Some schemes for detecting moving point targets against structured backgrounds from observations on the output of an imaging system are investigated. When the velocity of the target and the background image are considered as known, it is shown that the uniformly most powerful detector, invariant with respect to image intensity variations, consists of specific spatial-temporal differencing schemes. This places such schemes on a rigorous foundation.

Let\cal ybe a mean zero complex stationary Gaussian signal process depending on a vector parameter\theta \prime = \{ \theta_{1}, \theta_{2}, \theta_{3} \}whose components represent parameters of the covariance function R(r) of\cal y. These parameters are chosen as\theta_{1} = R(0), \theta_{2} = |R( \tau )| /R(0), \theta_{3} =phase ofR( \tau), and they are simply related to the parameters of the spectral density of\cal y. This paper is concerned with the determination of most powerful (MP) tests that distinguish between random signals having different covariance functions. The tests are based uponNcorrelated pairs of independent observations on\cal y. Although the MP test that distinguishes between\theta = \theta_{o}and the alternative hypothesis\theta = \theta_{1}has been solved previously [11], the problem of identifying the random signals is often complicated by the fact that the signal power\theta_{1} = R(0)is not a distinguishing feature of either hypothesis. This paper determines the MP invariant test that delineates between the composite hypothesis\lambda \equiv R( \tau)/R(0) = \lambda_{0}and the composite alternative\lambda = \lambda_{1}. In addition, the uniformly MP invariant test that distinguishes between the composite hypotheses\theta_{2} <_{=} | \lambda_{o} |and\theta_{2} > | \lambda_{0} |has also been found. In all cases, exact probability distributions have been obtained.

Methods for estimating the normalized moments of the power spectrum of a stationary stochastic process via both spectral and covariance approaches are outlined. Various relationships between these two techniques are examined and typical extensions which submit to similar analyses are listed.

We are interested in estimating the moments of the spectral density of a comp[ex Gaussian signal process\{ q^{(1)} (t) \}when the signal process is immersed in independent additive complex Gaussian noise\{q^{(2)} (t) \}. Using vector samplesQ = \{ q(t_1),\cdots ,q(t_m)\}, whereq(t) = q^{(1)}(t) + q^{(2)}(t), estimators for determining the spectral moments or parameters of the signal-process power spectrum may be constructed. These estimators depend upon estimates of the covariance functionR_1 (h)of the signal process at only one value ofh \neq 0. In particular, ifm = 2, these estimators are maximum-likelihood solutions. (The explicit solution of the likelihood equations form > 2is still an unsolved problem.) using these solutions, asymptotic (with sample size) formulas for the means and variances of the spectral mean frequency and spectral width are derived. It is shown that the leading term in the variance computations is identical with the Cramér-Rao lower bound calculated using the Fisher information matrix. Also considered is the case Where the data set consists ofNsamples Of continuous data, each of finite duration. In this case asymptotic (withN) formulas are also derived for the means and variances of the spectral mean frequency and spectral width.

Let\{q^(1) (t)\}, the signal, be a complex Gaussian process corrupted by additive Gaussian noise\{q^(2) (t) \}. Observations onp(t)q(t)andp(t) q^(2) (t)are assumed to be available wherep(t)is a smooth weighting function andq = q^(1) + q^(2). Using the Fourier transform of the samples ofp(t)q(t)andp(t) q^(2) (t), estimators are derived for estimating the mean frequency and spectral width of the unknown power spectrum of the unweighted signal process. The means and variances of these statistics are computed in general, and explicitly for nontrivial practical examples. Asymptotic formulas for the moment estimators as a function of the number of realizations, frequency resolution, signal-to-noise ratio and spectral width, and consistency of the estimators are some of the results that are discussed in detail.