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55
Deterministic Annealing for Clustering, Compression, Classification, Regression, and Related Optimization Problems
 Proceedings of the IEEE
, 1998
"... this paper. Let us place it within the neural network perspective, and particularly that of learning. The area of neural networks has greatly benefited from its unique position at the crossroads of several diverse scientific and engineering disciplines including statistics and probability theory, ph ..."
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Cited by 247 (11 self)
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this paper. Let us place it within the neural network perspective, and particularly that of learning. The area of neural networks has greatly benefited from its unique position at the crossroads of several diverse scientific and engineering disciplines including statistics and probability theory, physics, biology, control and signal processing, information theory, complexity theory, and psychology (see [45]). Neural networks have provided a fertile soil for the infusion (and occasionally confusion) of ideas, as well as a meeting ground for comparing viewpoints, sharing tools, and renovating approaches. It is within the illdefined boundaries of the field of neural networks that researchers in traditionally distant fields have come to the realization that they have been attacking fundamentally similar optimization problems.
Adaptive fuzzy segmentation of magnetic resonance images
 IEEE TRANS. MED. IMAG
, 1999
"... An algorithm is presented for the fuzzy segmentation of twodimensional (2D) and threedimensional (3D) multispectral magnetic resonance (MR) images that have been corrupted by intensity inhomogeneities, also known as shading artifacts. The algorithm is an extension of the 2D adaptive fuzzy Cme ..."
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Cited by 89 (8 self)
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An algorithm is presented for the fuzzy segmentation of twodimensional (2D) and threedimensional (3D) multispectral magnetic resonance (MR) images that have been corrupted by intensity inhomogeneities, also known as shading artifacts. The algorithm is an extension of the 2D adaptive fuzzy Cmeans algorithm (2D AFCM) presented in previous work by the authors. This algorithm models the intensity inhomogeneities as a gain field that causes image intensities to smoothly and slowly vary through the image space. It iteratively adapts to the intensity inhomogeneities and is completely automated. In this paper, we fully generalize 2D AFCM to threedimensional (3D) multispectral images. Because of the potential size of 3D image data, we also describe a new faster multigridbased algorithm for its implementation. We show, using simulated MR data, that 3D AFCM yields lower error rates than both the standard fuzzy Cmeans (FCM) algorithm and two other competing methods, when segmenting corrupted images. Its efficacy is further demonstrated using real 3D scalar and multispectral MR brain images.
A Fast Clustering Algorithm to Cluster Very Large Categorical Data Sets in Data Mining
 In Research Issues on Data Mining and Knowledge Discovery
, 1997
"... Partitioning a large set of objects into homogeneous clusters is a fundamental operation in data mining. The kmeans algorithm is best suited for implementing this operation because of its efficiency in clustering large data sets. However, working only on numeric values limits its use in data mining ..."
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Cited by 82 (2 self)
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Partitioning a large set of objects into homogeneous clusters is a fundamental operation in data mining. The kmeans algorithm is best suited for implementing this operation because of its efficiency in clustering large data sets. However, working only on numeric values limits its use in data mining because data sets in data mining often contain categorical values. In this paper we present an algorithm, called kmodes, to extend the kmeans paradigm to categorical domains. We introduce new dissimilarity measures to deal with categorical objects, replace means of clusters with modes, and use a frequency based method to update modes in the clustering process to minimise the clustering cost function. Tested with the well known soybean disease data set the algorithm has demonstrated a very good classification performance. Experiments on a very large health insurance data set consisting of half a million records and 34 categorical attributes show that the algorithm is scalable in terms of ...
Vector Quantization with Complexity Costs
, 1993
"... Vector quantization is a data compression method where a set of data points is encoded by a reduced set of reference vectors, the codebook. We discuss a vector quantization strategy which jointly optimizes distortion errors and the codebook complexity, thereby, determining the size of the codebook. ..."
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Cited by 53 (17 self)
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Vector quantization is a data compression method where a set of data points is encoded by a reduced set of reference vectors, the codebook. We discuss a vector quantization strategy which jointly optimizes distortion errors and the codebook complexity, thereby, determining the size of the codebook. A maximum entropy estimation of the cost function yields an optimal number of reference vectors, their positions and their assignment probabilities. The dependence of the codebook density on the data density for different complexity functions is investigated in the limit of asymptotic quantization levels. How different complexity measures influence the efficiency of vector quantizers is studied for the task of image compression, i.e., we quantize the wavelet coefficients of gray level images and measure the reconstruction error. Our approach establishes a unifying framework for different quantization methods like Kmeans clustering and its fuzzy version, entropy constrained vector quantizati...
A modified fuzzy Cmeans algorithm for bias field estimation and segmentation of MRI data
 IEEE Trans. on Medical Imaging
, 2002
"... Abstract—In this paper, we present a novel algorithm for fuzzy segmentation of magnetic resonance imaging (MRI) data and estimation of intensity inhomogeneities using fuzzy logic. MRI intensity inhomogeneities can be attributed to imperfections in the radiofrequency coils or to problems associated ..."
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Cited by 51 (1 self)
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Abstract—In this paper, we present a novel algorithm for fuzzy segmentation of magnetic resonance imaging (MRI) data and estimation of intensity inhomogeneities using fuzzy logic. MRI intensity inhomogeneities can be attributed to imperfections in the radiofrequency coils or to problems associated with the acquisition sequences. The result is a slowly varying shading artifact over the image that can produce errors with conventional intensitybased classification. Our algorithm is formulated by modifying the objective function of the standard fuzzy cmeans (FCM) algorithm to compensate for such inhomogeneities and to allow the labeling of a pixel (voxel) to be influenced by the labels in its immediate neighborhood. The neighborhood effect acts as a regularizer and biases the solution toward piecewisehomogeneous labelings. Such a regularization is useful in segmenting scans corrupted by salt and pepper noise. Experimental results on both synthetic images and MR data are given to demonstrate the effectiveness and efficiency of the proposed algorithm. Index Terms—Bias field, fuzzy logic, image segmentation, MR imaging. I.
An Adaptive Fuzzy CMeans Algorithm for Image Segmentation in the Presence of Intensity Inhomogeneities
 Pattern Recognition Letters
, 1998
"... We present a novel algorithm for obtaining fuzzy segmentations of images that are subject to multiplicative intensity inhomogeneities, such as magnetic resonance images. The algorithm is formulated by modifying the objective function in the fuzzy Cmeans algorithm to include a multiplier field, whic ..."
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Cited by 48 (6 self)
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We present a novel algorithm for obtaining fuzzy segmentations of images that are subject to multiplicative intensity inhomogeneities, such as magnetic resonance images. The algorithm is formulated by modifying the objective function in the fuzzy Cmeans algorithm to include a multiplier field, which allows the centroids for each class to vary across the image. First and second order regularization terms ensure that the multiplier field is both slowly varying and smooth. An iterative algorithm that minimizes the objective function is described, and its efficacy is demonstrated on several test images. Key words: image segmentation, fuzzy cmeans, intensity inhomogeneities, magnetic resonance imaging 1 Introduction Image segmentation plays an important role in a variety of applications such as robot vision, object recognition, and medical imaging. There has been considerable interest recently in the use of fuzzy segmentation methods, which retain more information from the original im...
Constrained clustering as an optimization method
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1993
"... AbstractOur deterministic annealing approach to clustering is derived on the basis of the principle of maximum entropy, is independent of the initial state, and produces natural hierarchical clustering solutions by going through a sequence of phase transitions. This approach is modified here for a ..."
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Cited by 43 (7 self)
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AbstractOur deterministic annealing approach to clustering is derived on the basis of the principle of maximum entropy, is independent of the initial state, and produces natural hierarchical clustering solutions by going through a sequence of phase transitions. This approach is modified here for a larger class of optimization problems by adding constraints to the free energy. The concept of constrained clustering is explained, and then, three examples are given in which it is used as means to introduce deterministic annealing. First, the previous clustering method is improved by adding cluster mass variables and a total mass constraint. Second, the traveling salesman problem (TSP) is reformulated as constrained clustering, yielding the elastic net (EN) approach to the problem. More insight is gained by identifying a second Lagrange multiplier that is related to the tour length add can also be used to control the annealing process. Finally, the “open path ” constraint formulation is shown to relate to dimensionality reduction by selforganization in unsupervised learning. A similar annealing procedure is applicable in this case as well. Index TermsAnnealing, clustering, maximum entropy, neural networks, nonconvex optimization, selforganization.
Spatial models for fuzzy clustering
 Computer Vision and Image Understanding
, 2001
"... A novel approach to fuzzy clustering for image segmentation is described. The fuzzy Cmeans objective function is generalized to include a spatial penalty on the membership functions. The penalty term leads to an iterative algorithm that is only slightly different from the original fuzzy Cmeans alg ..."
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Cited by 20 (5 self)
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A novel approach to fuzzy clustering for image segmentation is described. The fuzzy Cmeans objective function is generalized to include a spatial penalty on the membership functions. The penalty term leads to an iterative algorithm that is only slightly different from the original fuzzy Cmeans algorithm and allows the estimation of spatially smooth membership functions. To determine the strength of the penalty function, a criterion based on crossvalidation is employed. The new algorithm is applied to simulated and real magnetic resonance images and is shown to be more robust to noise and other artifacts than competing approaches.
Knowledge Discovery from Sequential Data
, 2003
"... A new framework for analyzing sequential or temporal data such as time series is proposed. It differs from other approaches by the special emphasis on the interpretability of the results, since interpretability is of vital importance for knowledge discovery, that is, the development of new knowl ..."
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Cited by 17 (0 self)
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A new framework for analyzing sequential or temporal data such as time series is proposed. It differs from other approaches by the special emphasis on the interpretability of the results, since interpretability is of vital importance for knowledge discovery, that is, the development of new knowledge (in the head of a human) from a list of discovered patterns. While traditional approaches try to model and predict all time series observations, the focus in this work is on modelling local dependencies in multivariate time series. This
Differential evolution methods for unsupervised image classification
 in Proc. 7th CEC, 2005
"... Abstract A clustering method that is based on ..."