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

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 (CISM International Centre for Mechanical Sciences)

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Let us suppose that l is an one-to-one projection of into. 4) are unique but not reciprocal. Any realizations of the relation Ru is then supposed to be projected into one and only one realization of Rv and any realization of the relation Rv is supposed to be a projection of at least one realization of Ru . In this case Rv will be called homomorphic to Ru . The following example will illustrate the concept of the homomorphism of the relations. Let U, ~ =1,2, ... ,9 be some trinary sets, consisting of the elements o, i,~ and let v~,!

The pro~ lem of proving the formal identity of expressions plays an important role in the structural pattern recognition technique. It is clear that a given pattern can be formally described in sev- eral ways and it is necessary to prove the fact that a given e~ pression obtained from a recognitive experiment describes the same pattern as defined by a theoretically deducted expression. The problem of proving the formal identity of the expressions resembles this one of automatic proving the theorems and can be solved using the same tebnique.

Let us proceed the former example. 22) = Taking m = 2,4,6,8, ... , n • = 2,4,6,8, ... [z•(4,2)] 1al[z'(4,4)] The noisy version of the picture given in the ex ample gives us then the matrix: 0 5 2 6 4 6 5 1 3 2 2 t 3 1 1 t Let us shortly denote the components of the last matrix by 1At~j'~·1,2,3, ... ,~=1,2,3, .... Once more the 3 x 3 components subvectors will be considered and the following relations will be defined: l(.. 24a) 46 Chap. i-1 + \t~,j + W"~,~+i > 'llt~-1,i-i + 'llt~-t,j.

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