Pattern classification theory involves an error criterion, optimal classifiers, and a theory of learning. For clustering, there has historically been little theory; in particular, there has generally (but not always) been no learning. The key point is that clustering has not been grounded on a probabilistic theory. Recently, a clustering theory has been developed in the context of random sets. This paper discusses learning within that context, in particular, k- nearest-neighbor learning of clustering algorithms.
Granulometric features have been widely used for classification, segmentation and recently in estimation of parameters in shape models. In this paper we study the inference of clustering based on granulometric features for a collection of structuring probes in the context of random models. We use random Boolean models to represent grains of different shapes and structure. It is known that granulometric features are excellent descriptors of shape and structure of grains. Inference based on clustering these features helps to analyze the consistency of these features and clustering algorithms. This greatly aids in classifier design and feature selection. Features and the order of their addition play a role in reducing the inference errors. We study four different types of feature addition methods and the effect of replication in reducing the inference errors.
This paper presents a toolbox for analyzing inferences drawn from clustering. Often the implication is that points in different clusters come from different underlying classes, whereas those in the same cluster come from the same class. These classes represent different random vectors. Each random vector is modeled as its mean plus independent noise, sample points are generated, the points are clustered, and the clustering error is the number of points clustered incorrectly according to the generating random vectors. Clustering algorithms are evaluated based on class variance and performance improvement with respect to increasing numbers of experimental replications. The study is presented on a website, which includes error tables and graphs, confusion matrices, principle-component plots, and validation measures. There, the toolbox is applied to gene- expression clustering based on cDNA microarrays using real data.
This paper studies pyramidal multiresolution design of aperture and W-operators for grayscale images. The multiresolution approach has been used previously to design binary filters with good results, which motivated us to extend the theory for grayscale. The initial results, theoretical and practical are also motivating.
The performance of a designed digital filter is measured by the sum of the errors of the optimal filter and the estimation error. Viewing an image at a high resolution results in optimal filters having smaller errors than at lower resolutions; however, higher resolutions bring increased estimation error. Hence, choosing an appropriate resolution for filter design is important. This paper discusses estimation of optimal filters in a pyramidal multiresolution framework. To take advantage of data at all resolutions, one can use a hybrid multiresolution design. In hybrid design, a sequence of filters is designed using data at increasing resolutions. With hybrid multiresolution design, the value of the designed filter at a given observation is based on the highest resolution at which conditioning by the observation is considered significant.
A basic paradigm in Mathematical Morphology is the construction of set operators by concatenations of dilations and erosions via the operations of composition, union, intersection and complementation. Since its introduction, in the sixties by Matheron and Serra, this paradigm has been applied on Image Analysis for designing set operators, that were called morphological operators. Classically morphological operators are constructed based on the experience and intuition of human beings. Recently, an approach for the automatic design of morphological operators, based on statistical optimization from the observation of collections of image pairs, was proposed. The two approaches have drawbacks: usually, the first approach is slow and depends on an expert in Mathematical Morphology, while the second requires large amounts of observed data. This paper proposes a symbiosis between the human and the statistical design approaches. The idea is that the design procedure be composed of simplified forms of both. Thus, avoiding difficulties that arise when applying each one independently
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