Alexander V. Pavlov

Optical holography and computational intelligence: algebraic foundations

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 DeMorgans law using, 4-f Fourier-holography setup constructs sequence of models elements, corresponding to Peanos 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.

An algebraic model for application of optical neuro-fuzzy systems to networking problems

Abstract Recent developments in architectures, devices, and protocols have dramatically increased the traffic capacity of telecommunication networks. New control and management techniques are needed for the potential, by this progress promising, to be achieved. Soft computing (SC) is the approach that gives an attractive way to this goal. From the hardware point of view, the main problem is SC based methods are computationally expensive. Optics gives a reasonable alternative for purely electronic hardware. Our main interest in this paper is to establish connections between optics as a physical reality and fuzzy set theory as an abstract construction and develop a model of Fourier-holography set-up as a neuro-fuzzy system. We formulate an algebraic description of both geometrical and Fourier-approximations of optics by using t-norms based approach to take into consideration non-linearity of real optical devices and recording media. We design logic for both approximations. We consider Fourier-holography set-up as a neuro-fuzzy system. We present experimental realisation of General Modus Ponens rule and develop model of partial associations.

Optical technologies for computational intelligence

Optics has a number of deep analogies with main principles of Computational Intelligence. We can see strong analogies between basic optical phenomena, used in Fourier-holography, and mathematical foundations of Fuzzy Set Theory. Also, analogies between optical holography technique and principles of Neural Networks Paradigm can be seen. Progress in new holographic recording media with self-developing property leads to Evolutionary Computations holographic realization. Based on these analogies we review holographic techniques from two points of view: Fuzzy Logic and Fuzzy relations.

Plural partial associations

The idea of the Plural Partial Associations is proposed to designate the case of associatability having the prototypes in ordinary life, but not described by the Associative Memory principle. The method based on computation of the correlation functions of the input vector with the back of the reference vectors is discussed as a one of the possible way to realize Plural Partial Associations and neural network model to implement this way is discussed. The possibility of this NN model implementation by two-layered bidirectional optical neural network used the Van der Lught correlator is discussed by the published before theoretical and experimental results.

Implementation of linguistic models by holographic technique

In this paper we consider linguistic model as an algebraic model and restrict our consideration to the semantics only. The concept allows “natural-like” language to be used by human-teacher to describe for machine the way of the problem solving, which is based on human’s knowledge and experience. Such imprecision words as “big”, “very big”, “not very big”, etc can be used for human’s knowledge representation. Technically, the problem is to match metric scale, used by the technical device, with the linguistic scale, intuitively formed by the person. We develop an algebraic description of 4-f Fourier-holography setup by using triangular norms based approach. In the model we use the Fourier-duality of the t-norms and t-conorms, which is implemented by 4-f Fourier-holography setup. We demonstrate the setup is described adequately by De-Morgan’s law for involution. Fourier-duality of the t-norms and t-conorms leads to fuzzy-valued logic. We consider General Modus Ponens rule implementation to define the semantical operators, which are adequate to the setup. We consider scales, formed in both +1 and -1 orders of diffraction. We use representation of linguistic labels by fuzzy numbers to form the scale and discuss the dependence of the scale grading on the holographic recording medium operator. To implement reasoning with multi-parametric input variable we use Lorentz function to approximate linguistic labels. We use an example of medical diagnostics for experimental illustration of reasoning on the linguistic scale.

Holographic neuro-predictor for fractional Brownian motion

In this paper we discuss our approach to based on holographic techniques implementation of neuro-fuzzy predictor for processes, described by Fractal Brownian Motion (FBM) model. We use the model of the predictor as a Riemann - Stieltjes integral over the observed traffic of specific weight function. We discuss two-layered bi-directional optical neural network to find our solution. To find the weight function we use non-linearity in the correlation layer of the neural network. In our experiments we used air-photograph of forest as this kind of images demonstrates self-similarity property and can be described by the FBM model. As a first step we used approximate solution for the weight function, achieved by using binary filtering function in the correlation layer. We demonstrate experimental results and discuss directions of our future investigations.

Linguistic modeling by Fourier holography technique

Linguistic modeling is the way, which allows human-like language to be used to teach machine. The task is to match metric scale of the machine with linguistic scale of the teacher. We discuss application of Fourier-holography technique to this task. We formulate logical description of 4-f Fourier-holography setup. Restrictions on the model, determined by property of real recording media are discussed. An example of linguistic measurement by the setup is presented.

Algebraic model of optical intelligence systems: Geometrical optics approximation

We consider algebraic foundations of geometrical optics approximation. The consideration is aimed at optical implementation of computational intelligence models. Theory of triangular norms and measure means are used to formulate the description. The process of negative photo-registration is considered as the implementation of the negation, which generates the algebra. Three approximations of negative recording media transmittance are considered: linear, involutive, and non-involutive one. Optically realizable orders and relations of fuzzy numbers, fuzzy sets and images are considered.

Application of holographic systems to networking problems

We discuss a way to implement predictor for stochastic process with stationary increments by using optical neural network.

Construction of fuzzy logic by optical techniques

Our main interest is to establish connections between Optics and Fuzzy Set Theory. We formulate the t-norms based algebraic description of both geometrical and Fourier- approximations of optics. Geometrical optics implements probabilistic operators under the linear approximation of negative recording process. For real recording media not Zadehs, but Sugeno negation is more appropriate approximation. It gives dual to the product t-norm family of t-conforms, parameterized by the recording medium and developing process properties. Fourier-optics allows Fourier-duality to be used in addition to N-duality. Fourier-holography setup implements semiring with product t- norm and F-dual family of t-conorms - sum-product convolutions, parameterized by holographic recording medium operator. Implication operator, implemented by Fourier- holography technique, is defined. Experimental realization of General Modus Ponenes rule by holographic fuzzy interference engine is presented.