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.

/pdf/data-clustering-a-review-4c452f5cab.pdf

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.

/pdf/normalized-cuts-and-image-segmentation-2ru3ikxuj5.pdf

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.

/pdf/normalized-cuts-and-image-segmentation-1q390dk21h.pdf

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

The Fisher information matrix (FIM) plays a key role in the analysis and applications of statistical image reconstruction methods based on Poisson data models. The elements of the FIM are a function of the reciprocal of the mean values of sinogram elements. Conventional plug-in FIM estimation methods do not work well at low counts, where the FIM estimate is highly sensitive to the reciprocal mean estimates at individual detector pairs. A generalized error look-up table (GELT) method is developed to estimate the reciprocal of the mean of the sinogram data, which achieves a bias of less than 2% for mean sinogram values greater than 0.7. This approach is also extended to randoms precorrected data. Based on these techniques, an accurate FIM estimate is obtained for both Poisson and randoms precorrected data. As an application, the new GELT method is used to improve resolution uniformity and achieve near-uniform reconstructed image resolution in both high and low count situations.

/pdf/accurate-estimation-of-the-fisher-information-matrix-for-the-49t9f397nj.pdf

Accurate estimation of the Fisher information matrix for the PET image reconstruction problem

http://sipi.usc.edu/~ajoshi/HBM2009_abstract_Dimitrios.pdf

Evaluating the accuracy of cortical registration using landmark-based and automatic methods

Sensor Weighted Multiple Sphere Head Model for MEG

Spatiotemporal Analysis of Visually Evoked Potentials in the Cortex of a Blind Subject With a Chronic Intraocular Retinal Prosthesis

Despite the popularity and broad range of spectral clustering algorithms, there is little work addressing the statistical significance of clustering results. Spectral clustering uses the eigenvalues of matrices, such as the Laplacian graph or the adjacency matrix minus a null model, to partition a network. Even though the distribution of the largest eigenvalue for these matrices is not known, random matrix theory provides analytical formulas for a family of matrices called Gaussian random ensembles. We demonstrate that the Tracy-Widom mapping of the largest eigenvalue of Gaussian random ensembles can be modified to predict the distribution of the largest eigenvalue of matrices used for modularity-based spectral clustering. Using this finding we derive formulas that control the type I error rate on modularity-based partitions.

Richard M. Leahy

Papers

Accurate estimation of the Fisher information matrix for the PET image reconstruction problem

Evaluating the accuracy of cortical registration using landmark-based and automatic methods

Sensor Weighted Multiple Sphere Head Model for MEG

Spatiotemporal Analysis of Visually Evoked Potentials in the Cortex of a Blind Subject With a Chronic Intraocular Retinal Prosthesis

Of the largest eigenvalue for modularity-based partitioning