Reconstruction in a generalized model for translation invariant systems

Abstract

We consider translation invariant (TI) operators on Φ, the set of maps from an abelian group G to Ω ∪ {−∞} , called LG-fuzzy sets, where 0 is a complete lattice ordered group. By defining Minkowski and morphological operations on Φ and considering order preserving operators, we prove a reconstruction theorem. This theorem, which is called the Strong Reconstruction Theorem (SRT), is similar to the Convolution Theorem in the theory of linear and shift invariant systems and states that for an order preserving TI operator Y one can explicitly compute Y( A), for any A, from a specific subset of Φ called the base of Y. The introduced framework is a general model for the theory of translation invariant systems, and SRT shows the consistency of it. For the special cases when G, Ω ϵ { R,Z }, SRT implies the results of Maragos and Schafer