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AN APPROACH TO OPTICAL IMPLEMENTATION OF FUZZY INFERENCE SYSTEMS
Alexander V.Pavlov
Laboratory for Optical Neural Networks, S.I.Vavilov State Optical Institute.
12, Birgevaya line, St.Petersburg, 199034, Russia.
E-mail: pavlov@soi.spb.su
Abstract — Our main interest in this paper 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 op tics. Geometrical optics implements probabilistic operators. We consider Fourier-approximation as an extension of N-duality to Fourier-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. We discuss a holographic fuzzy inference engine. Implication operator, implemented by Fourier-holography technique, is defined. Experimental results are presented.
Our main interest is to establish connections between Fuzzy Set Theory and Optics. From a practical point of view the consideration is useful due to the ability of optics to solve a number of problems, which are essential for elect ronic hardware. We restrict our consideration to two approximations: geometrical and Fourier- optics. Our study is motivated by the analogies [1,2] between optics as a physical reality and FST as an abstract construction :
- we consider the basic phenomenon of diffraction as the cause for fuzziness springing up;
- we consider the basic phenomenon of interference as the mechanism of fuzzy relation forming.
Definition 1. Let V:[0,1]2 ® [0,1] be a commutative, associative, non-decreasing binary operation with neutral element e, i.e. V(a,e) = V(e,a) = a for all a Î [0,1]. Then
(i) if e = 1, V is called a t-norm (V=T),
(ii) if e = 0, V is called a t-conorm (V = S).
Definition 2. A mapping N:[0,1] ®[0,1] is called a negation operator if
N(1)=0, N(0)=1, (1)
N(a) £ N(b) if a³b for all a,bÎ[0,1]. (2)
Negation is strict if N(a) is strictly decreasing and continuous. Negation is involutive if
N(N(a))=a (3)
The introduction of a negation allows a general De Morgan's law relating t-norms and t-conorms to be provided.
Definition 3. A mapping j: [0,1] ® [0,+ ¥ ] is called an additive generator of negation if j (0 =0 and j (1) is limited. Then
N(a) = j *( j (1) - j (a)), (4)
where j * is defined by:
a Î [0, j (1)], j *(a) = j -1(a)
a ³ j (1), j *(a) = 1 ,
where j -1 is inverse function of j.
Definition 4. Let an unlimited plane wave be universe X. Then any image Im, i.e. optical field in any plane, or a transparency, is a subset of X. Pixels are members and membership function is Im: X ® [0,1] ´PIm. In general case, the size of the member is determined by the diffractive limit.
3. Geometrical optics
Recording of image and modulation of light are the two operations, that to be considered. The latter operation is based on the first one: modulation is realized by illuminating a transparency, fabricated by recording of image Im. I t is the recording is to be a starting-point for our consideration.
Let us consider a conventional negative photo-process. The dependence of amplitude transmittance t on the exposuring amplitude Im, typical for conventional silver halide recording media, has a quasi- linear range [tmin,tmax]. The process implements Zadeh's complementation
N(Im)= = tmax - Im,tmax £1. (5)
The additive generator is j = Imtg-1, where constant t is an exposure time, g is a contrast ratio. The value of para meter tg-1 is chosen for the condition j(1) = 1 to hold. It has a physical sense of optimal exposure for the amplitude Im to be contained into the linear dynamical range. Condition tmax =1 can be met by using conventional normalizing procedure.
Negation (5) generates strict t-norm
T(ImA,ImB) = ImAImB. (6)
t-norm (6) describes the modulation of coherent illuminating light by the transparency. The neutral element for t-norm (6) is an unlimited plane wave X.
N-duality gives t-conorm
S(ImA,ImB) = ImA+ ImB - ImAImB. (7)
Hence, geometrical optics constructs probabilistic operators [3,4] under the assumption of linearity of the medium.
The fact, that an additive generator for negation (5) is a multiplicative generator for t-norm, in optics means that a two-stage negative-positive process is needed to prepare a modulating transparency.
Taking into account the diffraction phenomenon as a cause for image blurring retains the definition of t-norm (6) (ImC is to be used in (6) instead of ImB), and allows the duality principle to be e xtended from N-duality to Fourier-duality.
Let us consider a partial case - an amplitude distribution Im is described by Gaussian function. Then Fourier-transform operator F gives Gaussian distribution in the Fourier-plane too. In this case a Fourier-operator fits exactly the axiomatic definition of negation and can be assumed as the last one.
If the amplitude distribution is not Gaussian, definition N:[0,1] ® [0,1] is substituted by F:[0,1] ´PIn ®
[0,1]
In reality, the F-operator is not involutive because of the fact that the dynamical range is essentially narrower than it is needed for the Fourier-spectrum to be recorded. Unlike geometrical optics, this restriction cannot be ignored and operator h has to be taken into consideration. The usage of DeMorgan's law leads to t-conorm
S(ImC,ImA) = ImC*F(h(F(ImA))), (7)
where symbol * denotes sum-product convolution, realized in -1 diffraction order of the setup in Fig.1.. t-conorm (7) has an additive neutral element d, which is F-dual to the multiplicative neutral element X. Operator h depends on a number of parameters [6,7] and expression (7) describes a family of t-conorms, parameterized by h.
Opposite element ImAo for ImA(i,s) is defined by ImAo = ImA(-s,-i), where i and s are inverse functions for increasing and decreasing parts o f ImA. For real ImA,
F(ImA(-s,-i)) = F* (ImA(i,s)). (8)
The usage of the definition of subtraction as an addition with opposite [8] leads to the following definition
ImA-ImB = ImA**ImBo = ImA Ä ImB, (9)
where symbol Ä denotes correlation, realized in the +1 diffraction order of the setup in Fig.1..
Let us consider fuzzy inference [9]
Implication: if C is ImA, then B is ImB
Premise: C is ImC = ImA'
_______________________________
Conclusion: B is ImB'
where A, B, and C are linguistic variables, and ImA, ImA', ImB, ImB', and ImC are fuzzy sets representing linguistic labels. To implement the rule, an implicat ion function, defined as the fuzzy relation between ImA and ImB, is required.
As it was demonstrated in [1,2], Fourier-hologram is the fuzzy relation. The consequent ImB' is the result of an antecedent ImC on the fuzzy relation mapping. This inference is implemented by the setup in Fig.1. i n the +1 diffraction order, i.e. in the terms of t-conorm
S(ImA',T(A®B))=ImA'*F(h(F*(ImB)F(ImA))).(10)
where symbol ® denotes implication operator, and T is a truth function. Thus, implication is defined by the expression
T(A®B) = F(h(F*(ImA)F(ImB))), (11)
Expression (11) leads to a fuzzy-valued logic. If conclusion is defined by
(S(ImA',T(A®B)))max2=
=ImA'*F(h(F*(ImB)F(ImA)))max2, (12)
it restricts inference to the multi-valued logic, realized in the conventional practice of image correlation processing [7].
Experimental illustration. The setup in Fig.1. was used. The reference image ImA was an air-photograph of a forest, sloped at 35o around the x-axis by using photo-transformer. The antecedent was distorted-f ree reference image. To get correlation signal, the antecedent was sloped in the other transformer, connected with the setup input plane, to achieve maximum value of (12). Sections on the x-axis for both consequents ImB for ImC = I mA (modus ponens) and ImB' for ImC = ImA' (generalized modus ponens), normalized by ImBmax = ImB’max=1, are presented in Fig.2..
Fig.2. Experimental implementation of generalized modus ponens rule. Consequents: ImB for antecedent ImC = ImA (MP) and ImB' for antecedent ImC = ImA' (G MP).
In the framework of multi-valued logic the ratio
(S(ImA',T(A®B)))max2
-------------------------- Î
[0,1] (13)(S(ImA,T(A®B)))max2
is the conclusion. In the framework of fuzzy logic, an analysis of consequent ImB’ in respect to ImB allows us to make conclusion on the character of the residual distortions in the transformed ImA’, concentrated in the spectral range of high spatial frequences [7].
6. Conclusion
Thus, optics under geometrical approximation forms the algebra with Zadeh's complementation and N-dual pair of probabilistic operators. Fourier-optics leads to an extension of the negation operator to a non-monotonic mapping from a n image space to a Fourier-space. It allows to consider the sum-product convolution as the t-conorm, that is F-dual to the product t-norm. This algebra leads to a fuzzy-valued logic. The implication operator, realized by Fourier-holography setup with a p lane reference wave, is defined.
The operator of the holographic recording medium parameterizes the family of the t-conorms. It allows the rigour of the conclusion to be controlled and addition factors to be taken into account by the operator driving. The next step of our research will be developing of the special properties of Liquid Crystal Holographic Recording Media:
- direct and inverse operators;
- real-time driving of the operator;
- adaptive driving of the operator.
7. Acknowledgments
Author thanks Dr. A.N.Averkin for the discussion of the idea of this research.
This work was supported by the Russian Foundation for Basic Research, grant 98-02a-18189.
8. References
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