Knowledge of the bandwidth of a signal allows for extrapolation over a time interval that can be many times the length of the given segment. Computer results, utilizing a version of a one-step small-matrix minimum-energy procedure, have shown that such extrapolation is feasible in the presence of noise and coarse (5-bit) quantization. It is shown here, however, that such a minimum-energy procedure cannot produce enhanced spectral resolution even in the absence of noise. By generalizing a unifying model for such procedures to include non-minimum energy bandlimited extrapolation, a new algorithm is proposed that allows for tradeoffs to be made between enhanced spectral resolution and reduced sensitivity to out-of-band noise. This algorithm may also be utilized in two dimensions to enhance the spatial resolution of an aperture-limited sensor when viewing a region of limited extent. However, resolution enhancement by bandlimited extrapolation appears to be practical only if the resolution is extremely poor to begin with.

In various pattern-recognition problems such as classification of photographic data, preprocessing operations result in a two-dimensional array of binary random variables. An optimal recipe for classifying such patterns is described. It combines the use of an orthonormal expansion for the logarithm of probability functions, with the generation of a joint probability distribution from lower-order marginals. The unwieldy nature of the optimal recipe leads to the consideration of dependence represented by a Markov chain and a two-dimensional analog called a Markov mesh. The Markov chain has a "reflecting" property in that therth order nonstationary chain implies that a point depends on itsrnearest neighbors on each side. By scanning along a grid-filling curve so that the2rneighbors of a point are geometrically close to it, certain spatial dependencies are obtained. The grid-filling curves are sequences of functions which have Hilbert "space-filling" curves as their limit. This simple approach does not provide for many of the spatial dependencies that should be considered. A novel extension of Markov chain methods into two dimensions leads to the Markov mesh which economically takes care of a much larger class of spatial dependencies. Likelihood ratio classification based on Markov chain and Markov mesh assumptions requires the estimation of a much smaller number of parameters than the general case. The development presented in this paper is new and need not be restricted to binary variables.