On Patterns and Pattern Recognition

A set theoratic model for representing patterns and pattern classes is presented. Accordingly, a pattern P is defined as a finite non-empty set of features where feature element F is a 3-tupple, <X_i,X_j,q_k>. The first two components X_i and X_j of the feature tupple F are either primitive patterns or sub-patterns appearing in a given pattern, and the third component q_k is a binary predicate satisfied by X_i and X_j. It is then possible to depict P as a semantic net where nodes represent the components X_i and X_j of FεP, and the directed edge from X_i to X_j represent the predicate q_k.Depending on the values of X_i and X_j, it is possible to define a given complex pattern P in more than one way such that if X_i and X_j are primitives, then the representation P° is called the zero-order definition. The n-order definition of P is obtained by utilizing the sub-patterns $$ {X_{{{i_n}}}},{X_{{{j_n}}}} \subset {P^{{n - 1}}} $$.Different order representation of patterns lead into the notions of object-equivalence and closure of patterns. Further, with the aid of a probability function, a modeling scheme for pattern classes become possible.The concepts of null-pattern and null-relation help define simple and complex patterns, which in turn provide a path to the previous work done by the proponents of the statistical and structural approaches to the problem of pattern recognition.