Nikhil R. Pal

A review on image segmentation techniques

Many image segmentation techniques are available in the literature. Some of these techniques use only the gray level histogram, some use spatial details while others use fuzzy set theoretic approaches. Most of these techniques are not suitable for noisy environments. Some works have been done using the Markov Random Field (MRF) model which is robust to noise, but is computationally involved. Neural network architectures which help to get the output in real time because of their parallel processing ability, have also been used for segmentation and they work fine even when the noise level is very high. The literature on color image segmentation is not that rich as it is for gray tone images. This paper critically reviews and summarizes some of these techniques. Attempts have been made to cover both fuzzy and non-fuzzy techniques including color image segmentation and neural network based approaches. Adequate attention is paid to segmentation of range images and magnetic resonance images. It also addresses the issue of quantitative evaluation of segmentation results.

On cluster validity for the fuzzy c-means model

Many functionals have been proposed for validation of partitions of object data produced by the fuzzy c-means (FCM) clustering algorithm. We examine the role a subtle but important parameter-the weighting exponent m of the FCM model-plays in determining the validity of FCM partitions. The functionals considered are the partition coefficient and entropy indexes of Bezdek, the Xie-Beni (1991), and extended Xie-Beni indexes, and the Fukuyama-Sugeno index (1989). Limit analysis indicates, and numerical experiments confirm, that the Fukuyama-Sugeno index is sensitive to both high and low values of m and may be unreliable because of this. Of the indexes tested, the Xie-Beni index provided the best response over a wide range of choices for the number of clusters, (2-10), and for m from 1.01-7. Finally, our calculations suggest that the best choice for m is probably in the interval [1.5, 2.5], whose mean and midpoint, m=2, have often been the preferred choice for many users of FCM. >

Some new indexes of cluster validity

We review two clustering algorithms (hard c-means and single linkage) and three indexes of crisp cluster validity (Huberts statistics, the Davies-Bouldin index, and Dunns index). We illustrate two deficiencies of Dunns index which make it overly sensitive to noisy clusters and propose several generalizations of it that are not as brittle to outliers in the clusters. Our numerical examples show that the standard measure of interset distance (the minimum distance between points in a pair of sets) is the worst (least reliable) measure upon which to base cluster validation indexes when the clusters are expected to form volumetric clouds. Experimental results also suggest that intercluster separation plays a more important role in cluster validation than cluster diameter. Our simulations show that while Dunns original index has operational flaws, the concept it embodies provides a rich paradigm for validation of partitions that have cloud-like clusters. Five of our generalized Dunns indexes provide the best validation results for the simulations presented.

A possibilistic fuzzy c-means clustering algorithm

In 1997, we proposed the fuzzy-possibilistic c-means (FPCM) model and algorithm that generated both membership and typicality values when clustering unlabeled data. FPCM constrains the typicality values so that the sum over all data points of typicalities to a cluster is one. The row sum constraint produces unrealistic typicality values for large data sets. In this paper, we propose a new model called possibilistic-fuzzy c-means (PFCM) model. PFCM produces memberships and possibilities simultaneously, along with the usual point prototypes or cluster centers for each cluster. PFCM is a hybridization of possibilistic c-means (PCM) and fuzzy c-means (FCM) that often avoids various problems of PCM, FCM and FPCM. PFCM solves the noise sensitivity defect of FCM, overcomes the coincident clusters problem of PCM and eliminates the row sum constraints of FPCM. We derive the first-order necessary conditions for extrema of the PFCM objective function, and use them as the basis for a standard alternating optimization approach to finding local minima of the PFCM objective functional. Several numerical examples are given that compare FCM and PCM to PFCM. Our examples show that PFCM compares favorably to both of the previous models. Since PFCM prototypes are less sensitive to outliers and can avoid coincident clusters, PFCM is a strong candidate for fuzzy rule-based system identification.

A robust self-tuning scheme for PI- and PD-type fuzzy controllers

Proposes a simple but robust model independent self-tuning scheme for fuzzy logic controllers (FLCs). Here, the output scaling factor (SF) is adjusted online by fuzzy rules according to the current trend of the controlled process. The rule-base for tuning the output SF is defined on error (e) and change of error (/spl Delta/e) of the controlled variable using the most natural and unbiased membership functions (MFs). The proposed self-tuning technique is applied to both PI- and PD-type FLCs to conduct simulation analysis for a wide range of different linear and nonlinear second-order processes including a marginally stable system where even the well known Ziegler-Nichols tuned conventional PI or PID controllers fail to provide an acceptable performance due to excessively large overshoot. Performances of the proposed self-tuning FLCs are compared with those of their corresponding conventional FLCs in terms of several performance measures such as peak overshoot, settling time, rise time, integral absolute error and integral-of-time-multiplied absolute error, in addition to the responses due to step set-point change and load disturbance and, in each case, the proposed scheme shows a remarkably improved performance over its conventional counterpart.

Generalized clustering networks and Kohonen's self-organizing scheme

The relationship between the sequential hard c-means (SHCM) and learning vector quantization (LVQ) clustering algorithms is discussed. The impact and interaction of these two families of methods with Kohonens self-organizing feature mapping (SOFM), which is not a clustering method but often lends ideas to clustering algorithms, are considered. A generalization of LVQ that updates all nodes for a given input vector is proposed. The network attempts to find a minimum of a well-defined objective function. The learning rules depend on the degree of distance match to the winner node; the lesser the degree of match with the winner, the greater the impact on nonwinner nodes. Numerical results indicate that the terminal prototypes generated by this modification of LVQ are generally insensitive to initialization and independent of any choice of learning coefficient. IRIS data obtained by E. Andersons (1939) is used to illustrate the proposed method. Results are compared with the standard LVQ approach.

A mixed c-means clustering model

We justify the need for computing both membership and typicality values when clustering unlabeled data. Then we propose a new model called fuzzy-possibilistic c-means (FPCM). Unlike the fuzzy and possibilistic c-means (FCM/PCM) models, FPCM simultaneously produces both memberships and possibilities, along with the usual point prototypes or cluster centers for each cluster We show that FPCM solves the noise sensitivity defect of FCM, and also overcomes the coincident clusters problem of PCM. Then we derive first order necessary conditions for extrema of the PFCM objective function, and use them as the basis for a standard alternating optimization approach to finding local minima. Three numerical examples are given that compare FCM to FPCM. Our calculations show that FPCM compares favorably to FCM.

Measuring fuzzy uncertainty

First, this paper reviews several well known measures of fuzziness for discrete fuzzy sets. Then new multiplicative and additive classes are defined. We show that each class satisfies five well-known axioms for fuzziness measures, and demonstrate that several existing measures are relatives of these classes. The multiplicative class is based on nonnegative, monotone increasing concave functions. The additive class requires only nonnegative concave functions. Some relationships between several existing and the new measures are established, and some new properties are derived. The relative merits and drawbacks of different measures for applications are discussed. A weighted fuzzy entropy which is flexible enough to incorporate subjectiveness in the measure of fuzziness is also introduced. Finally, we comment on the construction of measures that may assess all of the uncertainties associated with a physical system. >

Entropy: a new definition and its applications

Shannons definition of entropy is critically examined and a new definition of classical entropy based on the exponential behavior of information gain is proposed along with its justification. The concept is extended to defining the global, local, and conditional entropy of a gray-level image. Based on these definitions four algorithms for object extraction are developed. One of these algorithms uses a Poisson distribution-based model of an ideal image. A concept of positional entropy giving any information regarding the location of an object in a scene is introduced. >

Genetic programming for simultaneous feature selection and classifier design

This paper presents an online feature selection algorithm using genetic programming (GP). The proposed GP methodology simultaneously selects a good subset of features and constructs a classifier using the selected features. For a c-class problem, it provides a classifier having c trees. In this context, we introduce two new crossover operations to suit the feature selection process. As a byproduct, our algorithm produces a feature ranking scheme. We tested our method on several data sets having dimensions varying from 4 to 7129. We compared the performance of our method with results available in the literature and found that the proposed method produces consistently good results. To demonstrate the robustness of the scheme, we studied its effectiveness on data sets with known (synthetically added) redundant/bad features.