problem of minimizing a 4or 5-variable function (lengths 16 and 32, respectively) when K-maps are used. Second, there does not exist a fast implementation for the K-map approach, so the compression process is rather slow. In this paper, we propose a new technique for the lossless compression of images using logical transforms to achieve Boolean minimization. This new technique provides several important advantages over other Boolean minimization-based compression methods: a) here, a Boolean function of an arbitrary number of input variables can be minimized, b) fast implementations exist, c) attaining compression ratios greater than those of existing lossless methods is possible, d) no multiplications are needed, e) hardware implementations are practical, f) the technique can be extended to compress data sets other than images. The compression methods presented here clearly demonstrate the feasibility of lossless coding schemes based on logical transforms. We expect to find further optimizations and soon arrive at a definitive standard for a compression technique using logical transforms.