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| Paper IPM / M / 55 |
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| Abstract: | |
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We consider translation invariant (TI) operators on Φ, the
set of maps from an abelian group G to Ω∪{−∞},
called LG-fuzzy sets, where Ω 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 ∈ {\BbbR,\BbbZ}, SRT implies the results of
Maragos and Schafer (1985, 1987) for set-processing,
function-set-processing and function-processing filters.
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