There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Clustering is the unsupervised classification of patterns (observations, data items, or feature vectors) into groups (clusters). The clustering problem has been addressed in many contexts and by researchers in many disciplines; this reflects its broad appeal and usefulness as one of the steps in exploratory data analysis. However, clustering is a difficult problem combinatorially, and differences in assumptions and contexts in different communities has made the transfer of useful generic concepts and methodologies slow to occur. This paper presents an overview of pattern clustering methods from a statistical pattern recognition perspective, with a goal of providing useful advice and references to fundamental concepts accessible to the broad community of clustering practitioners. We present a taxonomy of clustering techniques, and identify cross-cutting themes and recent advances. We also describe some important applications of clustering algorithms such as image segmentation, object recognition, and information retrieval.

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Data clustering: a review

We propose a novel approach for solving the perceptual grouping problem in vision. Rather than focusing on local features and their consistencies in the image data, our approach aims at extracting the global impression of an image. We treat image segmentation as a graph partitioning problem and propose a novel global criterion, the normalized cut, for segmenting the graph. The normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups. We show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion. We applied this approach to segmenting static images, as well as motion sequences, and found the results to be very encouraging.

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Normalized cuts and image segmentation

We propose a novel approach for solving the perceptual grouping problem in vision. Rather than focusing on local features and their consistencies in the image data, our approach aims at extracting the global impression of an image. We treat image segmentation as a graph partitioning problem and propose a novel global criterion, the normalized cut, for segmenting the graph. The normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups. We show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion. We have applied this approach to segmenting static images and found results very encouraging.

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Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

Network models can be used to represent interacting subsystems in the brain or other biological systems These subsystems can be identified by partitioning a graph representation of the network into highly connected modules In this paper we describe a modularity-based partitioning method based on a Gaussian model of a directed graph Using the degrees of each node, we first compute the conditional expected value of the connection weights The resulting adjacency matrix forms a null model for the network which does not favor any particular partition By comparing this null model to the true adjacency graph, we can perform a statistically optimal partitioning that maximizes modularity Similarly to other modularity-based partitioning methods, the solution is found using spectral matrix decomposition The process can be iterated to find multiple subgraphs We demonstrate this approach through simulations and application to standard biological and other network data

Statistically optimal modular partitioning of directed graphs

In neuromagnetic source reconstruction, a functional map of neural activity is constructed from noninvasive magnetoencephalographic (MEG) measurements. The overall reconstruction problem is under-determined, so some form of source modeling must be applied. The authors review the two main classes of reconstruction techniques-parametric current dipole models and nonparametric distributed source reconstructions. Current dipole reconstructions use a physically plausible source model, but are limited to cases in which the neural currents are expected to be highly sparse and localized. Distributed source reconstructions can be applied to a wider variety of cases, but must incorporate an implicit source model in order to arrive at a single reconstruction. The authors examine distributed source reconstruction in a Bayesian framework to highlight the implicit nonphysical Gaussian assumptions of minimum norm based reconstruction algorithms. They conclude with a brief discussion of alternative non-Gaussian approaches.

Neuromagnetic source reconstruction

Sensory stimuli, such as auditory, visual, or somatosensory, evoke neural responses in very localized regions of the brain. A SQUID biomagnetometer can measure the very weak fields that are generated outside of the head by this response. A simple source and head model of current dipoles inside a conducting sphere is typically used to interpret these magnetic field measurements or magnetoencephalograms (MEGs). Locating dipole sources using data recorded from an array of biomagnetic sensors is distinguished from conventional array source localization techniques by the quasi-static transient nature of the data. The basic MEG model is reviewed, and then a localization example is given to show the need for partitioning the data to improve estimator performance. Time-eigenspectrum analysis is introduced as a means of partitioning and interpreting spatio-temporal biomagnetic data. Examples using both simulated and somatosensory data are presented. >

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Biomagnetic localization from transient quasi-static events

Manually labeled landmark sets are often required as inputs for landmark-based image registration. Identifying an optimal subset of landmarks from a training dataset may be useful in reducing the labor intensive task of manual labeling. In this paper, we present a new problem and a method to solve it: given a set of N landmarks, find the k(<; N) best landmarks such that aligning these k landmarks that produce the best overall alignment of all N landmarks. The resulting procedure allows us to select a reduced number of landmarks to be labeled as a part of the registration procedure. We apply this methodology to the problem of registering cerebral cortical surfaces extracted from MRI data. We use manually traced sulcal curves as landmarks in performing inter-subject registration of these surfaces. To minimize the error metric, we analyze the correlation structure of the sulcal errors in the landmark points by modeling them as a multivariate Gaussian process. Selection of the optimal subset of sulcal curves is performed by computing the error variance for the subset of unconstrained landmarks conditioned on the constrained set. We show that the registration error predicted by our method closely matches the actual registration error. The method determines optimal curve subsets of any given size with minimal registration error.

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Optimization of landmark selection for cortical surface registration

Dynamic images of functional activity in the brain offer the potential to measure connectivity between regions of interest. We want to measure causal activity between regions of interest (ROIs) with signals recorded from multiple channels or voxels in each ROI. Previous methods, such as Granger causality, look for causality between individual time series; hence, they suffer from local interactions or interferers obscuring signals of interest between two ROIs. We propose a metric that reduces the effect of interference by taking weighted sums of sensors in each ROI, as is done with canonical correlation. Hence, we measure region-to-region, rather than channel-to-channel or point-to-point, Granger causality. We show in simulation that our “canonical Granger causality” accurately mimics the underlying structure with few samples, unlike current methods of multivariate Granger causality. We then use anatomically relevant regions of interest in a visuomotor task in a multichannel intracortical EEG study to infer the direction of transmission in visual processing.

Richard M. Leahy

Papers

Statistically optimal modular partitioning of directed graphs

Neuromagnetic source reconstruction

Biomagnetic localization from transient quasi-static events

Optimization of landmark selection for cortical surface registration

Canonical Granger causality applied to functional brain data