IntroductionThe concept of transform appears often in the literature of image processing and data compression. 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

Introduction

As a matter of fact, fuzzy relations  fit the problemof 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

IntroductionHence an interest concerning the algebraic structures related to Lukasiewicz logic arose.Once we  fixed the underlying algebras, Moreover, recent developments in the theory of MV-algebras, provide us with some tools both interesting from a theoretical point of view and useful for applications.More precisely, making use of the theory of semi modules over semi rings we prove that the structures of semi module defined in a natural way on the ( finite) Cartesian power of an MV-chain has exactly one basis,

and this fact leads to a natural definition of dimension for this class of semi modules; furthermore, over a  fixed MV-chain, there exists only one of these semi modules of a given dimension, up to isomorphisms. These properties of uniqueness mean that, given an MV-chain A and a natural number n, the Cartesian power with the operations defined point wise - is a general example of n-dimensional Lukasiewicz semi module over the semi ring reducts of A, and they also allow a very general and simple definition of a transform having the additional properties of being a semi module homomorphism and a residuated map

Introduction

The main topic of this work, the Lukasiewicz transform, is defined by means of a partition of the unit of the MV-algebra . It turns out that it is also a lattice-based fuzzy transform,Furthermore the maps attached to the pair compression/reconstruction are well coupled as mathematical objects, since they yield an adjoint pair.

Introduction

Definition :- A semi ring is an algebraic structure (S, +, . , 0 , 1), with two internal binary operations, + and . , and two constants 0 ,

1 S such that

)S1) (S , + , 0 (is a commutative monoid ;)S2) (S , . , 1 (is a monoid ;

)S3 (x.(y + z) = x y + x z and (x + y).z = x z + y z for all x, y, z S; (S4) 0x = x0 = 0 for all x S.

A semi ring is said to be commutative iff  the commutative property holds for the multiplication too.

Some definitions

Definition :-let S be a semi ring .a left S-semi module is a commutative monoid (M,+M,0M) with the external operation,

called scalar multiplication ,with coefficients in S

Some definitions

an MV-algebra is an algebraic structure with a binary operation ,a unary operation , and the constant 0, satisfying certain axioms. MV-algebras are models of Łukasiewicz logic; the letters MV refer to many-valued logic of Łukasiewicz.

Definitions An MV-algebra is an algebraic structure

consisting of a non-empty set A, a binary operation on A, a unary operation on A, and a constant 0 denoting a fixed element of A, which satisfies the following identities:

and

A simple numerical example is A = [0,1], with operations and In mathematical fuzzy logic, this MV-algebra is called the standard MV-algebra, as it forms the standard real-valued semantics of Łukasiewicz logic.

The trivial MV-algebra has the only element 0 and the operations defined in the only possible way , and

The two-element MV-algebra is actually the two-element Boolean algebra {0,1}, with coinciding with Boolean disjunction and with Boolean negation.

Other finite linearly ordered MV-algebras are obtained by restricting the universe and operations of the standard MV-algebra to the set of n + 1 equidistant real numbers between 0 and 1 (both included), that is, the set which is closed under the operations and of the standard MV-algebra.

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Chang devised MV-algebras to study many-valued logics, introduced by Jan Łukasiewicz in 1920. In particular, MV-algebras form the algebraic semantics of Łukasiewicz logic, as described below.

Given an MV-algebra A, an A-valuation is a homomorphism from the algebra of propositional formulas (in the language consisting of and 0) into A.

Formulas mapped to 1 (or 0) for all A-valuations are called A-tautologies. If the standard MV-algebra over [0,1] is employed, the set of all [0,1]-tautologies determines so-called infinite-valued Łukasiewicz logic.

Chang's (1958, 1959) completeness theorem states that any MV-algebra equation holding in the standard MV-algebra over the interval [0,1] will hold in every MV-algebra.

Algebraically, this means that the standard MV-algebra generates the variety of all MV-algebras. Equivalently, Chang's completeness theorem says that MV-algebras characterize infinite-valued Łukasiewicz logic, defined as the set of [0,1]-tautologies.

Now let we take another Equivalent definition of MV-algebra as follows:-

Definition :- An algebraic structure (A, , ⊕ * , 0) with an internal binary operation , an internal unary ⊕

operation* and a constant 0 is called an MV-algebra if the following hold

on every MV-algebra it is possible to define another constant 1 and two further operation as follows:

The following properties follow directly from the above definitions:

Lemma : Let A be an MV-algebra and x, y ∈A .Then

the following conditions are equivalent-:

Let A be an MV-algebra. For any two elements x and y of A let us agree to write

iff x and y satisfy the above equivalent conditions (i)–(iv).

It follows that is a partial order, called the natural order of A.

This relation also determines a lattice structure, with 0 and 1 respectively infimum and supremum elements, and and defined as follow∨ ∧

T HE LUKASIEWICZ TRANSFORM BASED (LTB) ALGORITHM FOR IMAGE PROCESSING

The Lukasiewicz Transform has been defined for functions f : [0, 1] → [0, 1], and this fact implies that the first step of its

application to image processing consists in “adapting” the image to the domain of our operator. In other words, each image (i.e. each fuzzy matrix) must be seen as a [0, 1]-valued function defined on (a subset of) [0, 1].

Every grey image we processed has been treated as

a fuzzy matrix , and each matrix has been divided in

blocks. After these preliminary operations, we

applied the Lukasiewicz Transform to each block

separately.

A PPLYING (LTB) ALGORITHM TO GREY IMAGES

We tested three processes of compression/decompression; in these processes we have divided the fuzzy matrix associated to the images in square blocks of sizes

The blocks we obtained have been afterwards decompressed to blocks of the respective original sizes.