Similarity and prototype-based approach for classi cation of microcalci cations a
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M. Rifqi , S. Bothorel , B. Bouchon-Meunier , S. Muller
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a LIP6 { Universit e Pierre et Marie Curie Case 169 { 4, Place Jussieu, 75252 Paris Cedex 05 fMaria.Rifqi,
[email protected] b General Electric Medical Systems Europe 283, route de la Miniere, 78533 Buc Cedex fbothosy,
[email protected]
Abstract
Our aim is to show the utility of a formal framework of measures of comparison, especially for a similarity based classi cation. We present both theoretical and practical arguments and we apply this approach to a real world problem.
1 INTRODUCTION An 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 the easy exploitation of the results given by these similarity-based methods that they are so successful. Indeed, the understanding of results is often an important constraint that all methods do not satisfy. For instance, as this paper shows, the easy interpretation of results in medical applications is essential. This paper focuses on the problem of the choice of a measure of comparison. The measure used to compare objects is often a distance. But, more and more, a similarity or 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 a classi cation task. In this paper, we use the formalization and the framework introduced in (BouchonMeunier et al., 1996) to deal with measures of comparison. We test this framework in a challenging classi cation problem: the classi cation of microcalci cations in mammographic images (Rifqi et al., 1997). Furthermore, we test a classi cation method based on fuzzy prototypes proposed in (Rifqi, 1996).
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2 APPLICATION
2.1 Description of the problem
One woman in 8 in the United States and one woman in 10 in Europe have a breast cancer during her life. Nowadays, mammography is the primary diagnostic procedure for the early detection of breast cancer. Microcalci cation clusters are an important element in the detection of breast cancer. This kind of nding is the direct expression of pathologies which may be benign or malignant. The objective is to help radiologists to increase their sensitivity 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... In this 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, providing a good description of the relevant classes to the radiologists enables them to improve their performances. The description of microcalci cations is not an easy task, even for an expert. If some of them are easy to detect and to identify, some others are more ambiguous. The texture of the image, the small size of objects to be detected (less than one millimeter), the various aspects they have, the radiological noise, are parameters which impact the detection and the characterization tasks. More generally, mammographic images present two kinds of ambiguity: imprecision and uncertainty. The imprecision on the contour of an object comes from the fuzzy aspect of the borders: the expert can de ne approximately the contour but certainly not with a high spatial precision. The uncertainty comes from the microcalci cation superimpositions: because objects are built from the superimpositions of several 3D structures on a single image, we may have a doubt about the contour position. Because of this uncertainty and imprecision, we are using a fuzzy segmentation of the microcalci cations (Bothorel, 1996), (Bothorel et al., pear). The values of attributes describing microcalci cations, obtained by this method, are fuzzy. For instance, the microcalci cation 1 in Figure 1 can be seen as a small and round microcalci cation superimposed on an elongated microcalci cation (hypothesis 1), or as a single complex elongated microcalci cation (hypothesis 2). The values of the attributes surface and compacity of the fuzzy contour of this object are shown in Figure 2. The given membership function of surface means that they are two possible contours for the considered microcalci cation: one has a surface of approximately 18 pixels (hypothesis 1) and the other approximately 67 pixels (hypothesis 2). In the same way, the uncertainty of 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. triangular or trapezoidal). It implies a delicate management of membership functions. 3
2.2 Our goal
The 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, not small, elongated, not elongated) thanks to a fuzzy prototype classify an unknown microcalci cation by comparing it with each obtained fuzzy prototype. A learning database and a test database for each kind of characterization (round or not, small or not, elongated or not) are at our disposal. The classi cation of each microcalci cation of these databases have been carried out by an expert. The sizes of learning and test databases are given in Table 1. Round Not round Long Not long Small Not small
Learning database Test database 28 39 66 69 42 43 100 93 107 118 43 41
Table 1: Sizes of learning and test databases. Each microcalci cation is described by means of 7 fuzzy attributes. These 7 attributes enable 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 cation taken in the learning database and classi ed as \round" by the expert.
3 CONSTRUCTION OF A FUZZY PROTOTYPE Before the very step of classi cation, objects has to be compared in order to construct a prototype for each class. According to E. Rosch (Rosch, 1978), all objects do not represent in a same manner the category they belong to. They are spread along a scale of typicality. According to Rosch 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 prototypical instances that contain the attributes most representative of items inside and least 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 that the concept of prototype is a fuzzy concept. In our method, we need to determine the typicality of each value appearing in a learning database in order to construct a fuzzy prototype.
3.1 Degree of typicality
According to (Rosch, 1978), we consider that the degree of typicality of an object depends positively on its total resemblance to other objects of its class (internal resemblance) and on its total dissimilarity to objects of other classes (external dissimilarity). Objects of a given class are compared by pairs in order to determine their total resemblance. This situation of comparison is based on the assumption that objects are considered to have the same level of generality, and no value can be taken as a reference. This situation needs a symmetrical measure because all objects have the same level of generality and a re exive measure because there does not exist any pair of distinctive objects which perfectly look like each other.
3.2 Measures for the computation of degrees of typicality
A similarity based-classi cation method has to solve the problem of the choice of a measure of similarity or, more generally, a family of measures of comparison. In (Bouchon-Meunier et al., 1996), we propose to formalize a measure of comparison between two fuzzy sets as a 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 , fA the 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 of degrees 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 comparison
such that FS (u; v; w) is:
{ independent of u and non decreasing in v and w { minimal: FS (:; 0; 0) = 0
A de nite symmetrical M -measure of dissimilarity satisfying the triangular inequality is a distance. A measure of resemblance is used for a comparison between the descriptions of two objects, 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 such
that 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, for a 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 obtain extensions of measures of similarity. Examples of M -measures of resemblance are the following: { SP(A; B ) = exp(? jdr (A; B )j) (Ovchinnikov, 1984) where > 0 and dr (A; B ) = ( jfA ? fB jr )1=r , for r 1, the generalized geometric distance for fuzzy sets. This quantity is a product-transitive indistinguishability relation. { S (A; B ) = M (A \ B )=M (A [ B ) (Dubois and Prade, 1980) for M such that : M (A [ B ) = M (A \ B ) + M (A ? B ) + M (B ? A). P (A ? B ) + M (B ? A)) (Dubois { S (A; B ) = 1 ? j 1 j x jfA (x) ? fB (x)j = 1 ? j 1 j (M P and Prade, 1980), with the sigma-count M (A) = x fA(x) as a fuzzy set measure. 6
3.2.1 Fuzzy prototype
Let X be a set of objects. We suppose that there exists a partition given on X composed by crisp classes Cj . For an object O of the class Ci , the typicality of the value v of an attribute A is computed as follows: Step 1. Compute the resemblance r(v; vj ) between v and the value vj of the attribute A for any example of the same class Ci . The global resemblance R(v) relative to the set of values of A present in examples, is obtained by aggregating the degrees r(v; vj ). Step 2. Compute the dissimilarity d(v; vj ) between v and the value vj of the attribute A 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 aggregating the 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 given class. For an attribute A, the degree of typicality of each value of A is computed for each class. Then, the fuzzy prototype of any given class is characterized by the most typical value(s) of each attribute. This means that a fuzzy prototype is a virtual object described by means of the same attributes as the objects pertaining to the learning database. The values 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 of objects. It is more a fuzzy schema enabling us to generate a set of objects because of the synthesized information it contains. The prototype is intrinsically interesting because of its power of description. This power can also be used for a classi cation process. In the application we deal with, this power of description is used in two directions: rst of all, it is provided to the radiologist as a help to formalize his reasoning and to better understand the structure of microcalci cations.
4 CLASSIFICATION
4.1 Description of the classi cation process
A new object we do not know tha class is classi ed thanks to a comparison with the prototype 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, and compacity = between 5 and 6 then class = round. The classi cation process is based on the question: does the new object satisfy a prototype? This question entails the use of a measure which is maximal when the object is included in the prototype and can be considered as a particular case of the prototype, and 7
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 ability computed attribute by attribute.
4.2 Measures for the classi cation process
A measure of satis ability corresponds to a situation in which we consider a reference object or a class and we need to decide if a new object is compatible with it or satis es the reference. De nition 4 An M -measure of satis ability on is an M -measure of comparison S such
that 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 ) = inf x min(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 RESULTS We have tested several measures of resemblance, dissimilarity and satis ability on the test databases and it appears that the following measures provide the best prototypes regarding the rate of classi cation: measure of resemblance: S (A; B ) =R (2=) arctan(2 M (A \ B ) sup(A \ B )=M (A [ B )) with M the surface (M (A) = x 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 B j = 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 and not 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 of degrees 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
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Figure 4 shows the prototypes obtained for the class \round" and \not round". The results of our method and of k-NN method of classi cation for di erent classes are given in Table 2. Our method k-NN Round/Not round 82.41 79.63 Long/Not long 80.88 73.53 Small/Not small 93.71 91.82 Table 2: Results of classi cation in percent of well-classi ed microcalci cations.
6 CONCLUSION The formal framework of comparison measures we have proposed has been tested on a real world 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 of fuzzy prototypes. These prototypes are also e ective for a classi cation problem.
Acknowledgment We are grateful to Doctor Levy (Institut de Radiologie { Scanner Hoche, Paris { France) for providing us with the original lms used in this study.
References Aha, 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 decision ou. Application a la classi cation semiologique des amas de microcalci cations. PhD thesis, Universite Paris 6. Bothorel, S., Bouchon, B., and Muller, S. (1997, to appear). A fuzzy logic-based approach for semiological analysis of microcalci cation in mammographic images. International Journal 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 of comparison of objects. Fuzzy Sets and Systems, 84(2):143{153. Bouchon-Meunier, B. and Valverde, L. (1993). Analogy relations and inference. In Proceedings 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 techniques. IEEE Computer Society Press. Diday, E. (1980). Introduction a la methode des nuees dynamiques. In Analyse des donnees, volume I, pages 121{132. A. P. M. E. P (Association des Professeurs de Mathematiques 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. Reidel Publishing Company. Rifqi, M. (1996). Constructing prototypes from large databases. In IPMU'96, pages 301{306, Granada. Rifqi, M., Bothorel, S., Bouchon-Meunier, B., and Muller, S. (1997). Similarity and prototype based approach for classi cation of microcalci cations. In 7th IFSA World Congress, 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 Associates. Rosch, E. and Mervis, C. B. (1975). Family resemblances: Studies in the internal structure of categories. Cognitive Psychology, 7:573{605. Trillas, E. and Valverde, L. (1984). On implication and indistinguishability in the setting of fuzzy logic. In Kacprzyk, J. and Yager, R. R., editors, Management Decision Support Systems Using Fuzzy Sets and Possibility Theory. Verlag TUV, Rheinland. Valverde, L. (1985). On the structure of t-indistinguishability operators. Fuzzy Sets and Systems, 17:313{328. Zadeh, L. A. (1982). A note on prototype theory and fuzzy sets. Cognition, 12:291{297.
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