Interval-valued restricted equivalence functions

IFSA-EUSFLAT 2009

Interval-valued restricted equivalence functions applied on Clustering Techniques Aranzazu Jurio1

Miguel Pagola1

Daniel Paternain1

Carlos Lopez-Molina1

Pedro Melo-Pinto2

1.Departamento de Autom´atica y Computaci´on, Universidad P´ublica de Navarra Pamplona, Spain 2.Department of Engineering, CITAB, Universidade de Tr´as-os-Montes e Alto Douro Vila Real, Portugal Email: [email protected], [email protected], [email protected]

Abstract— In this work we use interval-valued fuzzy sets in the Fuzzy C-Means algorithm for image segmentation. We introduce interval-valued restricted equivalence functions as a way of measuring the equivalence between the intervals associated to different pixels. We propose two construction methods for those new functions. We also prove experimentally that with these new concepts, the algorithm provides better results in ultrasound image segmentation that than those not using intervals. Keywords— Decomposability, Fuzzy C-Means, Interval-Valued Fuzzy Sets, Restricted Equivalence Functions, Ultrasound image.

1 Introduction Ultrasound images are used to evaluate breast anomalies during the diagnosis in breast exams. When any breast anomaly is studied to determine whether it is benign or not, the radiologist studies its size, area and other morphological characteristics [12]. In this sense, the development of algorithms that automatically separates the lesion and the rest of the image is very important for a quick and correct diagnosis. Ultrasound images segmentation is very limited by its bad quality. Speckle noise, shades and attenuation make the segmentation problem a very difficult topic [10]. There exist some methods used for this purpose; a comprehensive survey can be found in [9]. A very common method for segmenting images is clustering. The purpose of this work is to adapt the well known algorithm Fuzzy C-Means (FCM) [1, 2] to ultrasound image segmentation. This algorithm tries to classify a set of objects, in this case image pixels, in a determined number of clusters. That number must be chosen a priori. The main idea is to find as many centers as clusters must be, trying to minimize the distance between each pixel and the cluster centroids. Due to the complexity of ultrasound images, there exists a lot of uncertainty within them. This uncertainty is also present in the images obtained after processing the ultrasound images. We are going to use interval-valued fuzzy sets [13] in the FCM algorithm to deal with this uncertainty. As already mentioned, our proposed algorithm is based on the FCM. The new contributions we propose are: the use of interval-valued fuzzy sets and the search of the greatest interval equivalence between the image pixels and the centroids of the clusters. To find the greatest equivalence between intervals we introduce the concept of interval-valued restricted equivalence function. This paper is organized in the following way: first of all we present some preliminary concepts. In Section 3 we define the ISBN: 978-989-95079-6-8

concept of interval-valued restricted equivalence functions. In Section 4, we explain the way to apply these functions to ultrasound images segmentation and we show some experimental results. Finally some conclusions and future researchs are shown.

2 Preliminaries We know that in fuzzy set theory a function n : [0, 1] → [0, 1] with n(0) = 1, n(1) = 0 that is strictly decreasing and continuous is called strict negation. If, in addition, n is involutive, then it is said that it is a strong negation. We denote by L([0, 1]) the set of all closed subintervals of the closed interval [0, 1]; that is, L([0, 1]) = {x = [x, x] | (x, x) ∈ [0, 1]2 and x ≤ x} L([0, 1]) is a partially ordered set with respect to the relation ≤L defined in the following way: given x, y ∈ L([0, 1]), x ≤L y if and only if x ≤ y and x ≤ y; x =L y if and only if x = y and x = y; The relation above is transitive, antisymmetric and it expresses the fact that x strongly links to y, so that (L([0, 1]), ≤L ) is a complete lattice, where the smallest element is 0L = [0, 0], and the largest is 1L = [1, 1]. Definition 1 An interval-valued fuzzy set A on the universe U = ∅ is a mapping A : U → L([0, 1]). Definition 2 An IV negation is a function N : L([0, 1]) → L([0, 1]) that is decreasing (with respect to ≤L ) and such that N (1L ) = 0L and N (0L ) = 1L . If for all x ∈ L([0, 1]), N (N (x)) = x, it is said that N is involutive. Theorem 1 [7, 3] A function N : L([0, 1]) → L([0, 1]) is an involutive IV negation if and only if there exists an involutive negation n such that N (x) = [n(x), n(x)]. Definition 3 [4] A function REF : [0, 1]2 → [0, 1] is called a restricted equivalence function, if it satisfies the following conditions: (1) REF (x, y) = REF (y, x) for all x, y ∈ [0, 1]; (2) REF (x, y) = 1 if and only if x = y;

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IFSA-EUSFLAT 2009 (3) REF (x, y) = 0 if and only if x = 1 and y = 0 or x = 0 and y = 1; (4) REF (x, y) = REF (n(x), n(y)) for all x, y ∈ [0, 1], n being a strong negation; (5) For all x, y, z ∈ [0, 1], if x ≤ y ≤ z, then REF (x, y) ≥ REF (x, z) and REF (y, z) ≥ REF (x, z). Definition 4 [8] A function F : L([0, 1])2 → L([0, 1]) is said decomposable if there exist two functions F1 , F2 : [0, 1]2 → [0, 1] such that for all [x, x], [y, y] ∈ L([0, 1]) they satisfy that F ([x, x], [y, y]) = [F1 (x, y), F2 (x, y)] Definition 5 [6] A binary aggregation function is defined as a function Ag : [0, 1]2 → [0, 1] such that:

(2) As REFIV ([x, x], [y, y]) = 1L if and only if x = y and x = y, then F1 (x, x) = 1 and F2 (x, x) = 1, so for all x ∈ [0, 1] F1 (x, x) = F2 (x, x) = 1. (3) As REFIV ([x, x], [y, y]) = 0L if and only if x = 0 and y = 1 or vice versa, then F1 (0, 1) = 0 and F2 (0, 1) = 0. (4) As REFIV ([x, x], [y, y]) = REFIV ([n(x), n(x)], [n(y), n(y]) then [F1 (x, y), F2 (x, y)] = [F1 (n(x), n(y)), F2 (n(x), n(y))] . Separating each term, F1 (x, y) = F1 (n(x), n(y)) and F2 (x, y) = F2 (n(x), n(y)). In particular, if x = 1 and y = 0, and x = y = 1 then F2 (x, y) = F2 (1, 1) = 1 and F2 (n(x), n(y)) = F2 (0, 1) = 0. 1 = 0, so there is not any decomposable REFIV .

Theorem 3 Let REF be a restricted equivalence function, and Ag1 and Ag2 be two symmetric, idempotent aggregation (1) Ag(x1 , x2 ) ≤ Ag(y1 , y2 ) whenever x1 ≤ y1 and x2 ≤ functions such that: y2 ; (i) Ag1 ≤ Ag2 ; (2) Ag(0, 0) = 0 and Ag(1, 1) = 1. (ii) Ag1 (a, b) = 1 if and only if a = b = 1; This is an idempotent aggregation function if (iii) Ag2 (a, b) = 0 if and only if a = b = 0. (3) Ag(x, x) = x for all x ∈ [0, 1] Under these conditions, the function

3

Interval Valued Restricted Equivalence Functions

Definition 6 An Interval Valued Restricted Equivalence Function (REFIV ) is a function REFIV : L([0, 1]) × L([0, 1]) → L([0, 1]) such that: (1) REFIV (x, y) = REFIV (y, x) for all x, y ∈ L([0, 1]); (2) REFIV (x, y) = 1L if and only if x = y; (3) REFIV (x, y) = 0L if and only if x = 1L and y = 0L or x = 0L and y = 1L ; (4) REFIV (x, y) = REFIV (N (x), N (y)) being N an involutive IV negation; (5) For all x, y, z ∈ L([0, 1]), if x ≤L y ≤L z, then REFIV (x, y) ≥L REFIV (x, z) and REFIV (y, z) ≥L REFIV (x, z). Theorem 2 There is not any decomposable REFIV . Proof. We suppose there exists a decomposable REFIV . This means that there exist two functions F1 , F2 : [0, 1] × [0, 1] → [0, 1] such that: REFIV ([x, x], [y, y] = [F1 (x, y), F2 (x, y)] in such form that REFIV fulfills the five properties demanded in Definition 6. It means: (1) As REFIV (x, y) = REFIV (y, x), then F1 and F2 must satisfy that F1 (a, b) = F1 (b, a) and F2 (a, b) = F2 (b, a) for all a, b ∈ [0, 1]. ISBN: 978-989-95079-6-8

REFIV : L([0, 1]) × L([0, 1]) → L([0, 1]) given by REFIV (x, y) =

[Ag1 (REF (x, y), REF (x, y)), Ag2 (REF (x, y), REF (x, y))]

is an interval-valued restricted equivalence function in the sense of Definition 6. Proof. By hypothesis, Ag1 ≤ Ag2 , so the REFIV is a well defined interval. (1) REF , Ag1 and Ag2 are commutative functions, so REFIV is a commutative function too. (2) (Sufficiency) If REFIV (x, y) = 1L , then So, by (ii), Ag1 (REF (x, y), REF (x, y)) = 1. REF (x, y) = REF (x, y) = 1. Any REF is 1 if and only if both arguments are equal, so x = y and x = y. This means x = y. (Necessity) If x = y, it means x = y and x = y, then REFIV (x, x) = [Ag1 (REF (x, x), REF (x, x)), Ag2 (REF (x, x), REF (x, x))] [Ag1 (1, 1), Ag2 (1, 1)] and by the property (2) in Definition 5 this is [1, 1]. (3) (Sufficiency) If REFIV (x, y) = 0L , then Ag2 (REF (x, y), REF (x, y)) = 0, therefore, by (iii), REF (x, y) = REF (x, y) = 0. Any REF is 0 if and only if one of the arguments is 0 and the other is 1, so x = 1 and y = 0 or vice versa, and x = 1 and y = 0 or vice versa. This means x = x = 1 and y = y = 0 or vice versa. (Necessity) If x = 1L and y = 0L , then REFIV (x, y) = [Ag1 (REF (1, 0), REF (1, 0)), Ag2 (REF (1, 0), REF (1, 0))] = [Ag1 (0, 0), Ag2 (0, 0)], which is equal to [0, 0] by property (2) in Definition 5.

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IFSA-EUSFLAT 2009 (4) REFIV (N (x), N (y)) = [Ag1 (REF (n(x), n(y)), REF (n(x), n(y))), Ag2 (REF (n(x), n(y)), REF (n(x), n(y)))]. As REF (x, y) = REF (n(x), n(y)) and REF (x, y) = = REF (n(x), n(y)), then REFIV (N (x), N (y)) [Ag1 (REF (x, y), REF (x, y)), Ag2 (REF (x, y), REF (x, y))]. As Ag1 and Ag2 are commutative functions, this expression is equal to [Ag1 (REF (x, y), REF (x, y)), Ag2 (REF (x, y), REF (x, y))] = REFIV (x, y). (5) If x ≤L y ≤L z, then x ≤ y ≤ z and x y ≤ z. By (5) in Definition 3, REF (x, y) REF (x, z) and REF (x, y) ≥ REF (x, z). (1) in Definition 5 Ag1 (REF (x, z), REF (x, z)) Ag1 (REF (x, y), REF (x, y)) and Ag2 (REF (x, z), REF (x, z)) ≤ Ag2 (REF (x, y), REF (x, y)), so REFIV (x, z) ≤ REFIV (x, y). The reasoning for REFIV (x, z) ≤ REFIV (y, z) is analogous.

4 Application of REFIV to clustering algorithm The algorithm proposed in this paper is a modification of the Fuzzy C-Means (FCM) algorithm [1, 2], which aims to find the most characteristic point of each cluster, considered its centroid. Such a way, the membership degree of every object to each cluster is achieved via minimizing the target function: Jm (U, V ) =

n
K


m wij  xj − Ci 2

j=1 i=1

≤ ≥ In the new algorithm we propose, the function to minimize is By quite similar. The only difference is the way of calculating the ≤ distance between each pixel and the cluster centroids. Instead of using the euclidean distance, we use a negation of an interval restricted equivalence function. Therefore, our objective function to minimize is: Jm (U, V ) =

n
K


m wij (1 − Ag(REFIV (xj , Ci )))

j=1 i=1

Corollary 1 Given a restricted equivalence function REF , The main purpose of the new algorithm is to segment the imand given T and S any t-norm and any t-conorm, age into K areas represented by K interval centroids, maximizing the equivalence in each area, using the interval valued REFIV (x, y) = [T (REF (x, y), REF (x, y)), restricted equivalence functions, that we have previously deS(REF (x, y), REF (x, y))] fined. As the algorithm is partially supervised, the number of areas must be selected by the user before the execution. Notice is an interval-valued restricted equivalence function. that the REFIV is an interval. As we want to maximize the equivalence, we are going to use an aggregation of the bounds Proof. We need to see that any t-norm fulfills the properties of the equivalence interval. demanded to Ag1 and any t-conorm fulfills the properties deThe algorithm starts with the initialization of all the remanded to Ag2 . quired parameters. First of all it is necessary to build the For the t-norm T it is necessary that T (a, b) = 1 if and only interval-valued fuzzy set associated with the image. For that, if a = b = 1. (Sufficiency) If the t-norm is the minimum, each pixel is assigned an interval depending on the values of if min(a, b) = 1, then a = b = 1. As the minimum is the its neighborhood. The rest of the parameters to be initialized greatest of all t-norms, to be 1 the arguments of any t-norm are the number of areas to be found, the fuzzification degree, must be greater than or equal to 1. As they are defined in the the weight matrix, the maximum number of iterations and the unit interval, both arguments must be 1. finishing threshold. (Necessity) We know that T (1, x) = x for any x ∈ [0, 1]. After the initialization, the algorithm runs in a loop. In this If x = y = 1, then T (1, 1) = 1. loop the centroid of each cluster and the weight matrix are For the t-conorm S it is necessary that S(a, b) = 0 if and calculated until the improvement is not significant. To update only if a = b = 0. (Sufficiency) If the t-conorm is the maxi- that weight matrix, we use the interval-valued restricted equivmum, if max(a, b) = 0, then a = b = 0. As the maximum is alence functions. the smallest of all t-conorms, to be 0 the arguments of any tWhen the loop is finished, the algorithm labels each pixel conorm must be smaller than or equal to 0. As they are defined to show the results. For this purpose, each pixel is associated in the unit interval, both arguments must be 0. to the cluster with the biggest weight. (Necessity) We know that S(x, 0) = x for any x ∈ [0, 1]. If 4.1 Algorithm structure x = y = 0, then S(0, 0) = 0. Example 1 If we choose T = min, S = max and REF (x, y) = 1 − |x − y|, we have that: REFIV (x, y) =

[min(1 − |x − y|, 1 − |x − y|), max(1 − |x − y|, 1 − |x − y|)] =

1. Given an image X with n pixels, build its interval-valued fuzzy set A = {[xi , xi ]|xi ∈ X} with i = 1 . . . n. 2. Initialize 2.1 K = number of areas to be segmented. 2.2 T = maximum number of iterations.

[1 − max(|x − y|, |x − y|), 1 − min(|x − y|, |x − y|)] = N ([min(|x − y|, |x − y|), max(|x − y|, |x − y|)].

ISBN: 978-989-95079-6-8

2.3 m = fuzzification degree, 1 ≤ m ≤ ∞. 2.4 ε = finishing threshold, ε > 0. 2.5 W = weight matrix, 0 ≤ wij ≤ 1 with i = 1 . . . n and j = 1 . . . K.

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IFSA-EUSFLAT 2009 2.6 t = current iteration = 0. 3. REPEAT 3.1 Increase current iteration 3.2 Calculate the centroids n wm [xi , xi ] n ij m Cj = i=1 i=1 wij 3.3 Update the weight matrix wij = K  1 −1
1 − (Ag(REFIV (xi , Cj ))) m−1 k=1

1 − (Ag(REFIV (xi , Ck )))

4.2.2 False Edges image The false edges image represents to what degree each pixel is part of an edge. In [5] a method is presented to extract false edges from t-norms and t-conorms. For each pixel, the first step is to build a submatrix (3x3, 5x5, etc.). Applying any t-norm to the elements of that submatrix, the lower bound of an interval is obtained. Analogously, the upper bound is got applying any t-conorm to the same submatrix. The width of that interval, it means, the upper bound minus the lower bound, is called false edge. If the t-norm and the t-conorm chosen are the minimum and maximum respectively, the false edge is basically the substraction of the biggest element in the submatrix minus the lowest one. 4.3 Experiment

The experiment we show in this section is based on 8 ultrasound images, shown in Fig. 2. They are breast images in which there exist some zones that can be tumors. To create the interval-valued fuzzy set associated to the image, we use a net of 5x5 neighbors for every pixel. In this sense, each pixel 3.4 Calculate the error. is corresponded with an interval value that depends not only on its value but also on its neighbors’ values. To construct the UNTIL t = T or error ≤ ε. lower bound we take the minimum of the values of the mesh, and to calculate the upper bound we take the greatest one. 4. Label the image. The REFIV we use is the one shown in example 1: 4.2 Preprocessing REFIV (x, y) = [min(1 − |x − y|, 1 − |x − y|), max(1 − |x − y|, 1 − |x − y|)] = To prove the performance of the new algorithm, we are going N ([min(|x − y|, |x − y|), max(|x − y|, |x − y|)]. to segment some ultrasound images. In this case we work with breast ultrasound images, where the goal is to classify every To calculate the performance of the segmentation method pixel of the image. It means, decide whether they belong to we propose, we use the areas overlapping measure (SA ) bethe lesion or not. tween the image we have obtained and the ideal one. The Due to the characteristics of these images, which may conlatter is created by an expert radiologist. tain several similar regions, it is necessary to select the central pixel of the desired zone. Since the user has to contribute with |Ideal & Obtained| (1) SA = some information, we say that the proposed system is partially |Ideal + Obtained| supervised. To improve the results the images are preprocessed. In or- where Ideal is the binary image segmented by the expert rader to do so, we create two new images for every ultrasound diologist and Obtained is the binary image got by the studied image. The first one is a false edges image while the second method. We are going to compare the results from our algorithm one is a enhanced image. An example of these new images with the ones obtained with the FCM algorithm by comparcan be seen in Fig. 1. ing their accuracy with respect to the ideal images. In Fig. 2 some results are illustrated. The first column matches the original image, while the second and the third columns match the segmentation using the FCM algorithm and the proposed one respectively. Finally, the fourth column shows the ideal segmentation. As it can be seen, the results from our algorithm are more Figure 1: False edges and enhanced images. similar to the ideal ones than the ones got by the FCM. This improvement can be numerically observed by the areas overlapping percentage between each image and its corresponding 4.2.1 Enhanced image ideal one (see table 1, where A, B, . . . H refer to the images To create the enhanced image we use the algorithm proposed shown in Fig. 2). by Sahba et al. [11] for ultrasound images. The main idea is With these results, our proposed algorithm improves the reto remove the image noise and to enhance the gray level in sults obtained by the FCM in 8%. This percentage is the difthe selected area. First of all the noise is removed using the ference between the average overlap of the ideal image with median filter (7x7 or 9x9). After that, every pixel is fuzzified the one obtained by the simple FCM and with the one obtained depending on its gray level using a membership function con- by our method in the images of Fig. 2. We can say this is a sidering the average gray level in the pixel neighborhood and very good result, as it means a significant progress towards the the position of the central pixel selected. correct segmentation of this kind of images. where xi = [xi , xi ], Cj = [Cj , Cj ], Ck = [Ck , Ck ] and REFIV (x,y)+REFIV (x,y) . Ag(REFIV (x, y)) = 2

ISBN: 978-989-95079-6-8

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IFSA-EUSFLAT 2009 Original

FCM

Proposed

Ideal

A

B

C

D

E

F

G

H

Figure 2: Obtained results.

Table 1: Analytic solutions. A B C FCM 75.17 76.73 73.28 Proposed 70.81 89.43 76.37 E F G FCM 74.11 72.65 57.87 Proposed 76.77 79.63 71.68

ISBN: 978-989-95079-6-8

D 84.48 91.47 H 27.55 51.33

The use of interval-valued fuzzy sets in this sort of images increases the accuracy of the segmentation. In some cases, it allows us to discover regions that are not easily identified at first sight, as it can be seen in the images we have segmented.

5 Conclusions and future research In this work we show the advantage of using interval-valued fuzzy sets in the FCM algorithm. For this reason, we have introduced the concept of interval-valued restricted equivalence function and we have presented two different construction

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IFSA-EUSFLAT 2009 methods. The effectiveness of this method has been proved with the experimental results. As future research we want to work with intervals within the FCM instead of aggregating the bounds of the equivalence interval. Also, we are going to study other processings of ultrasound images to improve the results. Acknowledgment - This paper has been partially supported by the National Science Foundation of Spain, Reference TIN2007-65981. References [1] J.C. Bezdek. Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum, NY (1981). [2] J.C. Bezdek, J. Keller, R. Krisnapuram, N.R. Pal. Fuzzy Models and Algorithms for Pattern Recognition and Image Processing. (Kluwer Academic Publishers) (1999). [3] H. Bustince, E.Barrenechea, M. Pagola. Generation of IntervalValued Fuzzy and Atanassov’s Intuitionistic Fuzzy Connectives from Fuzzy Connectives and from Kα Operators: Laws for Conjunctions and Disjunctions, Amplitude. International Journal of Intelligent Systems, vol. 23 (2008) 680-714. [4] H. Bustince, E. Barrenechea, M. Pagola. Restricted equivalence functions. Fuzzy Sets and Systems 157 (2006) 23332346. [5] H. Bustince, E. Barrenechea, M. Pagola, R. Orduna. Construction of interval type 2 fuzzy images to represent images in grayscale. False edges. FUZZY IEEE 2007. London, U.K. [6] T. Calvo, A. Kolesarova, M.Komornikova, R. Masiar. Aggregation operators: properties, classes and construction methods. In T. Calvo, G. Mayor and R. Mesiar (eds): Aggregation Operators New Trends and Applications (Physica-Verlag, Heidelberg, 2002) 3-104. [7] G. Deschrijver, C. Cornelis, EE.Kerre. On the representation of intuitionistic fuzzy T-norms and T-conorms. IEEE Trans Fuzzy Syst. 12(1) (2004) 45-61. [8] P. Drygas. The problem of distributivity between binary operations in bi-fuzzy set theory. Proceedings of IPMU’08. L. Magdalena, M. Ojeda-Aciego, J.L. Verdegay (eds). (2008) 16481653. [9] J.A. Noble, D. BouKerroui. Ultrasound image segmentation: A survey. IEEE Transactions on Medical Imaging 25 (2006) 987-1010. [10] K. Ogawa, M. Fukushima, K. Kubota, N. Hisa. Computeraided diagnosis system for diffuse liver-diseases with ultrasonography by neural networks. IEEE Transactions on Nuclear Science. 45(6) (1998) 3069-3074. [11] F. Sahba, H.R. Tizhoosh, M.M. Salama. A coarseto-fine approach to prostate boundary segmentation in ultrasound images. BioMedical Engineering OnLine. (2005) 4-58. [12] M.H. Yap, E.A. Edirisinghe, H.E. Bez. A novel algorithm for initial lesion detection in ultrasound breast images.Journal of applied clinical medical physics (Fall 2008). Vol. 9, number 4. [13] L.A. Zadeh. The concept of a linguistic variable and its application to approximate reasoning-I. Information Science. 8 (1975) 199-249.

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