Lie methods in optics: An assessment -- Holographic image processing, coherent optical computing, and neural computer architecture for pattern recognition -- Canonical integration and analysis of periodic maps using non-standard analysis and Lie methods -- Concatenation of Lie algebraic maps -- Dispersion-diffraction coupling in anisotropic media and ambiguity function generation -- Elements of euclidean optics -- The map between Heisenberg-Weyl and Euclidean optics is comatic.

Recent developments in Lie methods applied to various problems in optics and computer design are surveyed in this volume, based on lectures given and work done at the 1988 workshop held in Cocoyoc, Mexico. Topics discussed include perturbation expansions, the mathematical foundations of coherent optical computing, holographic image and interferometry, neural architecture for pattern recognition, recent progress in symbolic calculations with Lie structures together with applications, the operations of convolution and correlation of signals performed by optical means, wide-angle optics based on the Euclidean group of motions and its relation to the Heisenberg-Weyl approach to canonical quantization. Applications discussed include computer design, particle optics in the Superconducting Supercollider, and neural networks. Computational techniques are emphasized. This volume is an excellent introduction to a rather active field of research and can be recommended to graduate students as well as to researchers.