The problem of classifying an observation into one of several different categories, or patterns, is considered. The observation consists of a sample function of a continuous-time parameter stochastic process observed over a finite-time interval. When only two categories are involved the general pattern recognition problem reduces to the signal detection problem. The methods used are based upon results from the theory of reproducing kernel Hilbert spaces. This theory has been developed within the last few years and the application of these results to problems of statistical inference for stochastic processes has taken place only recently. Therefore, a reasonably serf-contained exposition of the results required from the theory of reproducing kernel Hilbert spaces is presented. It is pointed out that the decision rule employed by the optimum pattern recognition system is based on the likelihood ratio. This quantity exists fi, and only if, the probability measures are equivalent, i.e., mutually absolutely continuous with respect to each other. In the present work only Gaussian processes are considered, in which case it is well known that the probability measures can only be either equivalent or perpendicular, i.e., mutually singular. It is shown that the reproducing kernel Hilbert space provides a natural tool for investigating the equivalence of Gaussian measures. In addition, this approach provides a convenient means for actually evaluating the likelihood ratio. The results are applied to two pattern recognition problems. The first problem involves processes which have the same covariance function but different mean-value functions and the second problem concerns processes with different covariance functions and zero mean-value functions.

The problem of designing a linear predetection filter for the detection of a sampled random signal in additive noise is considered. The design of the filter is based on an optimality criterion which maximizes the signal-to-noise ratio enhancement. The optimum weighting function obtained in this manner has the advantage that it is independent of signal characteristics and depends only on the covariance function of the noise. The optimum filter, for general covariance functions, is obtained forN = 2, 3and4samples. The asymptotic solution for largeNis also presented by employing results from the theory of Teeplitz forms. In addition, the complete solution for allNis given for several particular covariance matrices. An application of the results is made to the problem of designing a linear predetection filter in a moving target indication (MTI) radar system. The optimum weighting function forN = 2is a single-cancellation unit, while that forN = 3is similar but not quite the same as a double-cancellation unit. It is shown that the signal-to-noise ratio enhancement provided by the double-cancellation scheme is1.76db worse than that of the optimum filter when the noise has a Gaussian covariance function.

The eigenfunctions and eigenvalues of a certain integral equation are of importance in the Karhunen-Loéve expansion of second-order stationary random functions. In this note the asymptotic eigenfunctions and eigenvalues of this integral equation are derived for the case where the kernel is the Fourier transform of a rational function of\omega^2.

A detector examines an unknown waveform to determine whether it is a mixture of signal and noise, or noise alone. The Neyman-Pearson detector is optimum in the sense that for given false alarm probability, signal-to-noise ratio, and number of observations, it minimizes the false dismissal probability. This detector is optimum for all values of the signal-to-noise ratio, and its implementation is usually quite complicated. In many situations it is desired to detect signals which are very weak compared to the noise. The locally optimum detector is defined as one which has optimum properties only for small signal-to-noise ratios. It is proposed as an alternative to the Neyman-Pearson detector, since in practice it is usually only necessary to have a near-optimum detector for weak signals, since strong signals will be detected with reasonable accuracy even if the detector is well below optimum. In order to evaluate the performance of the locally optimum detector, it is compared to the Neyman-Pearson detector. This comparison is based on the concept of asymptotic relative efficiency introduced by Pitman for comparing hypothesis testing procedures. On the basis of this comparison, it is shown that the locally optimum detector is asymptotically as efficient as the Neyman-Pearson detector. A number of applications to several detection problems are considered. It is found that the implementation of the locally optimum detector is less, or at most as complicated as that of the Neyman-Pearson detector.

A first-order Markoff process representation for pictures is proposed in order to study the picture coding system known as run-length coding (differential-coordinate encoding). A lower bound for the saving in channel capacity is calculated on the basis of this model, and is compared with the results obtained by previous investigators. In addition, this representation is shown to yield an insight into the run-length coding system which might not otherwise be obtained. The application of this probabilistic model to an "elastic" system of run-length coding is also discussed.