Statistically Optimum Optical Data Processing with Automatic Focus Estimation

The problem of optimally processing data with unknown focus is investigated. Optimum data processors are found by the method of maximum likelihood under a variety of assumptions that apply to most of the situations arising in practice. The unknown focus may be either an unknown parameter or an unknown random variable; the signal may be of known form or a random function; it is further assumed that the signal is received in additive, white, Gaussian noise. The problems of jointly estimating other unknown parameters and, in the case of a random signal, jointly estimating the signal, are also treated. The asymptotic variance and correlation of the estimators is discussed. Electrooptical realizations of the maximum likelihood computers are given. An iterative method of solution of the likelihood equation is also discussed. The discussion and results are directly applicable to the processing of synthetic aperture radar data.     

Signal Sequence Detection Given Noisy, Common Background Image Sets

The optimum processing (likelihood functional) is found for a set of M images {Zm = Sm + Y + Nm}, each the sum of a member Sm of a signal sequence {Sm}, due to an object to be detected and its parameters estimated, a sample function Nm of a noise field {Nm}, and a sample function Y of a common background field {Y}. The noise fields {{Nm}} are independent, zero mean, white Gaussian fields, all independent of the background field {Y}; the latter is assumed to be either 1} completely unknown or of known mean and covariance functions with 2) a certain fluctuation property or 3) Gaussian. Three equivalent forms of the optimum processing are found: 1) a summation of generalized matched filterings of the images, 2) a summation of matched filtering of certain generalized differences of the images, 3) a summation of ?estimator-correlator? type filterings. The detection performance and optimum signal/image selection under the Neyman-Pearson criterion is given and the singularity of the ({{Nm = O}} and M > 1) case noted. It is shown that optimum processor and signal design can completely eliminate any effect of the background on detectability (M > 1). The Cramer-Rao lower bound for the signal parameter estimates meansquare error is given along with an example; optimum signal/image selection in the single parameter case is discussed.     

Synthetic Aperture Radar System Design for Random Field Classification

The optimum design of synthetic aperture radar (SAR) systems intended to classify randomly reflecting areas, such as agricultural fields, characterized by a reflectivity density spectral density is studied. Assuming areas of known shape and location, the binary case, and a certain Gaussian signal field property, and ignoring interfield interference, the problem solution is given. The optimum processor includes conventional matched filter processing, but is nonlinear; a coherent optical system realization is outlined. The performance is approximated using a x2 assumption and bounded by the Cernov bound. A fundamental design problem involves the system bandwidth analogously, in a special case, as in diversity communication systems; a solution is given based on the Cernov bound. A set of summary design curves is given and exemplified by a satellite SAR system design. Also discussed is the measurement of reflectivity spectral density amplitude with imaging sidelooking (synthetic or ?brute-force?) radars and the maximum likelihood estimator's accuracy and realization with a coherent optical system. It is also shown that a CW modulation is useable if the random reflectivity is, effectively, isotropic. Finally, the reflectivity density spectral density amplitude, when constant over the spatial bandpass of the measuring system, is related to the scattering cross-section density commonly measured.     

On Processing Optical Images Propagated Through the Atmosphere

The problem of how to process optical images that have propagated through a turbulent atmosphere is considered. It is assumed that either the object or its (free-space) scattered field at the receiving aperture is known, except for unknown parameters that are to be estimated. Isotropic and homogeneous turbulence as discussed by Tatarski and Chernov is assumed. It is assumed that a very short (in time) reception is made, that the turbulence-induced complex phase errors are either small or have a correlation distance short relative to the receiving aperture extent, and that the object is within an isoplanatic region. The method of maximum likelihood (ML) is used as a criterion; its applicability is discussed. The ML image-processing structure is found and is nonlinear. Asymptotic cases are examined; among other conclusions, it is shown that the ML processing of independent receptions is more complex than an ML processing of the sum of the receptions. A coherent optical system that realizes the ML image processor is described; it includes the capability to generate the scattered field. Simplifications are pointed out.     

Linear Minimum Variance Estimation with Complex Phase Errors

The linear minimum variance estimator of a random signal, received multiplied by a complex Gaussian phase error and added to random noise, is investigated. The results apply to the propagation of images through the turbulent atmosphere, fading channels, and synthetic-aperture radar. Among others, a result is that the multiplicative error can be replaced by an additive error, usually white. The best signal modulation is found in two important special cases.     

On the Mapping Associated with the Complex Representation of Functions and Processes

The mapping between function spaces that is implied by the representation of a real ?bandpass? function by a complex ?low-pass? function is explicitly emphasized. The discussion is extended to the representation of random processes where the mapping is between spaces of random processes.     

Radar Altimeter Optimization for Geodesy Over the Sea

The optimum processor and its accuracy limit for radar altimetry for geodetic use over the sea are studied with a model accounting for random surface reflectivity, sea height variation, additive noise, and pointing errors, and allowing for arbitrary antenna patterns, signal modulations, and other system parameters. The ?threshold? case solution (which can have any specified accuracy) dictates a signal modulation bandwidth just shy of resolving the sea height variation and/or illuminated sea area (as scaled into time delay and ?smeared? by pointing errors). For such a modulation a relatively complete solution is obtained. These results are used to determine practical radar altimeter designs, additionally accounting for antenna size, stability, and peak power restraints. Conditions allowing neglecting of limiting or complicating effects due to temporally varying reflectivity, sea height, and vehicle position are given and shown to be satisfied for a typical satellite.