Discretization by adjoint pairs: Lukasiewicz and Q-module transforms

We define the Lukasiewicz Transform as a residuated map and a homomorphism between semimodules over the semiring reducts of an MV-algebra. Then we describe the “ Lukasiewicz Transform Based” ( LTB) algorithm for image processing. We also propose a strong extension of such concepts and results in the framework of quantales and quantale modules. 1 The Lukasiewicz transform and its applications The concept of transform appears often in the literature of image processing and data compression (see, for instance, [6,10]). Indeed a suitable discrete representation of a problem seems to be the best way in terms of computability and accuracy of results to approach many different tasks. On the other hand, the theory of fuzzy relations is widely used in many applications and particularly in the field of image processing ([5], [7] – [9]). As a matter of fact, fuzzy relations fit the problem of processing the representation of an image as a matrix with the range of its elements previously normalized in [0, 1]. In such techniques, however, the approach is mainly experimental and the algebraic context is seldom clearly defined. For these reasons we focused our attention on the algebraic structures involved, more or less explicitly, in some of these approaches. Hence an interest concerning semimodule structures related to Lukasiewicz semirings arose. 1.1 MV-semimodules Once we fixed the underlying algebras, it turned out immediately that the method proposed in [7] could be somehow generalized. Moreover, recent developments in the theory of MV-algebras, provide us with some tools that proved to be useful for our scope. More precisely, making use of the theory of semimodules over semirings ([4]), and the results presented in [1], we prove that the structures of semimodule