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.

/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

We describe a matched subspace detection algorithm to assist in the detection of small tumors in dynamic positron emission tomography (PET) images. The algorithm is designed to differentiate tumors from background using the time activity curves (TACs) that characterize the uptake of PET tracers. TACs are modeled using linear subspaces with additive Gaussian noise. Using TACs from a primary tumor region of interest (ROI) and one or more background ROIs, each identified by a human observer, two linear subspaces are identified. Applying a matched subspace detector to these identified subspaces on a voxel-by-voxel basis throughout the dynamic image produces a test statistic at each voxel which on thresholding indicates potential locations of secondary or metastatic tumors. The detector is derived for three cases: using a single TAC with white noise of unknown variance, using a single TAC with known noise covariance, and detection using multiple TACs within a small ROI with known noise covariance. The noise covariance is estimated for the reconstructed image from the observed sinogram data. To evaluate the proposed method, a simulation-based receiver operating characteristic (ROC) study for dynamic PET tumor detection is designed. The detector uses a dynamic sequence of frame-by-frame 2-D reconstructions as input. We compare the performance of the subspace detectors with that of a Hotelling observer applied to a single frame image and of the Patlak method applied to the dynamic data. We also show examples of the application of each detection approach to clinical PET data from a breast cancer patient with metastatic disease.

Lesion Detection in Dynamic FDG-PET Using Matched Subspace Detection

BrainStorm is a collaborative project to build a software suite for EEG and MEG data visualization, mod- eling, and source imaging, with integration of MRI and fMRI information. BrainStorm is a Matlab-based toolbox distributed under the GNU public licensing and runs on any platform supporting Matlab. Forward models include multi-concentric spheres, fast EEG multisphere solutions, overlapping spheres, boundary elements with constant or linear approximations, collocation or Galer- kin weighting, and optional isolated skull approach. Finite element methods and an atlas-based generic head model for use with EEG data have recently been added. Inverse methods include least-squares, RAP-MUSIC, LCMV, variations of regularized weighted minimum norms, and statistical testing for regions of statistically significant activity. Results can be viewed as locations in MRI slices or as 3-D renderings onto cortical surfaces. The data format architecture is well-documented to allow the advanced user access to any point in the workflow. The software, documents, and publications may be found at http://neuroimage.usc.edu/brainstorm.

/pdf/brainstorm-electromagnetic-imaging-software-5a8d5wmujd.pdf

BrainStorm Electromagnetic Imaging Software

Randoms precorrected positron emission tomography (PET) data is formed as the difference of two Poisson random variables. Its exact probability mass function (PMF) is inconvenient for use in likelihood-based iterative image reconstruction as it contains an infinite summation. The shifted Poisson model is a tractable approximation to this PMF but requires that negative values are truncated, resulting in positively biased reconstructions in low count studies. Here we analyze the properties of the exact PMF and propose a simple but accurate approximation that allows negative valued data. We investigate the properties of this approximation and demonstrate its application to penalized maximum likelihood image reconstruction

Statistical Modeling and Reconstruction of Randoms Precorrected PET Data

We present a framework for registering and analyzing functional neuroimaging data constrained to the cortical surface of the brain. We assume as input a set of labeled data points that lie on a set of parameterized topologically spherical surfaces that represent the cortical surfaces of multiple subjects. To perform analysis across subjects, we first co-register the coordinates from each surface to a cortical atlas using labeled sulcal maps as constraints. The registration minimizes a thin plate spline energy function on the deforming surface using covariant derivatives to solve the associated PDEs in the intrinsic geometry of the individual surface. The resulting warps are used to bring the functional data for multiple subjects into a common surface atlas coordinate system. We then present a novel method for performing statistical analysis of points on this atlas surface. We use the Green’s function of the heat equation on the surface to model probability distributions and thus demonstrate the use of PDEs for statistical analysis in Riemannian manifolds. We describe methods for estimating the mean and variance of a set of points, such that the mean also lies in the manifold. We demonstrate the utility of this framework in the development of a maximum likelihood classifier for parcellation of somatosensory cortex in the atlas based on current dipole fits to MEG data, simulated to represent a somatotopic mapping of S1 sensory areas in multiple subjects.

A framework for registration, statistical characterization and classification of cortically constrained functional imaging data

An array of SQUID biomagentometers may be used to measure the spatio-temporal neuromagnetic field produced by the brain in response to a given sensory stimulus. A popular model for the neural activity that produces these fields is a set of current dipoles. We present here a common linear algebraic framework for three common spatio-temporal dipole models: i) moving and rotating dipoles, ii) rotating dipoles with fixed location, and iii) dipoles with fixed orientation and location. Our intent here is not to argue the merits of one model over another, but rather show how each model may be solved efficiently, and within the same framework as the others. In all cases, we assume that the location, orientation, and magnitude of the dipoles are unknown. We present the parameter estimation problem for these three models in a common framework, and show how, in each case, the problem may be decomposed into the estimation of the dipole locations using nonlinear minimization followed by linear estimation of the associated moment time series. Numerically efficient means of calculating the cost function are presented, and problems of model order selection and missing moments are also investigated. The methods described are demonstrated in a simulated application to a three dipole problem.

Richard M. Leahy

Papers

Lesion Detection in Dynamic FDG-PET Using Matched Subspace Detection

BrainStorm Electromagnetic Imaging Software

Statistical Modeling and Reconstruction of Randoms Precorrected PET Data

A framework for registration, statistical characterization and classification of cortically constrained functional imaging data

Multiple dipole modeling of spatiotemporal MEG data