Pattern matching and lossy data compression on random fields

We consider the problem of lossy data compression for data arranged on two-dimensional arrays (such as images), or more generally on higher dimensional arrays (such as video sequences). Several of the most commonly used algorithms are based on pattern matching: Given a distortion level D and a block of data to be compressed, the encoder first finds a D-close match of this block into some database, and then describes the data by describing the position of the match. We consider two idealized versions of this scenario. In the first, the database is taken to be a collection of independent realizations of the same size and from the same distribution as the original data. In the second, the database is assumed to be a single long realization from the same source as the data. We show that the compression rate achieved (in either version) is no worse than R(D/2) bits per symbol, where R(D) is the rate-distortion function. This is proved under the assumptions that (1) the data is generated by a Gibbs distribution, and (2) the distortion measure is a metric, generalizing the corresponding one-dimensional bound of Steinberg and Gutman (1993). Using large deviations results by Dembo and Kontoyiannis (see ibid., vol.48, p.1590-15, 2000) and by Chi (2001), we are able to give short proofs for the present results.