J. C. Dunn. A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters, J. Cybernet. 3 (3): 32-57, 1974  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... huge amount of literature on the subject, ranging from models, algorithms, algorithm parameter estimations to cluster validity studies [14] 33] The clustering methods can be divided up into exclusive and non exclusive methods [26] The best known non exclusive method is the fuzzy C means model [15]. In this method objects are soft clustered such that objects belong to all clusters to a certain degree. For an overview of fuzzy clustering methods see for example [3] and [4] In exclusive clustering methods, the objects are partitioned into a number of (crisp) subsets, such The authors are ....

J. C. Dunn. A fuzzy relative of the ISODATA process and its use in detecting compact wellseparated clusters. Journal of Cybernetics, 3:32--57, 1974.

....Other methods for extracting features include wavelets and Gabor filters [11] Essentially, the feature extraction problem is the image segmentation problem, for which a number of fuzzy methods have already been proposed. As an example, a recently published system [3] uses Fuzzy C Means clustering [7] followed by adaptive neural network thresholding to simultaneously segment an image into regions and find its edges. Fuzzy models of shape and relationships between objects have already been developed. Using these methods might be able to enhance the feature set available for image ....

....to determine which cluster is relevant. Clustering was introduced to the web as a method of limiting the number of documents that a user is shown. An early experiment in Web document clustering that allowing relevant documents to appear in multiple clusters is advantageous [15] Fuzzy clustering [7] is a well known generalization of clustering where each element can have non zero membership in multiple clusters. Cluster exemplars are then computed taking into consideration the relative membership of each member of the cluster. Given the complexity of the results of most internet searches, a ....

J.C. Dunn, "A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters," J. Cybernetics, vol. 3, 1973, pp. 32-57. Reprinted in Fuzzy Models for Pattern Recognition, J.C. Bezdek and S.K. Pal, eds., IEEE Press, 1992.

....to a given number of clusters such that each of them belongs to one or more clusters with di erent degrees of membership. The objective is to minimize the sum of squared distances to the centroids, weighted by the degrees of membership. The fuzzy clustering problem was initially formulated in [42] as a mathematical programming problem and later generalized in [13] The most popular heuristic for solving FCP is the so called Fuzzy C means (F CM) method [23] It alternatively nds membership matrices and centroids until there is no more improvement in the objective function value. A new ....

J.C. Dunn. A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters, J. Cybernet. 3 (3): 32-57, 1974.

....to subdivide a data set into subsets (clusters) Fuzzy clustering algorithms consider each cluster as a fuzzy set, while a membership function measures the possibility that each training vector belongs to this set. The fuzzy k means (FKM) algorithm also known as fuzzy ISODATA proposed by Dunn [2] and extended by Bezdek [3] and the fuzzy vector quantization (FVQ) algorithm proposed by Karayiannis [4] will be used for decision level fusion. The fuzzy clustering algorithms will be used to combine results coming from various single modality person authentication algorithms (e.g. from ....

....of the algorithm strongly depends on the initialization of the codebook vectors and on the presence of outliers. Once the codebook is designed, any data is classified into a cluster based on a classical distance criterion. C. The fzzy k means algorithm The fuzzy k means algorithm (FKM) [2] classifies each vector to all clusters with different values of membership between 0 and 1. This membership value indicates the association of a vector to each of the k dusters. Notice that the fuzzy k means algorithm does not classify fuzzy data, but crisp data into fuzzy dusters. The algorithm ....

J.C. Dunn, "A fuzzy relative of the isodata process and its use in detecting compact well-separated clusters," J. Cybernetics, vol. 3, pp. 32-57, March 1973.

....clustering and median radial basis function (MRBF) algo rithms for decision level fusion, is proposed. Classical clustering methods refer to a wide variety of methods that attempt to subdivide a data set into sub sets (clusters) Fuzzy clustering algorithms, such as the fuzzy k means (FKM) [2, 3] and the fuzzy vector quantization (FVQ) 4] consider each cluster as a fuzzy set, while a membership function measures the possibility that each training vector belongs to a cluster. The FKM and FVQ are used to combine results coming from various single modality person authentication algorithms ....

....depends on the initialization of the codebook vectors. Since the codebook is designed, any data is classified into a cluster based on a classical distance criterion. The fuzzy k means algorithm (FKM) classifies each vector to all clusters with different values of membership between 0 and i [2]. This membership value indicates the association of a vector to each of the k clusters. No tice that the fuzzy k means algorithm does not classify fuzzy data, but crisp data into fuzzy clusters. The algorithm is derived from the constrained minimization of the following objective function: k M ....

J. C. Dunn. A fuzzy relative of the isodata pro- cess and its use in detecting compact well-separated clusters. J. Cybernetics, 3:32-57, March 1973.

....some data vectors belong partially to several clusters. The fuzzy set theory [15] is a natural way to describe this situation. In this case. the membership degree of a vector x to the th cluster (zti) is a value from [0.1] interval. This idea was first introduced by Ruspini [12] and used by Dunn [5] to construct a fuzzy clustering method based on the criterion function minimization. In [1] Bezdek generalized this approach to an infinite family of fuzzy c means algorithm using a weighted exponent on the fuzzy memberships. Fuzzy c means clustering algorithm is successfully applied to a wide ....

....all real (c x N) dimeional matrices. function h the form [1] c N (u, v) 2) where U ,V p. di is the ier product induced norm: i) The fuzzy c means criterion where A is a positive definite matrix and m is a weighting exponent in [1, oc) Criterion (2) for m = 2 was introduced by Dunn [5]. An infinite family of fuzzy c means criterions for m [1, oc) were introduced by Bezdek. Using Lagrange multipliers the following theorem can be proved, via obtaining necessary conditions for minimization of (2) 1] Theorem 1: If m and c are fixed parameters, and I, k are sets defined as: ....

J.C. Dunn, "A Fuzzy Relative of the ISODATA Process and its Use in Detecting Compact Well-Separated Cluster" Journal Cybernetics Vol. 3, No.3, pp.32-57, 1973.

....Finally, an unsupervised segmentation method is obtained by adding to the segmentation rule above one of the parameter estimation methods of the previous sections. Let us briefly discuss the relation of such methods to the fuzzy means methods. The fuzzy means algorithm was first proposed by Dunn [13] for the case 2 [see (32) as an extension of hard classification ( 1) called Isodata. The general form of the fuzzy means algorithm, i.e. for any greater then one, was proposed by Bezdek [2] and studied by Hunstberger, Jacobs, and Canno [16] among others. In the latter methods the fuzzy ....

....same is true in the fuzzy context. Otherwise, it is possible to define fuzzy hidden Markov models, which include Lebesgue and Dirac measures in priors and consider the corresponding global methods [26] 28] The segmentation method we presented is different from the fuzzy means algorithm [2] [13], 16] and appears, according to the results of segmentation of a real image, as complementary. As for topics of further work, let us point out the possibility of merging of our algorithms with Krishnapuram and Keller s approach [19] This allows one to relax our hypothesis according to which the ....

J. C. Dunn, "A fuzzy relative of the Isodata process and its use in detecting compact well-separated clusters," J. Cybernet., vol. 31, pp. 32--57, 1974.

....X x k , k , x k by attributing each data point x k to a subset w j X , j , defined by its centroid y j . This attribution is made based on a given distance d . The most widely used clustering method is probably the Fuzzy c Means Fuzzy ISODATA [1][2][3] FCM) algorithm, which is a fuzzy relative to the simple c Means technique [4] FCM defines the w j as fuzzy partitions of the data set X . Variations over this basic scheme try to overcome some of its well known limitations. The Deterministic Annealing (or Maximum Entropy) approach [5] 6] ....

....view 2. 1 The c Means family We will now review some clustering algorithms derived from the basic c Means: hard ) c Means (HCM) 4] entropy constrained fuzzy clustering by Deterministic Annealing (DA) 5] Possibilistic c Means with an entropic cost term (PCM II) 14] Fuzzy c Means (FCM) [2]. All of these techniques are based on minimizing the following cost function: E (2) this includes also FCM, although in the usual formulation this is not evident; see the Appendix) We will refer collectively to these algorithms as the c Means (CM) family. Here u jk U is the degree of ....

J. C. Dunn, "A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters", Journal of Cybernetics, vol. 3, pp. 32--57, 1974.

....N. Otsu, A threshold selection method from gray level histogram, IEEE Trans. Syst. Man, Cybern. vol. SMC 9, pp. 62 66, Jan. 1979. 8] L. Gupta and T. Sortrakul, A Gaussian mixture based image segmentation algorithm, Pattern Recognit. vol. 31, no. 3, pp. 315 325, 1998. [9] E. R. Dougherty, An Introduction to Morphological Image Processing. Bellingham, WA: SPIE, 1992. 10] L. Gupta, T. Sortrakul, A. Charles, and P. Kisatsky, Robust automatic target recognition using a localized boundary representation, Pattern Recognit. vol. 28 10, pp. 1587 1598, 1995. 11] H. ....

....the appropriate clustering automatically. Variable string length genetic algorithm (VGA) 6] with real encoding of the cluster centers in the chromosome [7] is used as the underlying search tool for this purpose. Several cluster validity indices viz. Davies Bouldin (DB) index [8] Dunn s index [9], two of its generalized versions [10] and a newly developed validity index are utilized for computing the fitness of the chromosomes. The results provide a comparison of these indices in terms of their utility in determining the appropriate clustering of the data. Several artificial and ....

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J. C. Dunn, "A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters," J. Cybern., vol. 3, pp. 32--57, 1973.

....coefficients for a region of an image containing a spot and a portion of background. The centroid values and crossover point are shown. If auto calculate background is checked, the foreground and background pixel densities are computed from the selected region by the fuzzy k means algorithm [2 5]. As shown in the above figure, this algorithm estimates the contribution to each pixel from the foreground and background clusters. The centroid (modal value) of the resulting population of partition coeificients is the most probable density of the pixels in each cluster. If the algorithm ....

J. C. Dunn. A fuzzy relative of the isodata process and its use in detecting compact well-separated clusters. J. Cybern., 3:32, 1974.

....depend on all distances of point # to the model prototypes, i.e. # # # # # ## # # ##### # #. Well known algorithms minimizing energies contained in this framework are the snakes (with Cohen energy [3] and several Pattern Recognition methods (Kohonen maps [7] elastic nets [5] and fuzzy c means [4] [2] The weighting functions associated with these algorithms are shown in Tab l e 1 The minimization of (1) can be performed by several methods. A popular solution is the use of a gradient algorithm # . The gradient of the energy (2) can be expressed in terms of the differences between the ....

J. Dunn. A fuzzy relative of the ISODATA process and its use in detecting compact wellseparated clusters. Journal of Cybernetics, 3(3):32--57, 1973.

....In a recent letter [19] we presented some initial results on an unsupervised segmentation algorithm called the adaptive fuzzy C means algorithm (AFCM) designed for segmenting twodimensional (2 D) scalar images corrupted by intensity inhomogeneities. Based on the fuzzy C means algorithm (FCM) [20, 21], the advantages of 2 D AFCM are that it automatically produces soft segmentations, it is robust to inhomogeneities, and it computes a smooth gain field based on all pixels in the image. Although this algorithm is suitable for the segmentation of MR images obtained using single or multi slice ....

....Taking the first derivatives of Eq. 1) with respect to u jk and v k and setting those equations to zero yields necessary conditions for (1) to be minimized. Iterating through these two necessary conditions leads to a grouped coordinate 5 descent scheme for minimizing the objective function [20, 21]. This is the standard FCM algorithm. The resulting fuzzy segmentation can be converted to a hard or crisp segmentation by assigning each voxel solely to the class that has the highest membership value for that voxel. This is known as a maximum membership segmentation. The advantages of FCM are ....

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J.C. Dunn, "A fuzzy relative of the ISODATA process and its use in detecting compact well-sparated clusters," Journal of Cybernetics, vol. 3, pp. 32--57, 1973. 30

.... clustering algorithms There are many different types of clustering algorithms, every method has its own advantages and disvantages, from K means clustering algorithm (KM) Competitive Learning (CL) to Fuzzy c means algorithm (FCM) 15] However, almost all the common clustering algorithms [16,17,18,19, 20, 21, 22] nowadays can divide into two groups. They are probabilistic clustering [23] and possibilistic clustering [24] One of the great problem on Probabilistic Clustering algorithms is Outliers. Outliers are vectors, or called data point, in the data domain which are so distant from the rest of the ....

J. Dunn, "A fuzzy relative of the isodata process and its use in detecting compact wellseparated clusters," in Journal of Cybernetics, p. 3:32, 1974.

....each # # belongs to several di erent fuzzy events to a certain degree. As will be explained below, we are interested in a very special sample space # of fuzzy events, namely one for which the membership values of each fuzzy sample point sum up to one, so that # # ##=1###### form a fuzzy partition [7] of # . In mathematical terms, we suppose that ## # : # ## ### (# # ) 1# (12) If the last equation holds, we can proof an interesting property of a discrete fuzzy sample space. Theorem 2.1 Letasetoffuzzy events # # ## # ###### # #### be given whereeach event # # is described by its membership ....

J.C. Dunn, \A fuzzy relative of the isodata process and its use in detecting compact, well-seperated clusters," Journal of Cybernetics,vol. 3, no. 3, pp. 32-57, 1973.

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J. C. Dunn. A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters, J. Cybernet. 3 (3): 32-57, 1974

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J. C. Dunn. A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters, J. Cybernetics, 3, pp. 32-57, 1973.

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J. C. Dunn, "A fuzzy relative of the isodata process and its use in detecting compact well-separated clusters," in Journal of Cybernetics, 3, 1973.

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J.C.Dunn, A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters. Journal of Cybernetics 3:32-57, 1973.

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J.C. Dunn. A fuzzy relative of the isodata process and its use in detecting compact, well-seperated clusters. Journal of Cybernetics, 3(3):32-57, 1973.

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J. C. Dunn, "A fuzzy relative of the isodata process and its use in detecting compact well-separated clusters," in Journal of Cybernetics, 3, 1973.

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J. C. Dunn, "A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Seperated Clusters", Journal Cybernetics, vol. 3, no. 3, 1973, pp. 32-57.

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J. Dunn, "A fuzzy relative of the isodata process and its use in detecting compact well-separated clusters", in Journal of Cybernetics, page 3:32, 1974.

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J. Dunn, "A fuzzy relative of the isodata process and its use in detecting compact, well-separated clusters," J. Cybernet., vol. 3, no. 3, pp. 32--57, 1974.

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Dunn, J. (1973). A fuzzy relative of the ISODATA process and its use in detecting compact, well-separated clusters. Journal of Cybernetics, 3(3):32--57.

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DUNN, J. C. A fuzzy relative of the ISODATA process and its use in detecting compact wellseparated clusters. J. Cybern. 3, 32 (1974).

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