Compression of Two-Dimensional Images

Distortion-free compressibility of individual pictures, i.e., two-dimensional arrays of data, by finite-state encoders is investigated. For every individual infinite picture I, a quantity ρ(I) is defined, called the compressibility of I, which is shown to be the asymptotically attainable lower bound on the compression-ratio that can be achieved for I by any finite-state, information-lossless encoder. This is demonstrated by means of a constructive coding theorem and its converse that, apart from their asymptotic significance, might also provide useful criteria for finite and practical data-compression tasks. The proposed picture-compressibility is also shown to possess the properties that one would expect and require of a suitably defined concept of two-dimensional entropy for arbitrary probabilistic ensembles of infinite pictures. While the definition of ρ(I) allows the use of different machines for different pictures, the constructive coding theorem leads to a universal compression-scheme that is asymptotically optimal for every picture. The results of this paper are readily extendable to data arrays of any finite dimension. The proofs of the theorems will appear in a forthcoming paper.