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Download Algebraic Methods in Pattern Recognition: Course held at the by Juliusz Kulikowski PDF

By Juliusz Kulikowski

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Extra info for Algebraic Methods in Pattern Recognition: Course held at the Department of Automation and Information July 1971

Sample text

In particular, the de Morgan's identities hold: (2. 24) 32 Chap. 26b) as well as the distributi vity law of any operation·. with. ,x,-,> form a Boolean algebra with the empty relation as a null and the trivial relation Q as an unity element. This last statement may be very important in pra~ tice. It means that the well known methods of logical functions minimization and constructive methods of logical systems design can be applied for the design of picture interpretation and pa! tern recognition algorithms.

Um-t,n-i Um-t,n ... owing relation describ ed on it: R' = {<10 10>,<01 01>} . Then, let us consider the family of sets and the relation The relations R' and R" are not identical, of course. Nevertheless, they are mutually isomorphic. t it is an order-preserving proje_£ tion. Then, the following four projections can be defined: Chap. 4 - Recognition of Integral Features 54 The relation R I projected intc the family of sets gives us R" 1 = {<11 00>,<00 11>} which is identical with the relation R" R .

The algebra of relations makes it p~ssible to describe the meta-relations between the relations {R~} as well as those between the relations {Rt} . The meta-relations can be expressed in terms of subrelations, convolutions, cartesian pro~ ucts etc. re true: if A(R) denotes that there is given a realization sat isfying to the relation R, then a) A(R) <=====> A(R) ; b) A(R) ~ A(R A(R'xR") for any R" ; e) A(R'*R") ==> A(R'),A(R"); f) A(R) ~ 1 A(R) .

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