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Linguistic Interpretation of Mathematical Morphology Eureka-2013. Fourth International Workshop Proceedings Linguistic Interpretation of Mathematical Morphology Agustina Bouchet1,2, Gustavo Meschino3, Marcel Brun1, Rafael Espin Andrade4, Virginia Ballarin1 1 Universidad Nacional de Mar de Plata, Mar de Plata, Argentina. 2 CONICET, Mar de Plata, Argentina. 3 Universidad Nacional de Mar de Plata, Mar de Plata, Argentina. 4 ISPJAE, Universidad Técnica de La Habana, Cuba. [email protected], [email protected], [email protected], [email protected], [email protected] Abstract strains imposed by t-norms and s-norms, which are by themselves also conjunctions and disjunctions. Mathematical Morphology is a theory based on geometry, By replacing t-norm and s-norm by conjunction and algebra, topology and set theory, with strong application disjunction, respectively, we obtain the dilation and ero- to digital image processing. This theory is characterized sion operators for the Compensatory Fuzzy Mathematical by two basic operators: dilation and erosion. In this work Morphology (CMM), called compensatory dilation and we redefine these operators based on compensatory fuzzy compensatory erosion, respectively. Two different im- logic using a linguistic definition, compatible with previ- plementations of the CMM were presented at this moment ous definitions of Fuzzy Mathematical Morphology. A [12,21]. comparison to previous definitions is presented, assessing In this work new operators based on the definition of robustness against noise. the CMM are presented, but replacing the supreme and infimum by logical operators, which allow for a linguistic Keywords: Fuzzy Logic, Compensatory Fuzzy Logic, interpretation of their meaning. We call the New Com- Mathematical Morphology, Fuzzy Mathematical Mor- pensatory Morphological Operators. These operators are phology. compared to the existing ones (MM and CMM) on their robustness against noise. 1. Introduction 2. Theorical Concepts Mathematical Morphology (MM) is a theory based on concepts of geometry, algebra, topology and the theory of The theory of Mathematical Morphology (MM) is a pow- sets, created to characterize physical and structural prop- erful tool in the Digital Image Processing field. A key as- erties of various materials [1-3]. MM allows processing pect of this theory is the use of the structuring element images with different objectives, such as enhancing dif- (SE), a probe set that is used to test the image in several fuse zones, segmenting objects, detecting edges, analyze ways, generating information about its geometry. One of structures, among others. The central idea of the MM is to the reasons of the success of Mathematical Morphology is examine the geometric structures of an image by super- its simplicity of implementation: most of the operators posing with small localized patterns, called structuring can be built by combination of basic operators of nega- elements (SE), in different parts of it [1]. The two basic tion, complement, dilation and erosion. The last two are operations of the MM are dilation and erosion. Its compo- considered the pillars of MM, and most of its variants are sition generates filtering operators, for example, opening based on variants of these operators, as for example in the and closing. gray level morphology [14]. Another existing approach in the literature of Mathe- matical Morphology is the Fuzzy Mathematical Morphol- 2.1. Binary Mathematical Morphology ogy (FMM), which is supported by concepts of fuzzy log- ic and fuzzy set theory [4-11]. Unlike bimodal logic, Here we define the basic operators of the MM for binary vagueness and uncertainty are qualities of this approach. images. A binary X is a subset of 2 , represented by Particularly, in predicate fuzzy logic, a truth value be- its characteristic function : 2 0,1 . tween 0 and 1 can be associated with each predicate, ex- X tending the boolean concept that the predicates should be Let A and B two subsets of U 2 , the binary dila- only true or false. tion of A by the structuring element (SE) B , written as Compensatory fuzzy logic (CFL) is a multivariate DAB, , is the set of points xU such that B has non model, a particular case of the FMM, based on the re- x placement of t-norms and s-norms by conjunctions and empty intersection with A [2]: disjunctions [12-13]. This is possible by relaxing con- D A,/ B x U Bx A (1) © 2013. The authors - Published by Atlantis Press 8 where Bx b x/ b B is called the traslation of B . by applying fuzzification function over the gray scale im- ages. Dilating an image A by the structuring element consists on the removal, from the background, of all the The literature that studies the extension of basic opera- tors in binary images to gray levels image using the fuzzy points x for which B is not included in A. It is equiva- x set theory presents several approaches. Di Gesú, De lent to define the dilated image as the set of points such Baets, Kerre, Bloch, Maître y Nachtegael are some of the that intersects the image A. authors which have developed several theories and have Let A and B be two subsets of . The binary ero- defined different formulas of the basic operators of the sion of A by the structuring element B , written as FMM [4-8,10,11,15]. Bloch and Maître have achieved the EAB, , is the set of points xU such that B is includ- unification of all the models proposed by the authors x mentioned previously, by the use of t-norms and s-norms. ed in A [2]: The following are the definitions of the basic operators of E A,/ B x U B A x (2) the FMM given by these authors. To erode the image A by the structuring element B Fuzzy dilation of the image by the SE [4]: consists on reducing the set A via a process of removal of , x sup t y , y x (6) points. yU where t a, b is a t-norm [19]. 2.2. Fuzzy Mathematical Morphology Fuzzy erosion of the image by the EE [4]: Several methodologies have been developed to extend , x inf s y , c y x (7) yU binary MM to grayscale images. One of these extensions, where s a, b is a s-norm and ca is the fuzzy comple- based on fuzzy set theory, is the Fuzzy Mathematical Morphology (FMM) [4]. The FMM has proven to be a ment operator [20]. solid theory and has been applied successfully in biomed- ical image segmentation [15-18]. 2.3. Compensatory Fuzzy Mathematical Morphology Operations between fuzzy sets are defined based on the conjunction and disjunction operators, applied to the Compensatory Fuzzy Mathematical Morphology (CMM) membership values of these sets [19]. The values of the is a particular case of the FMM, based on the replacement membership functions are numbers in the interval 0,1 . of t-norms and s-norms by conjunctions and disjunctions of compensatory fuzzy logic (CFL) [13]. This is possible In most cases the images in gray levels are defined so that by relaxing the constraints imposed by t-norms and s- the gray level intensity at each pixel is an integer value norms, which are by themselves also conjunctions and belonging to the natural range 0,255 . Therefore, to be disjunctions. able to apply the FMM operators, gray level images must By replacing t-norm and s-norm by conjunction and be modeled as fuzzy sets, with a change of scale to the disjunction, respectively, we obtain the dilation and ero- range . This process of scaling is called sion operators for the CMM: "fuzzification", while the reverse process is called , x sup C y , y x (8) "defuzzification". Usually the fuzzification function used, yU , x inf D y , c y x (9) g : 0,1,2,...,255 0,1 , is defined by: yU x These operators are called compensatory dilation and gx (3) 255 compensatory erosion, respectively. The reverse process by which the intensities of the gray levels of an image, belonging to the interval , are 2.4. New Operators of the CMM brought back to the set 0,1,2,...,255 is defined by the In this work we propose a new linguistic representation of function h : 0,1 0,1,2,...,255 defined by: CMM dilation and erosion operators, in a way that they h x 255 x (4) can be associated to colloquial language, offering the fol- lowing advantages: where : represents the integer part function, de- A colloquial expression for the operators, which gives fined by: clearer understanding of the effects of transformations, a sup k / k a (5) and the operations carried out in the process. A new paradigm for the morphological2 operators. In the following sections and will denote two U A broader range of functions for the logical connec- fuzzy sets, with membership functions :U 2 0,1 tives, providing more implementation alternatives. and :U 2 0,1 , where the first one corresponds to a grayscale image and the second one determines the The proposal consists on replacing the definitions of the structuring element. Membership functions are obtained dilation and erosion operators, replacing the supremum 9 and infimum by logic operations that model linguistically Considering the functions and x described previ- their meaning. Specifically, supremum is replaced by the ously, this last linguistic expression can be written as: “existential” quantifier and infimum is replaced by the y U: y y “for all” quantifier. The whole process is described in the x next section. y U: y x y 2.4.1. New Dilation Operator y U: c y x y Binary morphological dilation is defined by equation (1): y U:, D y c y x D A,/ B x U Bx A In this equation, the non-empty intersection between B As a consequence, the new erosion operator is defined as: and A, BAx , can be expressed linguistically as “the , x y:, D y c y x (12) intensity of a pixel in the output image is high when, after yU placing the SE over the original image, centered on such pixel, at least in one pixel of the region covered by the SE, the intensity of such pixel is high for both the original im- 2.4.3.
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