Compression of two-dimensional data

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
\rho(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
\rho(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 are readily extendable to data arrays of any finite dimension.