KR101866522B1 - Object clustering method for image segmentation - Google Patents

Abstract

The present invention relates to an object clustering method to segment an image, capable of increasing correctness of clustering. According to the present invention, the object clustering method to segment an image comprises: a step (a) of setting the number of cluster prototypes, a fuzzing parameter, and an interrupt condition; a step (b) of initializing the cluster prototypes; a step (c) of calculating fuzzy membership values for each object with respect to clusters; a step (d) of assigning an approximate region corresponding to a rough set of each object based on the fuzzy membership value; a step (e) of calculating a prototype for each cluster based on a weight parameter dependent on distribution characteristics of each cluster; a step (f) of determining the prototypes determined in the current repetitive step as the prototype for each cluster and, if not, repeating the steps (c) to (e) until the prototype for each cluster is determined; and a step (g) of assigning each object to the corresponding cluster in accordance with the fuzzy membership value based on the determined prototype for each cluster.

Description

{OBJECT CLUSTERING METHOD FOR IMAGE SEGMENTATION}

The present invention relates to an object clustering method for image segmentation.

Clustering analysis is a fundamental technique for discovering the inherent structural complexity of a dataset. The main task of the clustering analysis is to divide a data set without labels into several groups so that objects in the same group share more similarity with each other than objects belonging to different groups. The clustering process attempts to mimic how a person assigns objects to different classes and categories. One of the most widely used methods in data mining (Mitra, 2003) is pattern recognition, machine learning, web mining (Mecca, 2007), biology (Xu, 2010); It has been applied to a wide range of engineering such as Valafar (2002), image processing (Jain, 1988), and market segmentation (Bigus, 1996). Image segmentation is one of the main techniques in image processing and computer vision. Many clustering-based methods have been proposed for image segmentation, because the task of image segmentation is to partition the image into a number of non-overlapping areas with the same characteristics such as gray level, color and texture.

The k-means algorithm is considered as a classical partitioned clustering method. In the k-means algorithm (Tou, 1974), each object is assigned to one cluster. The boundaries of the various regions are crisp or hard because each object is assigned to only one cluster. It represents the advantage of fast convergence speed, but the desired clustering results can not be easily achieved in handling redundant or skewed data distribution problems (Xiong, 2009). On the other hand, real world datasets are always characterized by uncertainty and overlapping boundaries. There is a significant need for a soft computing technique with flexible information processing capabilities that can accurately meet the needs of real applications.

In practical applications, it may not be possible to obtain complete information of a given pattern set. Incomplete information can lead to incomplete representations of the pattern set by using various pattern recognition methods. Moreover, in the field of image analysis, there is no precise definition of the concept of boundaries between regions or regions. This prompted researchers to consider various types of uncertainty when designing pattern recognition methods.

One of the most common methods of image segmentation is fuzzy clustering, which in some cases can hold more information than hard clustering. Since the range of pixel values of an image belonging to different regions is generally redundant, fuzzy clustering techniques appear to be a suitable and practical choice for identifying pixels. As a widespread soft computing technique, fuzzy set theory (Dubois (2012); Pedrycz (2012)) is widely used in real applications (Vieira (2012); Bermudez (2012); Niros (2012) FCM: fuzzy c-means algorithm) (Bezdek, 1981). In an FCM, an object may be assigned to many clusters simultaneously having different membership degrees, which alleviates the dependency requirement in the k-means algorithm. Under this condition, the FCM could obtain desirable results in handling the redundant environment. A weakness of the FCM is that if the object is an object with noise, the resulting fuzzy membership values may not always correspond to the actual belonging degree to which the object belongs to the clusters.

The rough sets theory is another effective soft computing technique for describing uncertainty or ambiguity. It uses a pair of lower approximations and upper approximations to form an approximation definition for the incorrect set. Recently, rough set theory has been combined with clustering algorithms that form some rough set-based clustering algorithms (Lingras (2004); Mitra (2004); Maji (2011)). Based on the rough set theory (Tiwari (2012); Chen (2012)), clusters are described together by prototype and pair of lower and upper bound approximations. Objects within the lower bound approximation have more certainty with respect to being explicitly in the cluster, while objects within the upper bound approximation have less confidence in being assigned to a definite cluster. The approximate pairs are aimed at finding an incorrect set of actual boundaries from two aspects that are more suitable for describing the properties of the recognition. For this reason, the rough set-based clustering algorithm (Mitra, 2004, 2006) was able to handle uncertainty well with the help of two approximations when the cluster was uncertain or ambiguous in the actual dataset.

Soft computing techniques show the flexibility of information processing ability to deal with uncertainty and duplication problems in pattern recognition. Fuzzy sets and rough sets are two types of soft computing methods, both of which have their respective advantages. A fuzzy set has good ability to resolve redundant partitions and a rough set handles uncertainty and ambiguity well. Combining the fuzzy set with the rough set provides an important direction in managing uncertainty. By combining their advantages, an integrated technique called a rough-fuzzy c-means algorithm (RFCM) has been proposed (Mitra (2006)); Maji (2007a). In order to combine the advantages of both probabilistic and possibilistic membership functions together, a new hybrid clustering algorithm called rough-fuzzy possibilistic c-menas (RFPCM) (Maji (2007b)). Much of the effort to find correlations among them was still underway (Mitra (2006); Peters (2006); Pawlak (1991); Maji (2007a, )).

It is attempted to express and manage uncertainties in the field of pattern recognition, especially clustering. If the uncertainty information embedded in a given data set can be appropriately identified, the performance improvement can become a reality. By integrating soft computing techniques, including fuzzy sets and rough sets, hybrid clustering algorithms lead to effective techniques for information particles through a clustering process. Although rough set-based clustering algorithms have been considered as effective soft computing methods, a common problem is weighted parameter selection. When updating a new prototype, weighting parameters that control the importance of the lower bound approximation and the upper bound approximation are always set manually. Under conditions where no prior knowledge is provided, suitable weighting parameters can not be readily selected. Furthermore, whatever the distribution characteristics of the data set, the ratio of the lower bound approximation to the upper bound approximation of each cluster is usually set to be the same. However, since the context varies from cluster to cluster, it is not reasonable to use the same weighting parameter when updating different prototypes.

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Available: http://www.ics.uci.edu/ mlearn / MLRepository.html

The object of the present invention is to provide an object clustering method for image segmentation, which improves the accuracy of clustering by adaptively applying weighting parameters based on distribution characteristics of each cluster.

According to an aspect of the present invention, there is provided an object clustering method for image segmentation,

)을 설정하는 단계;(a) the number of cluster prototypes (c), the fuzzy parameter (m)

);

(b) initializing cluster prototypes;

(c) calculating fuzzy membership values for each object for the clusters;

(d) assigning each object to a corresponding approximate region of the rough set based on the fuzzy membership value;

(e) computing a prototype for each cluster based on a weighting parameter that is dependent on a distribution characteristic of each cluster;

인 경우, 현재의 반복 단계에서 결정된 프로토타입들을 각 클러스터에 대한 프로토타입으로 결정하고, 그렇지 않은 경우, 각 클러스터에 대한 프로토타입이 결정될 때까지 단계 (c) 내지 단계 (e)를 반복하는 단계; 및(f)

, Determining prototypes determined in the current iteration step as prototypes for each cluster, and if not, repeating steps (c) through (e) until a prototype for each cluster is determined; And

(g) assigning each object to a corresponding cluster according to a fuzzy membership value based on the determined prototype for each cluster,

는 클러스터 프로토타입들의 벡터들이고,

는 현재의 반복 단계에서 획득된 프로토타입들이 이전의 반복 단계에서 생성된 프로토타입들과 동일하다는 것을 의미한다.In the above,

Are vectors of cluster prototypes,

Means that the prototypes obtained in the current iteration step are the same as the prototypes generated in the previous iteration step.

In an object clustering method for image segmentation according to an embodiment of the present invention, the distribution characteristic of each cluster may include compactness of each cluster.

Also, in the object clustering method for image segmentation according to an embodiment of the present invention, the prototype for each cluster in step (e)

에 의해 계산되고,

Lt; / RTI >

이며,In the above,

Lt;

이고,

ego,

이며,

Lt;

는 클러스터 i의 가중 파라미터로서,

에 의해 계산되고,In the above,

Is a weighting parameter of cluster i,

Lt; / RTI >

이며,

Lt;

는 클러스터 i를 나타내고,

이며,

는 클러스터 i의 하한 근사이고,

는 클러스터 i의 경계 영역이며,

는 오브젝트 j가 클러스터 i에 속하는 정도를 나타내는 퍼지 멤버십 값이고,

는 오브젝트 j를 나타내며,

이고, n은 양의 정수이며,

는 클러스터 i의 하한 근사에 있는 오브젝트들의 수일 수 있다.In the above,

Represents cluster i,

Lt;

Is the lower bound approximation of cluster i,

Is the boundary region of the cluster i,

Is a fuzzy membership value indicating the degree to which object j belongs to cluster i,

Represents an object j,

, N is a positive integer,

May be the number of objects in the lower bound approximation of cluster i.

According to the object clustering method for image segmentation according to an embodiment of the present invention, the accuracy of clustering can be improved by adaptively applying the weighting parameters based on the distribution characteristics of each cluster.

=0.51, 도 2b는

=0.6, 도 2c는

=0.7, 도 2d는

=0.8, 도 2e는

=0.9, 도 2f는

=0.99인 경우의 도면.
도 3은 RFCM II에 의한 합성 데이터세트에 대한 상이한 가중 파라미터들에 따른 CA, ARI 및 MS의 상자 그림으로서, 도 3a는 CA, 도 3b는 ARI, 도 3c는 MS의 도면.
도 4는 RFCM II에 의한 각 클러스터의 하한 근사와 상한 근사의 비주얼화를 도시한 도면으로, 도 4a는 최종 분할 결과, 도 4b는 3개의 클러스터들의 하한 근사들, 도 4c는 빨간색 클러스터의 경계 영역, 도 4d는 녹색 클러스터의 경계 영역, 도 4e는 파란색 클러스터의 경계 영역을 도시한 도면.
도 5는 4개의 합성 데이터세트의 산포도.
도 6은 4개의 테스트 문제들을 해결하는데 있어서 각 알고리즘에 의해 획득된 CA, ARI 및 MS의 통계값들을 도시한 도면으로서, 도 6a는 CA, 도 6b는 ARI, 도 6c는 MS의 도면.
도 7은 합성 데이터세트 2에 대한 ARFCM에 의해 획득된 각 발생에 따른 가중 파라미터들의 변화를 도시한 도면.
도 8은 4개의 알고리즘에 의해 획득된 통계값들을 도시한 도면으로서, 도 8a는 8 UCI 데이터세트를 해결하는데 있어서 4개의 알고리즘들에 의해 획득된 CA의 통계값들, 도 8b는 8 UCI 데이터세트를 해결하는데 있어서 4개의 알고리즘들에 의해 획득된 ARI의 통계값들, 도 8c는 8 UCI 데이터세트를 해결하는데 있어서 4개의 알고리즘들에 의해 획득된 MS의 통계값들을 도시한 도면.
도 9는 4개의 알고리즘들에 의해 획득된 분할 결과들을 도시한 도면으로서, 도 9a는 하우스(house), 도 9b는 라이스(rice), 도 9c는 카메라맨(cameraman), 도 9d는 브레인(brain), 도 9e는 두번째 브레인 MR 이미지, 도 9f는 세번째 브레인 MR 이지미를 도시한 도면.
도 10은 상이한 크기를 갖는 4개의 합성 데이터세트의 산포도.
도 11은 상이한 크기를 갖는 합성 데이터세트를 해결하는데 있어서 4개의 알고리즘들에 의한 실행 시간의 평균값을 도시한 도면.
도 12는 본 발명의 일 실시예에 의한 이미지 분할을 위한 오브젝트 클러스터링 방법의 흐름도.Brief Description of the Drawings Fig. 1 is a diagram showing a lower limit approximation and an upper limit approximation of a cluster. Fig.
Figure 2 shows the visualization of the result of the division according to different weighting parameters in the last iteration step according to RFCM II,

= 0.51, Figure 2b

= 0.6, Fig. 2C

= 0.7, Figure 2d

= 0.8, Figure 2e shows

= 0.9, Figure 2f

= 0.99.
FIG. 3 is a box diagram of CA, ARI and MS according to different weighting parameters for the composite data set by RFCM II; FIG. 3A is a CA, FIG. 3B is an ARI, and FIG.
FIG. 4A shows the final division result, FIG. 4B shows the lower bound approximations of the three clusters, FIG. 4C shows the boundary region of the red cluster, FIG. 4D is a boundary region of a green cluster, and FIG. 4E is a boundary region of a blue cluster.
5 is a scatter diagram of four synthetic data sets.
FIG. 6 shows statistical values of CA, ARI and MS obtained by each algorithm in solving four test problems, FIG. 6A being a CA, FIG. 6B being an ARI, and FIG.
7 shows a variation of weighting parameters according to each occurrence obtained by the ARFCM for composite data set 2;
FIG. 8 shows statistical values obtained by four algorithms. FIG. 8A shows statistical values of CA obtained by four algorithms in solving 8 UCI data sets, FIG. 8B shows statistical values of 8 UCI data sets Fig. 8C shows statistical values of an MS obtained by four algorithms in solving 8 UCI data sets; Fig.
9A is a house, FIG. 9B is a rice, FIG. 9C is a cameraman, FIG. 9D is a brain, and FIG. 9B is a diagram showing the results obtained by the four algorithms. , FIG. 9E shows a second brain MR image, and FIG. 9F shows a third brain MR image.
10 is a scatter diagram of four synthetic data sets having different sizes.
Fig. 11 shows average values of execution times by four algorithms in solving a composite data set having different sizes; Fig.
12 is a flowchart of an object clustering method for image segmentation according to an embodiment of the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS The objectives, specific advantages and novel features of the present invention will become more apparent from the following detailed description taken in conjunction with the accompanying drawings, in which: FIG.

Prior to that, terms and words used in the present specification and claims should not be construed in a conventional and dictionary sense, and the inventor may properly define the concept of the term in order to best explain its invention Should be construed in accordance with the principles and the meanings and concepts consistent with the technical idea of the present invention.

It should be noted that, in the present specification, the reference numerals are added to the constituent elements of the drawings, and the same constituent elements are assigned the same number as much as possible even if they are displayed on different drawings.

Also, the terms "first", "second", "one side", "other side", etc. are used to distinguish one element from another, It is not.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS In the following description of the present invention, a detailed description of known arts which may unnecessarily obscure the gist of the present invention will be omitted.

Hereinafter, preferred embodiments of the present invention will be described in detail with reference to the accompanying drawings.

For real pattern recognition applications, complete and accurate information for a given set is not always easy to obtain. This incomplete information can lead to an incomplete representation of the set by using many pattern recognition methods. The rough set theory is designed to approximate an inaccurate set by a pair of lower and upper bound approximations weighted by different parameters. Since distribution characteristics vary from set to set, it is not desirable to use a constant weighting parameter to control the importance of the lower and upper bound approximations when describing the various given sets. The object clustering method for image segmentation according to an embodiment of the present invention is a method for clustering objects for image segmentation, wherein the parameter selection strategy is an improved roughness filter that is designed to adaptively adjust weighting parameters depending on the distribution characteristics of each cluster, - Provides a fuzzy c-mean clustering algorithm. The weighting parameters are automatically selected according to the structural characteristics of the cluster and updated at each iteration step in an online fashion. Relatively accurate approximate regions of each cluster, which are important in the computation of the corresponding prototype, can be formed, which allows the formed prototype to be close to the desired location. Experimental results on synthetic data sets, real life data sets, and image segmentation problems confirm the effect of the adaptive parameter selection strategy proposed in the object clustering method for image segmentation according to an embodiment of the present invention. With the introduction of adaptive parameter selection strategies, the improved rough set-based clustering algorithm outperforms existing algorithms.

Rough set (rough set) -based clustering algorithm

According to rough set theory, an incorrect set can be approximated by both a lower approximation and an upper approximation. These two approximations provide an approximate definition of the inaccurate set on both sides. The objects in the lower bound approximation form a subset of the target set, so they certainly belong to that set. The objects in the upper bound approximations form a nonempty intersection with the target set. By introducing a set of roughs into the clustering algorithm, the prototype of each cluster can be applied to objects belonging to its lower and upper bound approximations instead of using all objects such as k-means or FCM , And then the useless information can be removed (Zhou (2011)).

Figure 1 provides a schematic of a cluster. The objects in the lower bound approximation are definitely in this cluster. The objects in the upper bound approximation probably belong to this cluster. Objects between a lower bound approximation and an upper bound approximation form a boundary region. The contribution of objects located in different approximations of the cluster is distinguished. In general, objects located in exclusion areas are almost ignored in the calculation of the corresponding prototype. Objects in the lower bound approximation exhibit a higher degree of importance when compared to objects in the boundary region in the calculation of the corresponding prototype.

For a rough set-based clustering algorithm, the following basic rough set properties need to be satisfied:

1. An object can be assigned to a lower bound approximation of only one cluster.

2. If the object is in the lower bound approximation of the cluster, it also belongs to the upper bound approximation of the same cluster.

3. If the object does not belong to the lower bound approximation of any cluster, it belongs to the upper bound approximations of some clusters, at least two of them, the upper bound approximations.

Rough c-mean clustering algorithm ( RCM : Rough c-means clustering algorithm)

에 할당된다고 가정하자. 프로토타입들은 하기와 같이 갱신된다:In RCM (Peters (2006)), clusters are described by prototypes and two approximations. n < / RTI >

Lt; / RTI > The prototypes are updated as follows:

이고,In the above,

ego,

이며,

Lt;

이다.

to be.

는 클러스터 i를 나타내고,

이다.

는 오브젝트 j를 나타내고,

이며,

와

는 각각 클러스터 i의 하한 근사 및 상한 근사이고,

는 클러스터 i의 하한 근사에 있는 오브젝트들의 수이며,

는 클러스터 i의 경계 영역이고,

와

는 각각 상기 하한 영역과 경계 영역의 가중 파라미터들이다. 그들은 대응하는 프로토타입을 계산할 때 하한 영역과 경계 영역의 중요도를 제어한다. 하한 근사는 경계 영역보다 더 기여를 하기 때문에,

이고

이다.In the above,

Represents cluster i,

to be.

Represents an object j,

Lt;

Wow

Are the lower bound approximation and upper bound approximation of cluster i, respectively,

Is the number of objects in the lower bound approximation of cluster i,

Is the boundary region of the cluster i,

Wow

Are the weighting parameters of the lower bound region and the boundary region, respectively. They control the importance of lower bounds and bounding regions when computing the corresponding prototype. Since the lower bound approximation contributes more than the boundary region,

ego

to be.

와

를 모든 클러스터들에 대한 오브젝트 j의 최소 거리 및 두번째 최소 거리라 하자.

는 임계값 파라미터이다.

인 경우, 오브젝트 j는 클러스터 i의 하한 근사에 속한다. 그렇지 않으면, 오브젝트 j는 클러스터 i와 클러스터 k 양자의 경계 영역들에 속한다.Objects belonging to the lower bound area and the border area of each cluster are determined by the following rules:

Wow

Let be the minimum distance and second minimum distance of object j for all clusters.

Is a threshold parameter.

, Then object j belongs to the lower bound approximation of cluster i. Otherwise, the object j belongs to the boundary areas of both cluster i and cluster k.

Rough-Fuzzy c-Mean Clustering Algorithm ( RFCM : rough-fuzzy c-means clustering algorithm)

Since no membership degrees are involved, uncertainties arising from overlapping cluster boundaries can not be effectively handled by the RCM. Since the fuzzy set and the rough set are complementary in some cases, it is natural to design an integrated method that combines the advantages of both the fuzzy set and the rough set. In this environment, S. Mitra et al. Introduced both the fuzzy set and the rough set into the framework of the c-means clustering method and proposed a rough-fuzzy c-means algorithm (referred to as RFCM I) (Mitra, 2006 ).

In RFCM I, clusters are represented by a purge lower approximation and a fuzzy upper bound approximation, which is different from the crisp approximation in RCM. Prototypes are updated in the following way:

이고,In the above,

ego,

이며,

Lt;

이고,

ego,

이다.

to be.

는 클러스터 i를 나타내며,

이고,

는 클러스터 i의 하한 근사이고,

는 클러스터 i의 경계 영역이며,

는 클러스터 i에 속하는 오브젝트 j의 멤버십 정도(membership degree)를 나타내는 퍼지 멤버십 값이고, m은 퍼지 계수이다. 두 근사들 어느 것도 비어 있지 않다면, 새로운 프로토타입이 하한 근사와 경계 영역 양자에 의해 계산되고, 그들의 중요도는 가중 파라미터들(

와

)에 의해 제어된다. 하한 근사는 새로운 프로토타입의 계산 시 경계 영역보다 더 많이 기여하기 때문에,

이고

이다.Where the number of clusters is c,

Represents cluster i,

ego,

Is the lower bound approximation of cluster i,

Is the boundary region of the cluster i,

Is a fuzzy membership value indicating the degree of membership of an object j belonging to cluster i, and m is a fuzzy coefficient. If neither of the two approximations is empty, a new prototype is computed by both the lower bound approximation and the bounding domain,

Wow

). Since the lower bound approximation contributes more than the bounded area in the computation of the new prototype,

ego

to be.

와

를 모든 클러스터들에 대한 오브젝트 j의 최대 및 두 번째 최대 멤버십 값들이라 하자.

인 경우, 오브젝트 j는 클러스터 i의 하한 근사에 속한다. 그렇지 않으면, 오브젝트 j는 클러스터 i와 클러스터 k 양자의 경계 영역들에 속한다.The objects are assigned to different approximations according to the following rules:

Wow

Let be the maximum and second maximum membership values of object j for all clusters.

, Then object j belongs to the lower bound approximation of cluster i. Otherwise, the object j belongs to the boundary areas of both cluster i and cluster k.

In determining which approximation an object belongs to, the absolute distance and relative distance adopted in the RCM has been replaced by a degree of fuzzy membership. The fuzzy membership function has the advantage of simultaneously belonging to all clusters with different degrees of ownership of the object, increasing the robustness of the clustering algorithm. The introduction of a fuzzy set will allow a rough set-based clustering algorithm to effectively handle duplicate problems.

RFCM's  Enhanced version

The basic feature of the rough set-based clustering algorithm is that the objects in the lower bound approximation are definitely in the cluster. Therefore, objects in the lower bound approximation must make the same contribution when updating the prototype, and their weights should not be affected by other prototypes. On the other hand, objects in bounded areas probably belong to one cluster and potentially belong to another cluster. Thus, objects in the bounding regions have different effects on updating the prototype. From the calculation of A ₁ in Equation (2), it can be seen that the lower bound approximation and the weights of the objects in the boundary region are fuzzy in RFCM I. To some extent, the criterion for updating prototypes in equation (2) can reduce the importance of objects in the lower bound approximation and cause the resulting prototypes to drift in the correct position (Maji, 2007a). A modified version of RFCM I was proposed by Maji (2007a), which is referred to as RFCM II. In RFCM II, the criteria for updating prototypes are summarized as follows:

이고,In the above,

ego,

이며,

Lt;

이다.

to be.

와

는 클러스터 i의 크리스프(crisp) 하한 근사 및 퍼지 경계 영역이고,

는 클러스터 i의 하한 근사에 있는 오브젝트들의 수이다. 수학식 3에서, 하한 근사에 있는 오브젝트들은 새로운 프로토타입의 계산 시 동일한 중요도를 갖고, 그들은 다른 프로토타입들을 고려하지 않는다. 경계 영역에 있는 오브젝트들은 여전히 프로토타입의 갱신시 상이한 중요도를 갖는다. RFCM II 알고리즘의 주 단계들과

와

의 파라미터 설정은 프로토타입들을 갱신하기 위한 기준을 제외하곤 RFCM I 알고리즘과 동일하다.In the above,

Wow

Is a crisp lower bound approximation and a fuzzy boundary region of cluster i,

Is the number of objects in the lower bound approximation of cluster i. In Equation 3, the objects in the lower bound approximation have the same importance in the computation of the new prototype, and they do not consider other prototypes. Objects in the bounds area still have different significance in updating the prototype. The main steps of the RFCM II algorithm and

Wow

Is the same as the RFCM I algorithm except for the criteria for updating prototypes.

In accordance with the commonness of the rough set-based c-means clustering algorithms such as RCM, RFCM I and RFCM II, the generalized framework of the rough set-based c-means clustering algorithms is as follows:

)을 설정한다.Step 1: The number of cluster prototypes (c), the fuzzification parameter (m), and the stop condition

).

Step 2: Initialize random prototypes for the c-clusters.

Step 3: Assign each object to corresponding approximations.

Step 4: Update the prototype for each cluster.

Step 5: Repeat steps 3 and 4 until the break criterion is reached.

In a rough set-based clustering algorithm, objects are divided into three regions for a certain cluster. During the update of the prototype and in the description of the cluster, the contribution of objects belonging to different regions is distinguished. The weighting parameters control the lower bound approximation and the contribution of the boundary regions when updating the prototypes. The values of the weighting parameters have a large effect on forming accurate approximate regions of a given cluster and directly affect the clustering results. Normally, the weighting parameter is manually set in advance without considering the distribution of the data set. Selecting a constant parameter can not reflect the structural characteristics of the clusters and the compactness of the objects belonging to any clusters. In this case, the formed approximate regions may be distorted and the resulting prototypes may deviate from the expected locations.

Weighting parameter Adaptive  Selection

In a classical rough set-based clustering algorithm, a weighted parameter is set manually without taking knowledge of the global relationship between objects. The object clustering method for image segmentation according to an embodiment of the present invention proposes an adaptive selection strategy of weighting parameters in a rough set-based clustering algorithm. In this way, each cluster will itself acquire a suitable weighting parameter that reflects the closeness of the objects. The formed approximate regions are determined in terms of individual clusters reflecting differences between all clusters in the data set.

라 하자.Conventionally, even if any rough set-based clustering algorithm is used, a user-defined weighting parameter is adopted. To see the effect of the weighting parameters on the clustering results, the weighting parameters of the different values versus the clustering result for the composite data set are compared. In view of the robustness of set-based clustering algorithms, RFCM II is adopted for weighted parameter analysis. It is not surprising that the importance of the lower bound approximation is greater than the importance of the boundary region in the prototype update

Let's say.

Figure 2 shows a scatter plot obtained by RFCM II in the final iterative step according to the change in weighting parameter. The final positions of quantitative measures and prototypes are shown in Table 1. Specifically, if the weighting parameter takes an unsuitable value, the lower bound approximation and contribution of the boundary region can not be well controlled at the time of updating the prototypes, and the resulting prototypes may deviate from their preferred position. Figure 2 and Table 1 show that different weighting parameters result in a wide variety of clustering results in solving the same problem.

(Clustering Accuracy, 1975), Adjusted Rand Index (ARI) (Hubert, 1985), and Minkowski Scores (Ben-Hur) , 2003) are employed to evaluate the effect of weighting parameters on clustering results versus different values. The statistical results of CA, ARI and MS are shown by box plot in Fig. The middle line in the box figure is used to represent the average result of the statistical samples, and the sign '+' is used to describe the stable characteristics of the sample distribution. The average results of CA, ARI and MS are shown in Table 2. From FIG. 3, it can be seen that when the weighting parameter takes a relatively large value, the box picture is very wide, indicating that the clustering results are not very stable as the weighting parameter changes. In addition, average results obtained from the CA, ARI and MS perspectives can also not be performed well with these values.

Figure 3 and Table 2 show that RFCM II is sensitive to the weighting parameters and that careful selection of the weighting parameters will result in satisfactory clustering. RFCM II performs relatively well with a weighting parameter of 0.51. FIG. 4 shows detailed approximate regions of each cluster when the weighting parameter is 0.51. The final segmentation result is shown in FIG. 4A, where prototypes are shown in yellow. The lower bound approximations of the three clusters are shown in FIG. 4B. 4C to 4E show boundary regions of respective clusters, respectively. The number of objects in the lower bound approximation of each cluster is different. It can be seen from FIG. 4 that the green cluster is the closest cluster with the largest number of objects in its lower bound approximation, while the blue cluster is the least dense cluster and has the smallest number of objects in its lower bound approximation have. The structural characteristics of each cluster are different.

From this analysis it has been proved that the use of certain weighted parameters to update different prototypes can not reflect the structural changes between clusters and can lead to unsatisfactory clustering results. Of course, the choice of weighting parameters depends on a given problem and is different for each cluster. The idea naturally comes to have an appropriate weighting parameter that controls the ratio of importance between its two approximations, so that each cluster reflects the closeness of the objects it belongs to. In order to form accurate approximate regions for each cluster, on-line determination of the weighting parameters is required, which allows the unique structural characteristics of the clusters to be well represented.

Once the fuzzy partitioning result is obtained, compactness is one of the main criteria in designing the clustering feasibility index to prove that the partitioning result accurately describes the data structure. The density is a measure of the closeness of objects in clusters and is a measure of the partition entropy (PE), partition coefficient (PC) index and partition coefficient and exponential partition (PCAES) ) Can be found in many clustering validity indices such as, for example, Sun (2004), Tsekouras (2004), Wu (2005), Bezdek (1974) and Trauwaert (1988).

The partitioned entropy (PE) index (Bezdek, 1974) measures the amount of fuzziness in the partitioning matrix, defined as:

는 퍼지 멤버십 값이다.Where a is the base of the log, n is the number of samples, c is the number of clusters,

Is a fuzzy membership value.

The partition coefficient (PC) index (Trauwaert, 1988) measures the average relative amount of membership sharing one of the pairs of fuzzy subsets in the partitioning matrix, defined as:

는 퍼지 멤버십 값이고, m은 퍼지 계수이다. 큰 PC 값은 더 나은 분할 결과를 나타낸다.Where n is the number of samples, c is the number of clusters,

Is a fuzzy membership value, and m is a fuzzy coefficient. A large PC value represents better splitting results.

The partitioning factor and exponent division (PCAES) index takes into account two factors with a normalized partitioning factor and exponentiation (Wu, 2005) for each cluster, thus collecting together these two factors of all clusters. The PCAES index is defined as:

In the above,

이다.

to be.

는 퍼지 멤버십 값이고, m은 퍼지 계수이다.Where n is the number of samples, c is the number of clusters,

Is a fuzzy membership value, and m is a fuzzy coefficient.

는 오브젝트 j가 클러스터 i에 속하는 정도를 측정하는 퍼지 멤버십 값이다. 그것은 상기한 3개의 클러스터링 타당성 지수들에서 나타난다. PCAES 지수에서,

로 지칭되는, 정규화된 분할 계수가 가장 컴팩트한 클러스터에 관한 클러스터 i의 밀집도(compactness)를 측정하는데 채택된다. 이 용어는 상대적인 값이고 PC 지수에서 사용되는 어떤 클러스터의 밀집도 척도와 유사하다. 차이는, PC 지수에서 밀집도 척도는 모든 클러스터들의 평균 값이고, PCAES 지수에서 각 클러스터의 밀집도 척도는 총 측정치를 형성하기 위하여 합산된다는 것이다. 퍼지 분할 결과를 획득한 이후에, 밀집도는 클러스터링 타당성의 중요한 측정치이다. 이 경우, 밀집도 척도는 또한 러프-퍼지 클러스터링 알고리즘에서 각 클러스터에 속하는 오브젝트들의 근접성을 스케일링하는데 사용될 수 있다.

Is a fuzzy membership value that measures the degree to which object j belongs to cluster i. It appears in the above three clustering validity indices. In the PCAES index,

Quot; is employed to measure the compactness of cluster i with respect to the most compact cluster. This term is a relative value and is similar to the cluster density measure of any cluster used in the PC index. The difference is that in the PC index, the density measure is the average value of all clusters, and in the PCAES index, the density measure of each cluster is summed to form the total measure. After obtaining the fuzzy partitioning result, the density is an important measure of clustering validity. In this case, the density measure may also be used to scale the proximity of objects belonging to each cluster in a rough-fuzzy clustering algorithm.

) 내의 오브젝트들의 수에 대한 하한 근사(

) 내의 오브젝트들의 수의 비율이다. 타깃 러프셋의 정확도가 높을수록, 형성될 그것의 2개의 근사들은 더 양호하다. 러프셋 X의 정확도는 하기와 같이 정의된다:Accuracy is a numerical feature to describe the inaccuracy of the target rough set. The accuracy of the rough set X is determined by the upper bound approximation (

Lt; RTI ID = 0.0 > (

Quot;). The higher the accuracy of the target rough set, the better the two approximations to be formed. The accuracy of the rough set X is defined as:

The definition in Equation (7) is based on the fact that each object is assigned to only one cluster. Since both the fuzzy set and the rough set are considered in the hybrid clustering algorithm, objects can be assigned to different clusters at the same time. To some extent, Equation 7 is the accuracy of the target set of the hard version, which is not suitable to describe the inaccuracy of the clusters in the hybrid clustering algorithms.

In rough set-based clustering algorithms, clusters are considered as a target set that is described together in terms of the lower bound approximation and the bounding domain. The weighting parameters control the lower bound approximation and the importance of the boundary region in representing clusters. In the object clustering method for image segmentation according to an embodiment of the present invention, since the structural characteristics of each cluster are different, the weight parameters of each cluster are adaptively selected. According to the above-mentioned analysis result, in order to describe the distribution characteristic in the hybrid clustering algorithms and to select the weight parameter of each cluster, the advantage of adopting the density for analyzing the fuzzy set and the accuracy for describing the rough set are used Some inspiration comes to mind by integrating the benefits together. The automatically selected weighting parameters are calculated as follows:

이다.And

to be.

는 클러스터 i를 나타내며,

이고,

와

는 각각 클러스터 i의 하한 근사 및 경계 영역이며,

는 오브젝트 j가 클러스터 i에 속하는 정도를 나타내는 퍼지 멤버십 값이고, m은 퍼지 계수이다.

는 클러스터 i의 프로토타입을 갱신할 때 클러스터 i의 하한 근사 내의 오브젝트들의 중요도를 반영한다. 그것은 전체 클러스터 i의 밀집도에 대한 클러스터 i에서의 하한 근사의 밀집도의 비율이다. 수학식 7과 수학식 8을 비교함으로써,

의 정의는 러프셋의 정확도의 표기와 유사하다는 것을 알 수 있다. 어느 정도까지,

는 퍼지 버전에서 러프셋의 정확도이다. 가중 파라미터의 계산은 퍼지셋을 분석하기 위한 밀집도의 이점과 러프셋을 기술하기 위한 정확도의 이점을 함께 통합한다.Where n is the number of objects,

Represents cluster i,

ego,

Wow

Are the lower bound approximation and boundary region of cluster i, respectively,

Is a fuzzy membership value indicating the degree to which object j belongs to cluster i, and m is a fuzzy coefficient.

Reflects the importance of the objects in the lower bound approximation of cluster i when updating the prototype of cluster i. It is the ratio of the density of the lower bound approximation in cluster i to the density of the entire cluster i. By comparing Equation 7 and Equation 8,

Is similar to the representation of the accuracy of the rough set. To some extent,

Is the accuracy of the rough set in the fuzzy version. The calculation of the weighting parameters together incorporates the advantages of density to analyze the fuzzy set and the accuracy to describe the rough set.

With adaptive weighting parameters, the disadvantage of manually selecting the weighting parameters in advance can be avoided and knowledge of the structural characteristics of each cluster is considered. In addition, the parameter selection scheme only uses the fuzzy membership values without any additional computation. Design criteria for computing prototypes are summarized as follows:

이고,In the above,

ego,

이며,

Lt;

이다.

to be.

는 클러스터 i를 나타내고,

이며,

는 클러스터 i의 하한 근사이고,

는 클러스터 i의 경계 영역이며,

는 오브젝트 j가 클러스터 i에 속하는 정도를 나타내는 퍼지 멤버십 값이고,

는 오브젝트 j를 나타내며,

이고, n은 양의 정수이며,

는 클러스터 i의 하한 근사에 있는 오브젝트들의 수이고,

는 클러스터 i의 가중 파라미터이다. 상기 가중 파라미터는 각 클러스터의 밀집도 특징에 의존하여 계산되고 클러스터 마다 상이하다.In the above,

Represents cluster i,

Lt;

Is the lower bound approximation of cluster i,

Is the boundary region of the cluster i,

Is a fuzzy membership value indicating the degree to which object j belongs to cluster i,

Represents an object j,

, N is a positive integer,

Is the number of objects in the lower bound approximation of cluster i,

Is a weighting parameter of cluster i. The weighting parameters are calculated depending on the density characteristics of each cluster and are different for each cluster.

In the object clustering method for image segmentation according to an embodiment of the present invention, a parameter set strategy is introduced to a rough set-based clustering algorithm and a rough set-based clustering algorithm, which adaptively selects a more suitable weighting parameter for each cluster during a clustering process, An improved version of the fuzzy c-mean clustering algorithm was developed. The main steps of the adaptive rough-fuzzy c-means clustering algorithm proposed in the object clustering method for image segmentation according to an embodiment of the present invention are as follows:

)을 설정한다.Step 1: The number of cluster prototypes (c), the fuzzification parameter (m), and the stop condition

).

Step 2: Initialize the cluster prototypes randomly.

Step 3: Set the loop counter b = 0.

Step 4: Compute the membership values for each object for the clusters.

Step 5: Allocate each object to corresponding approximations.

Step 6: Calculate the prototype by equation (9) for each cluster.

인 경우 중단한다. 그렇지 않으면

로 설정하고 단계 4로 진행한다.Step 7:

If it is. Otherwise

And proceeds to Step 4.

는 클러스터 프로토타입들의 벡터들이다. 단계 7에서의

는 현재의 반복 단계에서 획득된 프로토타입들이 이전의 반복 단계에서 생성된 것들과 동일하다는 것을 의미한다.From here,

Are vectors of cluster prototypes. In step 7,

Means that the prototypes obtained in the current iteration step are the same as those generated in the previous iteration step.

To obtain the final clustering result, each object is assigned to a cluster according to fuzzy membership values. In general, the maximum membership procedure method that can be summarized as follows is adopted:

, i≠k에 대해,

인 경우,

는 클러스터 i에 할당된다.-

, for i? k,

Quot;

Is assigned to cluster i.

This procedure assigns object j to cluster i with the highest membership value. A defuzzification process is performed to transform the fuzzy partition matrix into a final crisp partition.

12 is a flowchart of an object clustering method for image segmentation according to an embodiment of the present invention.

)을 설정한다.Referring to FIG. 12, in step S100, the number (c) of cluster prototypes, the fuzzification parameter (m)

).

In step S102, cluster prototypes are initialized.

In step S104, fuzzy membership values for each object for the clusters are calculated.

In step S106, each object is assigned to a corresponding approximate region of the rough set based on the fuzzy membership value.

In step S108, a prototype is calculated for each cluster based on weighting parameters that depend on the distribution characteristics of each cluster.

인지가 판단된다.In step S110,

.

는 클러스터 프로토타입들의 벡터들이고,

는 현재의 반복 단계에서 획득된 프로토타입들이 이전의 반복 단계에서 생성된 프로토타입들과 동일하다는 것을 의미한다. 상기에서

는 현재의 반복 단계에서 획득된 프로토타입들이고,

는 이전의 반복 단계에서 획득된 프로토타입들이다.In the above,

Are vectors of cluster prototypes,

Means that the prototypes obtained in the current iteration step are the same as the prototypes generated in the previous iteration step. In the above,

Are the prototypes obtained in the current iteration step,

Are the prototypes obtained in the previous iteration step.

인 경우, 단계 S112에서 현재의 반복 단계에서 결정된 프로토타입들을 각 클러스터에 대한 프로토타입으로 결정한다.

, The prototypes determined in the current iteration step are determined as prototypes for each cluster in step S112.

의 조건이 만족하지 않는 경우, 각 클러스터에 대한 프로토타입이 결정될 때까지 단계 S104 내지 단계 S108이 반복된다.

Is satisfied, steps S104 to S108 are repeated until a prototype for each cluster is determined.

In step S114, each object is assigned to a corresponding cluster according to the determined fuzzy membership value based on the prototype for each cluster.

The distribution characteristic of each cluster may include compactness of each cluster.

에 의해 계산되고,The prototype for each cluster in step < RTI ID = 0.0 > S108 &

Lt; / RTI >

이며,

이고,

이며,In the above,

Lt;

ego,

Lt;

는 클러스터 i의 가중 파라미터로서,

에 의해 계산되고,

이며, 상기에서

는 클러스터 i를 나타내고,

이며,

는 클러스터 i의 하한 근사이고,

는 클러스터 i의 경계 영역이며,

는 오브젝트 j가 클러스터 i에 속하는 정도를 나타내는 퍼지 멤버십 값이고,

는 오브젝트 j를 나타내며,

이고, n은 양의 정수이며,

는 클러스터 i의 하한 근사에 있는 오브젝트들의 수일 수 있다.In the above,

Is a weighting parameter of cluster i,

Lt; / RTI >

, And

Represents cluster i,

Lt;

Is the lower bound approximation of cluster i,

Is the boundary region of the cluster i,

Is a fuzzy membership value indicating the degree to which object j belongs to cluster i,

Represents an object j,

, N is a positive integer,

May be the number of objects in the lower bound approximation of cluster i.

Experiment result

The designed parameter selection strategy is introduced in a hybrid algorithm which forms an adaptive rough-fuzzy c-means clustering algorithm, briefly referred to as ARFCM. The ARFCM is compared with three rough set-based algorithms including RCM, RFCM I and RFCM II. Experimental work consists of three parts, including four composite data sets, eight real world data sets and six image partitioning problems. All compared algorithms are randomly initialized and run 30 times to compensate for the wrong starting point. Statistical results are shown by the box picture (McGill, 1978). The middle line in the box figure is used to represent the average result of the statistical samples, and the sign '+' is used to describe the stability characteristics of the sample distribution.

A summary of performance metrics is already available in several papers, Pakhira (2004); Wang (2007); Tasdemir (2011); Xu (2012). The evaluation indices, including CA, ARI and MS, are employed to evaluate the effectiveness of the proposed algorithm. The three metrics can only be used under conditions where the true label knowledge is available, but the actual verification is not readily obtained in the field of image segmentation. In this case, metrics that do not have a true reference set, such as PC Trauwaert (1988), PE Bezdek (1974), and Xie-Beni Index (XB) (Xie,

The parameter setting of the compared algorithms is described in the original reference, Mitra (2006); Peters (2006); Maji (2007a) has a consistent and small number of minor adjustments to the parameter settings. To simplify the representation, use the Arabic numerals 1, 2, 3, and 4 to denote RCM, RFCM I, RFCM II, and ARFCM. Specific parameter settings are shown in Table 3.

Results for composite datasets

Four composite data sets derived from three three-dimensional Gaussian classes are designed. Figure 5 shows a scatter diagram of four synthetic data sets. It is clear that each data set has three distinct classes. The coordinates of the three prototypes are (2.5, 4.5), (0.5, 1) and (3.5, 1), respectively. The size and density of the composite data sets change back and forth.

FIG. 6 shows statistical values of CA, ARI and MS achieved by each algorithm. The value of CA in solving the four composite data sets is 1 in nearly all independent 30 independent runs by object clustering method for image segmentation according to an embodiment of the present invention as shown in FIG. can do. The box pictures from the perspective of CA, ARI and MS obtained by RFCM I and RFCM II for composite data set 3 are very wide indicating that they lack effective stability in handling this problem, Can easily fall into the trap of local optimization. Although the RFCM II has also obtained a large value in terms of ARI for composite data set 2, the results achieved by RFCM II are not as stable as those obtained by ARFCM as shown in FIG. 6B. In general, the results obtained by RFCM I and RFCM II are better than those of RCM, so validation of fuzzy membership hybridization with rough approximation may result in better clustering results. The statistical results obtained by the ARFCM are more stable and superior in terms of ARI and MS than the results of the other three algorithms.

In the object clustering method for image segmentation according to an embodiment of the present invention, the weighting parameters are adaptively adjusted during the clustering process. Figure 7 shows that the weighting parameters vary with each occurrence in solving the composite data set 2. It can be seen that the weighting parameters change per iteration until a distinct and stable condition is reached for each cluster.

UCI  Results for datasets

A comparison test of four algorithms for 8 UCI data sets is provided. The data sets may be downloaded from the Asunction 2007. The statistical results in terms of CA, ARI and MS are shown in box in Fig.

Taking into account the statistical results, the following conclusions can be drawn:

1. An object clustering method for image segmentation according to an exemplary embodiment of the present invention provides the best result with the lowest MS value, highest ARI, and CA values for the New-thyroid, WBC, and Pima than the other three algorithms Respectively.

2. In solving the complex entangled problem called Balance, all four algorithms could not obtain stable clustering results (the results have a broad box picture). Even in this situation, the object clustering method for image segmentation according to an embodiment of the present invention has obtained the best result in terms of three indices.

3. In resolving Wine, Breast, Weather, and Vote, although three other algorithms have obtained results similar to those of the object clustering method for image segmentation according to an embodiment of the present invention, ARFCM, which is an object clustering method for image segmentation, provides the most stable solution among all four algorithms.

The statistical results shown in Figure 8 demonstrate that the solutions obtained by the ARFCM are more stable and superior in terms of CA, ARI and MS than the other three rough set-based algorithms.

Image segmentation result

Recently, soft computing techniques have been extensively studied for image segmentation because they can hold more information than hard clustering algorithms. Image segmentation can be viewed as a process of pixel intensity clustering, where pixel intensities are used as input samples. If the gray levels are scaled to be in the range [0, 1], the regions and relationships between them will similarly be regarded as a fuzzy subset of images. Pixels in the same region share more similarity with each other than pixels belonging to different regions. For preliminary experiments, only gray information is considered without other types of image information, such as spatial or texture information.

Experiments are performed on six images used in the field of image segmentation algorithms to compare segmentation performance of different rough set-based clustering algorithms. To demonstrate the performance of each algorithm, not only XB but also indexes such as partitioning factor, partitioning entropy are employed to evaluate partitioning results. Large PC values and small PE and XB values indicate better splitting results. A quantitative description of the segmentation results achieved by the compared algorithms is presented in Table 4. < tb > < TABLE >

Rough-fuzzy c-mean clustering algorithms integrate the advantages of both fuzzy sets and rough sets. RFCM I and RFCM II achieve good segmentation results in terms of three indices compared to the RCM for most images, as shown in Table 4. The object clustering method for image segmentation according to an embodiment of the present invention achieves much higher PC values and lower PE and XB values than RFCM I and RFCM II. In general, the average result obtained by the ARFCM is the best of the four algorithms.

A comparison of the segmentation results obtained by the relative algorithms is also investigated. Figure 9 shows a visual comparison of the segmentation results obtained by each algorithm. The loss of image detail caused by the RCM is more severe because the RCM miscategorizes in some locations on brain MR images. Two variants of RFCM still lack robustness, although they are better than RCM. The ARFCM can preserve image details better, which is evidenced by the photographer's right cross and the location under the left leg.

Run Time Research

The average execution time (in seconds) by four algorithms is examined in finding a solution for a composite data set of different sizes. In this experiment, the codes of the algorithms compared are programmed in MATLAB 9.0.1 and the machine is HP Workstation xw9300 (2.19 GHz, 16GB, RAM; Hewlett Packard, Palo Alto, Calif.). The operating system is Microsoft XP Professional x64 Edition. Composite data sets with different sizes are shown in FIG.

From Fig. 11, it can be seen that RCM is the best of the four algorithms in terms of computation time. This is because RCM does not use any membership values during implementation. The distinction between RFCM I and RFCM II is unclear. ARFCM calculation costs are lower than those of RFCM I and RFCM II. Table 5 shows the average number of iterations to reach the convergence condition of the different algorithms. From Table 5 and Figure 11, it can be seen that ARFCM requires fewer iterations than RFCM I and RFCM II to reach the convergence condition. With the aid of adaptive weighting parameters, prototypes can quickly be placed close to the desired position.

conclusion

In a rough set-based clustering algorithm, it is not desirable to adopt certain parameters to describe the lower and upper bound approximations of different clusters, since the distribution characteristics vary from cluster to cluster. In the object clustering method for image segmentation according to an embodiment of the present invention, a parameter selection strategy is designed to adaptively adjust the weighting parameters depending on distribution characteristics of each cluster, instead of manually selecting certain parameters. According to this approach, the calculated weighting parameters are different for each cluster and are updated at each iteration step. In the object clustering method for image segmentation according to an embodiment of the present invention, since the proposed parameter selection strategy utilizes the knowledge of the global relation between objects, the two generated approximations are used to describe the actual distribution characteristics of clusters It will be close. By combining various soft computing techniques, including fuzzy sets and rough sets, designed pattern recognition methods simultaneously gain their advantages.

For many modified versions of the c-means clustering algorithm, the Euclidean distance is widely used as a similarity measure. Its performance will deteriorate when looking for a solution of a data set with non-constant density or non-hyper spherical shape. The choice of the similarity measure is highly dependent on the given problem and has a significant impact on the clustering results.

While the present invention has been particularly shown and described with reference to exemplary embodiments thereof, it is to be understood that the invention is not limited to the disclosed exemplary embodiments. It is clear that the present invention can be modified or improved.

It will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.

S100: Initial setting value setting step
S102: prototype initialization step
S104: Fuzzy membership value calculation step
S106: Assigning the object to the approximate area
S108: Calculating the prototype for each cluster based on the weighting parameters dependent on the distribution characteristics of the cluster
S110: Repeat condition determination step
S112: Prototype determination step for each cluster
S114: Assigning each object to a corresponding cluster

Claims (3)

(a) the number of cluster prototypes (c), the fuzzy parameter (m)

);
(b) initializing cluster prototypes;
(c) calculating fuzzy membership values for each object for the clusters;
(d) assigning each object to a corresponding approximate region of the rough set based on the fuzzy membership value;
(e) computing a prototype for each cluster based on a weighting parameter that is dependent on a distribution characteristic of each cluster;
(f)

, Determining prototypes determined in the current iteration step as prototypes for each cluster, and if not, repeating steps (c) through (e) until a prototype for each cluster is determined; And
(g) assigning each object to a corresponding cluster according to a fuzzy membership value based on the determined prototype for each cluster,
In the above,

Are vectors of cluster prototypes,

Means that the prototypes obtained in the current iteration step are the same as the prototypes generated in the previous iteration step,
The distribution characteristic of each cluster includes the compactness of each cluster,
The prototype for each cluster in step (e)

Lt; / RTI >
In the above,

Lt;

ego,

Lt;
In the above,

Is a weighting parameter of cluster i,

Lt; / RTI >

Lt;
In the above,

Represents cluster i,

Lt;

Is the lower bound approximation of cluster i,

Is the boundary region of the cluster i,

Is a fuzzy membership value indicating the degree to which object j belongs to cluster i,

Represents an object j,

, N is a positive integer,

Is the number of objects in the lower bound approximation of cluster i. delete delete

Images