Extended \pmb{$\alpha \beta $} associative memories

Abstract

The $\alpha \beta $ associative memories recently developed in Ref. 10 have proven to be powerful tools for memorizing and recalling patterns when they appear distorted by noise. However they are only useful in the binary case. In this paper we show that it is possible to extend these memories now to the gray-level case. To get the desired extension, we take the original operators $\alpha $ and $\beta $, foundation of the $\alpha \beta $ memories, and propose a more general family of operators. We find that the original operators $\alpha $ and $\beta $ are a subset of these extended operators. For this we first formulate a set of functional equations in terms of the original properties of operators $\alpha $ and $\beta.$ Next we solve this system of equations and find a family of solutions. We show that the $\alpha $ and $\beta $ originally proposed in Ref. 10 are just a particular case of this new family. We present the properties of the new operators. We then use these operators to build a new set of extended memories. We also give the conditions under which the extended memories are able to recall a pattern either from the pattern's fundamental set or from altered versions of them. We give real examples with images where the proposed memories show their efficiency. We compare the proposal with other similar works, and show the ours performs much better.