<title>Optical holography and computational intelligence: algebraic foundations</title>

Abstract

Our main interest is to formulate algebraic description of both Geometrical and Fourier-approximations of Optics. We use triangular- norms based approach to formulate algebraic descriptions for both geometrical and Fourier-approximations of optics. To take into consideration real nonlinearity of recording media the measure theory is used. Unlimited plane wave as an universal set is considered. For geometrical optics an algebraic model as designed. Logical operators, parameterized by recording media operators, are defined. To extend dynamical range of negative recording media from linear to over-exposure one, non-additive measure is defined. Algebraic properties of the model in dependence on the approximating function choosing are discussed. Theoretical conclusions are illustrated by experimental measuring and numerical simulation for two-layered optical system. For Fourier- approximation Fourier-duality is used to design semi-ring by DeMorgan's law using, 4-f Fourier-holography setup constructs sequence of model's elements, corresponding to Peano's axioms. Convolution is an abstract addition and correlation is an abstract subtraction in the model Fuzzy- valued measure is defined and fuzzy-value logic is designed. Theoretical conclusions are confirmed by experimental demonstration of logical inference Generalized Modus Ponens realization.