Similarity and prototype-based approach for classi�cation ofmicrocalci�cationsM. Rifqia, S. Bothorelb, B. Bouchon-Meuniera, S. Mullerba LIP6 { Universit�e Pierre et Marie CurieCase 169 { 4, Place Jussieu, 75252 Paris Cedex 05fMaria.Rifqi, Bernadette.Bouchon-Meunierg@lip6.frb General Electric Medical Systems Europe283, route de la Mini�ere, 78533 Buc Cedexfbothosy,mulleseg@gemse.frAbstractOur aim is to show the utility of a formal framework of measures of comparison, es-pecially for a similarity based classi�cation. We present both theoretical and practicalarguments and we apply this approach to a real world problem.1 INTRODUCTIONAn important number of classi�cation methods are based on the comparisons of objects:the k-nearest neighbors method (k-NN) or instance based learning (Dasarathy, 1990),(Aha et al., 1991), clustering methods like (Diday, 1980). It is partly thanks to theeasy exploitation of the results given by these similarity-based methods that they are sosuccessful. Indeed, the understanding of results is often an important constraint that allmethods do not satisfy. For instance, as this paper shows, the easy interpretation of resultsin medical applications is essential.This paper focuses on the problem of the choice of a measure of comparison. Themeasure used to compare objects is often a distance. But, more and more, a similarityor a dissimilarity measure is chosen. It is not easy to choose an appropriate measure.The choice is linked to the problem of the characterization of relevant properties for aclassi�cation task.In this paper, we use the formalization and the framework introduced in (Bouchon-Meunier et al., 1996) to deal with measures of comparison. We test this framework ina challenging classi�cation problem: the classi�cation of microcalci�cations in mammo-graphic images (Rifqi et al., 1997). Furthermore, we test a classi�cation method based onfuzzy prototypes proposed in (Rifqi, 1996).1

2 APPLICATION2.1 Description of the problemOne woman in 8 in the United States and one woman in 10 in Europe have a breastcancer during her life. Nowadays, mammography is the primary diagnostic procedure forthe early detection of breast cancer. Microcalci�cation clusters are an important elementin the detection of breast cancer. This kind of �nding is the direct expression of pathologieswhich may be benign or malignant. The objective is to help radiologists to increase theirsensitivity in both detection and characterization task.The �rst step of a diagnosis is the description of microcalci�cations and their classi�-cation into the relevant classes usually used by radiologists: round, small, elongated... Inthis paper, we focus on this step of microcalci�cations classi�cation.An automatic classi�cation of microcalci�cations provides to the radiologist:� an objective description,� a systematic classi�cation of every microcalci�cation in the image.These two characteristics are the foundations for a good diagnosis. Furthermore, providinga good description of the relevant classes to the radiologists enables them to improve theirperformances.The description of microcalci�cations is not an easy task, even for an expert. If someof them are easy to detect and to identify, some others are more ambiguous. The textureof the image, the small size of objects to be detected (less than one millimeter), the variousaspects they have, the radiological noise, are parameters which impact the detection andthe characterization tasks.More generally, mammographic images present two kinds of ambiguity: imprecisionand uncertainty. The imprecision on the contour of an object comes from the fuzzy aspectof the borders: the expert can de�ne approximately the contour but certainly not witha high spatial precision. The uncertainty comes from the microcalci�cation superimposi-tions: because objects are built from the superimpositions of several 3D structures on asingle image, we may have a doubt about the contour position.Because of this uncertainty and imprecision, we are using a fuzzy segmentation ofthe microcalci�cations (Bothorel, 1996), (Bothorel et al., pear). The values of attributesdescribing microcalci�cations, obtained by this method, are fuzzy. For instance, the micro-calci�cation 1 in Figure 1 can be seen as a small and round microcalci�cation superimposedon an elongated microcalci�cation (hypothesis 1), or as a single complex elongated mi-crocalci�cation (hypothesis 2). The values of the attributes surface and compacity of thefuzzy contour of this object are shown in Figure 2.The given membership function of surface means that they are two possible contours forthe considered microcalci�cation: one has a surface of approximately 18 pixels (hypothesis1) and the other approximately 67 pixels (hypothesis 2). In the same way, the uncertaintyof contours appears in membership functions of the attributes concerning the compacity,with values approximately equal to 122 (hypothesis 1) or approximately equal to 201(hypothesis 2). 2

Figure 1: Imprecision and uncertainty of the contours of microcalci�cations

Figure 2: Description of a small microcalci�cation by means of fuzzy values.We can see that the obtained membership functions are not standard (i.e. triangularor trapezoidal). It implies a delicate management of membership functions.3

2.2 Our goalThe challenging problem is to design an algorithm enabling us to:� manage fuzzy non-standard values of attributes� learn from a database the characteristics of each class (round, not round, small, notsmall, elongated, not elongated) thanks to a fuzzy prototype� classify an unknown microcalci�cation by comparing it with each obtained fuzzyprototype.A learning database and a test database for each kind of characterization (roundor not, small or not, elongated or not) are at our disposal. The classi�cation of eachmicrocalci�cation of these databases have been carried out by an expert.The sizes of learning and test databases are given in Table 1.Learning database Test databaseRound 28 39Not round 66 69Long 42 43Not long 100 93Small 107 118Not small 43 41Table 1: Sizes of learning and test databases.Each microcalci�cation is described by means of 7 fuzzy attributes. These 7 attributesenable us to describe more precisely:� the contrast (1 attribute)� the shape (3 attributes) : elongation, compacity1, compacity2.� the dimension (2 attributes) : surface, perimeter.� the volume (1 attribute)Figure 3 gives an example of the values of 2 attributes concerning a microcalci�cationtaken in the learning database and classi�ed as \round" by the expert.3 CONSTRUCTION OF A FUZZY PROTOTYPEBefore the very step of classi�cation, objects has to be compared in order to construct aprototype for each class.According to E. Rosch (Rosch, 1978), all objects do not represent in a same mannerthe category they belong to. They are spread along a scale of typicality. According toRosch and Mervis (Rosch and Mervis, 1975) :4

Figure 3: Description of a round microcalci�cation by means of fuzzy values.[..] categories tend to become de�ned in terms of prototypes or prototypicalinstances that contain the attributes most representative of items inside andleast representative of items outside the category.(p.30)Then, the notion of prototype is linked to the notion of typicality. Zadeh (Zadeh, 1982)has also emphasized this aspect: the typicality is a matter of degree and it implies thatthe concept of prototype is a fuzzy concept.In our method, we need to determine the typicality of each value appearing in a learningdatabase in order to construct a fuzzy prototype.3.1 Degree of typicalityAccording to (Rosch, 1978), we consider that the degree of typicality of an object dependspositively on its total resemblance to other objects of its class (internal resemblance) andon its total dissimilarity to objects of other classes (external dissimilarity). Objects ofa given class are compared by pairs in order to determine their total resemblance. Thissituation of comparison is based on the assumption that objects are considered to have thesame level of generality, and no value can be taken as a reference. This situation needs asymmetrical measure because all objects have the same level of generality and a re exivemeasure because there does not exist any pair of distinctive objects which perfectly looklike each other.3.2 Measures for the computation of degrees of typicalityA similarity based-classi�cation method has to solve the problem of the choice of a measureof similarity or, more generally, a family of measures of comparison. In (Bouchon-Meunieret al., 1996), we propose to formalize a measure of comparison between two fuzzy sets asa function of the common features and the distinctive features.5

Formally, for any set of elements, let F () denote the set of fuzzy subsets of , fAthe membership function of any description A in F () and for any fuzzy set measure M .We consider the following de�nition:De�nition 1 An M -measure of comparison on is a mapping S : F ()�F ()! [0; 1]such that S(A;B) = FS(M(A \ B);M(B � A);M(A � B)), for a given mapping FS :IR+ � IR+ � IR+ ! [0; 1] and a fuzzy set measure M on F ().We focus on two types of M -measures of comparison useful for the computation ofdegrees of typicality : the measure of dissimilarity and the measure of resemblance.De�nition 2 An M -measure of dissimilarity S on is an M -measure of comparisonsuch that FS(u; v; w) is:{ independent of u and non decreasing in v and w{ minimal: FS(:; 0; 0) = 0A de�nite symmetrical M -measure of dissimilarity satisfying the triangular inequalityis a distance.A measure of resemblance is used for a comparison between the descriptions of twoobjects, of the same level of generality, to decide if they have many common characteristics.De�nition 3 An M -measure of resemblance on is an M -measure of comparison S suchthat FS(u; v; w) is{ non decreasing in u, non increasing in v and w{ re exive: FS(u; 0; 0) = 1{ symmetrical: FS(u; v; w) = FS(u;w; v)M -measures of resemblance S which satisfy an additional property of T -transitivity, fora triangular norm T , are extensions of indistinguishability relations (Trillas and Valverde,1984), (Valverde, 1985) to fuzzy sets. In the case where T is the minimum, we obtainextensions of measures of similarity.Examples of M -measures of resemblance are the following:{ S(A;B) = exp(��jdr(A;B)j) (Ovchinnikov, 1984) where � > 0 and dr(A;B) =(P jfA � fBjr)1=r, for r � 1, the generalized geometric distance for fuzzy sets. Thisquantity is a product-transitive indistinguishability relation.{ S(A;B) =M(A\B)=M(A[B) (Dubois and Prade, 1980) forM such that : M(A[B) =M(A \B) +M(A�B) +M(B �A).{ S(A;B) = 1 � 1jjPx jfA(x) � fB(x)j = 1 � 1jj(M(A � B) +M(B � A)) (Duboisand Prade, 1980), with the sigma-count M(A) =Px fA(x) as a fuzzy set measure.6

3.2.1 Fuzzy prototypeLet X be a set of objects. We suppose that there exists a partition given on X composedby crisp classes Cj. For an object O of the class Ci, the typicality of the value v of anattribute A is computed as follows:Step 1. Compute the resemblance r(v; vj) between v and the value vj of the attribute Afor any example of the same class Ci. The global resemblance R(v) relative tothe set of values of A present in examples, is obtained by aggregating the degreesr(v; vj).Step 2. Compute the dissimilarity d(v; vj) between v and the value vj of the attributeA for any example of class Ck di�erent from Ci. The total dissimilarity D(v)relative to the set of values of A present in examples, is obtained by aggregatingthe degrees d(v; vj).Step 3. The aggregation of this two values, R(v) et D(v), gives the typicality T (v) of v,according to the attribute A, for the class Ci.Degrees of typicality participate in the construction of a fuzzy prototype of a givenclass. For an attribute A, the degree of typicality of each value of A is computed for eachclass. Then, the fuzzy prototype of any given class is characterized by the most typicalvalue(s) of each attribute. This means that a fuzzy prototype is a virtual object describedby means of the same attributes as the objects pertaining to the learning database. Thevalues taken by the fuzzy prototype are the most typical.A prototype, as said L. A. Zadeh (Zadeh, 1982), is not a unique object or a group ofobjects. It is more a fuzzy schema enabling us to generate a set of objects because of thesynthesized information it contains.The prototype is intrinsically interesting because of its power of description. Thispower can also be used for a classi�cation process. In the application we deal with, thispower of description is used in two directions: �rst of all, it is provided to the radiologistas a help to formalize his reasoning and to better understand the structure of microcalci-�cations.4 CLASSIFICATION4.1 Description of the classi�cation processA new object we do not know tha class is classi�ed thanks to a comparison with theprototype of each class. Indeed, a prototype can be considered as a rule describing a class.For example, the prototype of the class \round" might be: around 15 pixels for surface,between 5 and 6 for compacity. In other words, if surface = around 15 pixels, andcompacity = between 5 and 6 then class = round.The classi�cation process is based on the question: does the new object satisfy a pro-totype? This question entails the use of a measure which is maximal when the object isincluded in the prototype and can be considered as a particular case of the prototype, and7

minimal when no common features are shared by the two objects. The total degree of satis-�ability of a new object for a prototype is obtained by aggregating degrees of satis�abilitycomputed attribute by attribute.4.2 Measures for the classi�cation processA measure of satis�ability corresponds to a situation in which we consider a referenceobject or a class and we need to decide if a new object is compatible with it or satis�esthe reference.De�nition 4 AnM -measure of satis�ability on is an M -measure of comparison S suchthat FS(u; v; w) is:{ independent of w{ non decreasing in u and non increasing in v.{ exclusive: FS(0; v; :) = 0 whatever v and w may be,{ maximal: FS(u; 0; :) = 1 whatever u 6= 0 may be.Analogy relations (Bouchon-Meunier and Valverde, 1993) such as: S(A;B) = infxmin(1�fB(x) + fA(x); 1), and fuzzy similitude (Bouchon-Meunier, 1993) such as: S(A;B) =1� supfA(x)=0 fB(x) are particular M -measure of satis�ability.5 RESULTSWe have tested several measures of resemblance, dissimilarity and satis�ability on thetest databases and it appears that the following measures provide the best prototypesregarding the rate of classi�cation:� measure of resemblance: S(A;B) = (2=�)�arctan(2�M(A\B)� sup(A\B)=M(A[B)) with M the surface (M(A) = Rx fA(x)dx).� measure of satis�ability: S(A;B) =M(A \B)=M(B) with M the surface.� measure of dissimilarity:S(A;B) = jA Bj = 1(Px=fA>fB fA�B(x) +Px=fB>fA fB�A(x))where A B describes the fuzzy set of elements that approximately belong to A andnot to B or inversely, with the sigma-count as the fuzzy set measure.The operator of aggregation is:� the median for the �rst and the second step of the algorithm of computation ofdegrees of typicality and for round microcalci�cations and small microcalci�cations.It is the mean for elongated microcalci�cations.� a t-conorm for the last step of the algorithm of computation of degrees of typicality:probabilistic t-conorm for round microcalci�cations and for elongated microcalci�-cation, Zadeh's t-conorm for small microcalci�cation.8

ROUND NOT ROUND

Figure 4: Prototypes of the classes \round" and \not round"

9

Figure 4 shows the prototypes obtained for the class \round" and \not round". Theresults of our method and of k-NN method of classi�cation for di�erent classes are givenin Table 2. Our method k-NNRound/Not round 82.41 79.63Long/Not long 80.88 73.53Small/Not small 93.71 91.82Table 2: Results of classi�cation in percent of well-classi�ed microcalci�cations.6 CONCLUSIONThe formal framework of comparison measures we have proposed has been tested on a realworld problem. This test has con�rmed that this framework is performant. Furthermore,degrees of typicality based on measures of comparison are e�ective for the construction offuzzy prototypes. These prototypes are also e�ective for a classi�cation problem.AcknowledgmentWe are grateful to Doctor Levy (Institut de Radiologie { Scanner Hoche, Paris { France)for providing us with the original �lms used in this study.ReferencesAha, D. W., Kibler, D., and Albert, M. K. (1991). Instance-based learning algorithms.Machine Learning, 6:37{66.Bothorel, S. (1996). Analyse d'image par arbre de d�ecision ou. Application �a la classi-�cation semiologique des amas de microcalci�cations. PhD thesis, Universit�e Paris6.Bothorel, S., Bouchon, B., and Muller, S. (1997, to appear). A fuzzy logic-based approachfor semiological analysis of microcalci�cation in mammographic images. InternationalJournal of Intelligent Systems.Bouchon-Meunier, B. (1993). Fuzzy similitude and approximate reasoning. In Wang, P. P.,editor, Advances in Fuzzy Theory and Technology, pages 161{166. Bookwrights Press.Bouchon-Meunier, B., Rifqi, M., and Bothorel, S. (1996). Towards general measures ofcomparison of objects. Fuzzy Sets and Systems, 84(2):143{153.Bouchon-Meunier, B. and Valverde, L. (1993). Analogy relations and inference. In Pro-ceedings of 2nd IEEE International Conference on Fuzzy Systems, pages 1140{1144,San Fransisco. 10

Dasarathy, B. V. (1990). Nearest Neighbors (NN) Norms: NN pattern classi�cation tech-niques. IEEE Computer Society Press.Diday, E. (1980). Introduction �a la m�ethode des nu�ees dynamiques. In Analyse desdonn�ees, volume I, pages 121{132. A. P. M. E. P (Association des Professeurs deMath�ematiques de l'Enseignement Public).Dubois, D. and Prade, H. (1980). Fuzzy Sets and Systems, Theory and Applications.Academic Press, New- York.Ovchinnikov, S. V. (1984). Representations of transitive fuzzy relations. In Skala, H. J.,Termini, S., and Trillas, E., editors, Aspects of Vagueness, pages 105{118. D. ReidelPublishing Company.Rifqi, M. (1996). Constructing prototypes from large databases. In IPMU'96, pages301{306, Granada.Rifqi, M., Bothorel, S., Bouchon-Meunier, B., and Muller, S. (1997). Similarity andprototype based approach for classi�cation of microcalci�cations. In 7th IFSA WorldCongress, pages 123{128, Prague.Rosch, E. (1978). Principles of categorization. In Rosch, E. and Lloyd, B. B., editors,Cognition and categorization, pages 27{48. Hillsdale, N. J. : Laurence Erlbaum As-sociates.Rosch, E. and Mervis, C. B. (1975). Family resemblances: Studies in the internal structureof categories. Cognitive Psychology, 7:573{605.Trillas, E. and Valverde, L. (1984). On implication and indistinguishability in the settingof fuzzy logic. In Kacprzyk, J. and Yager, R. R., editors, Management DecisionSupport Systems Using Fuzzy Sets and Possibility Theory. Verlag TUV, Rheinland.Valverde, L. (1985). On the structure of t-indistinguishability operators. Fuzzy Sets andSystems, 17:313{328.Zadeh, L. A. (1982). A note on prototype theory and fuzzy sets. Cognition, 12:291{297.

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