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

http://pages.stat.wisc.edu/~mchung/teaching/MIA/reading/multiple.comparision.pantazis.ni.2005.pdf

A comparison of random field theory and permutation methods for the statistical analysis of MEG data.

The human cerebral cortex is topologically equivalent to a sheet and can be considered topologically spherical if it is closed at the brainstem. Low-level segmentation of magnetic resonance (MR) imagery typically produces cerebral volumes whose tessellations are not topologically spherical. The authors present a novel algorithm that analyzes and constrains the topology of a volumetric object. Graphs are formed that represent the connectivity of voxel segments in the foreground and background of the image. These graphs are analyzed and minimal corrections to the volume are made prior to tessellation. The authors apply the algorithm to a simple test object and to cerebral white matter masks generated by a low-level tissue identification sequence. The authors tessellate the resulting objects using the marching cubes algorithm and verify their topology by computing their Euler characteristics. A key benefit of the algorithm is that it localizes the change to a volume to the specific areas of its topological defects.

/pdf/automated-graph-based-analysis-and-correction-of-cortical-556slbrste.pdf

Automated graph-based analysis and correction of cortical volume topology

We describe a practical statistical methodology for the reconstruction of PET images. Our approach is based on a Bayesian formulation of the imaging problem. The data are modelled as independent Poisson random variables and the image is modelled using a Markov random field smoothing prior. We describe a sequence of calibration procedures which are performed before reconstruction: (i) calculation of accurate attenuation correction factors from re-projected Bayesian reconstructions of the transmission image; (ii) estimation of the mean of the randoms component in the data; and (iii) computation of the scatter component in the data using a Klein - Nishina-based scatter estimation method. The Bayesian estimate of the PET image is then reconstructed using a pre-conditioned conjugate gradient method. We performed a quantitation study with a multi-compartment chest phantom in a Siemens/CTI ECAT931 system. Using 40 1 min frames, we computed the ensemble mean and variance over several regions of interest from images reconstructed using the Bayesian and a standard filtered backprojection (FBP) protocol. The values for the region of interest were compared with well counter data for each compartment. These results show that the Bayesian protocol can produce substantial improvements in relative quantitation over the standard FBP protocol, particularly when short transmission scans are used. An example showing the application of the method to a clinical chest study is also given.

https://iopscience.iop.org/article/10.1088/0031-9155/41/9/015/pdf

Bayesian reconstruction of PET images: methodology and performance analysis.

We describe a method for computing a continuous time estimate of tracer density using list-mode positron emission tomography data. The rate function in each voxel is modeled as an inhomogeneous Poisson process whose rate function can be represented using a cubic B-spline basis. The rate functions are estimated by maximizing the likelihood of the arrival times of detected photon pairs over the control vertices of the spline, modified by quadratic spatial and temporal smoothness penalties and a penalty term to enforce nonnegativity. Randoms rate functions are estimated by assuming independence between the spatial and temporal randoms distributions. Similarly, scatter rate functions are estimated by assuming spatiotemporal independence and that the temporal distribution of the scatter is proportional to the temporal distribution of the trues. A quantitative evaluation was performed using simulated data and the method is also demonstrated in a human study using /sup 11/C-raclopride.

/pdf/spatiotemporal-reconstruction-of-list-mode-pet-data-2q4gli3xp3.pdf

Spatiotemporal reconstruction of list-mode PET data

A method of computing the three-dimensional (3-D) velocity field from 3-D cine computer tomographs (CTs) of a beating heart is proposed. Using continuum theory, the authors develop two constraints on the 3-D velocity field generated by a beating heart. With these constraints, the computation of the 3-D velocity field is formulated as an optimization problem and a solution to the optimization problem is developed using the Euler-Lagrange method. The solution is then discretized for computer implementation. The authors present the results for both simulated images and clinical cine CT images of a beating heart. >

Richard M. Leahy

Papers

A comparison of random field theory and permutation methods for the statistical analysis of MEG data.

Automated graph-based analysis and correction of cortical volume topology

Bayesian reconstruction of PET images: methodology and performance analysis.

Spatiotemporal reconstruction of list-mode PET data

Computation of 3-D velocity fields from 3-D cine CT images of a human heart