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

The Radon transform converts an image into a sinogram, and is often used as a model of data acquisition for many tomographic imaging modalities. Although it is well-known that sinograms possess some redundancy, we observe in this work that they can have substantial additional redundancies that can be learned directly from incomplete data. In particular, we demonstrate that sinograms approximately satisfy multiple data-dependent shift-invariant local autoregression relationships. This autoregressive structure implies that samples from the sinogram can be accurately interpolated as a shift-invariant linear combination of neighboring sinogram samples, and that a Toeplitz or Hankel matrix formed from sinogram data should be approximately low-rank. This multi-fold redundancy can be used to impute missing sinogram values or for noise reduction, as we demonstrate with real X-ray CT data.

Autoregression and Structured Low-Rank Modeling of Sinograms

The authors describe inverse methods for using the magnetoencephalogram (MEG) to image neural current sources associated with functional activation in the cerebral cortex. A Bayesian formulation is presented that is based on a Gibbs prior which reflects the sparse, focal nature of neural activation. The model includes a dynamic component so that the authors can utilize the full spatio-temporal data record to reconstruct a sequence of images reflecting changes in the current source amplitudes during activation. The model consists of the product of a binary field, representing the areas of activation in the cerebral cortex, and a time series at each site which represents the dynamic changes in the source amplitudes at the active sites. The authors' estimation methods are based on the optimization of three different functions of the posterior density. Each of these methods requires the estimation of a binary field which the authors compute using a mean field annealing method. They demonstrate and compare their methods in application to computer generated and experimental phantom data.

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Dynamic MEG imaging of focal neuronal sources

We describe the use of the nonparametric bootstrap to investigate the accuracy of current dipole localization from magnetoencephalography (MEG) studies of event related neural activity. The bootstrap is well suited to analysis of event-related MEG data since the experiments are often repeated 100 or more times and averaged to achieve acceptable SNRs. The set of repetitions or "epochs" can be viewed as a set of i.i.d. realizations of the brain's response to the experiment. Sampling from these epochs and averaging can generate bootstrap resamples. In this study we applied the bootstrap resampling technique to MEG data from a somatotopic experiment. Four fingers of the right and left hand of a healthy subject were electrically stimulated, and about 400 trials per stimulation were recorded and averaged in order to measure the somatotopic mapping of the fingers in the SI area of the brain. Based on the single trial recordings for each finger, we performed 5000 bootstrap resamples. We reconstructed dipoles from these resampled averages, using the RAP-MUSIC source localization algorithm. To find the correspondences between multiple sources in each resample dipoles with similar time-series and forward fields were assumed to represent the same source. These dipoles were then clustered using a GMM (Gaussian mixture model) clustering algorithm, using their combined normalized time-series and topography as feature vectors. The mean and standard deviation of the dipole position and the dipole time-series in each cluster were computed to provide estimates of the accuracy of the reconstructed source locations and time-series.

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Investigations of dipole localization accuracy in MEG using the bootstrap

Objective Assessment of Image Quality.- On the Difficulty of Detecting Tumors in Mammograms.- Objective Comparison of Quantitative Imaging Modalities Without the Use of a Gold Standard.- Theory for Estimating Human-Observer Templates in Two-Alternative Forced-Choice Experiments.- Shape Modeling.- The Active Elastic Model.- A Minimum Description Length Approach to Statistical Shape Modelling.- Multi-scale 3-D Deformable Model Segmentation Based on Medial Description.- Automatic 3D ASM Construction via Atlas-Based Landmarking and Volumetric Elastic Registration.- Molecular and Diffusion Tensor Imaging.- A Regularization Scheme for Diffusion Tensor Magnetic Resonance Images.- Distributed Anatomical Brain Connectivity Derived from Diffusion Tensor Imaging.- Study of Connectivity in the Brain Using the Full Diffusion Tensor from MRI.- Poster Session I: Registration and Structural Analysis.- Incorporating Image Processing in a Clinical Decision support system.- Automated Estimation of Brain Volume in Multiple Sclerosis with BICCR.- Automatic Image Registration for MR and Ultrasound Cardiac Images.- Estimating Sparse Deformation Fields Using Multiscale Bayesian Priors and 3-D Ultrasound.- Automatic Registration of Mammograms Based on Linear Structures.- Tracking Brain Deformations in Time-Sequences of 3D US Images.- Robust Multimodal Image Registration Using Local Frequency Representations.- Steps Toward a Stereo-Camera-Guided Biomechanical Model for Brain Shift Compensation.- Poster Session I: Functional Image analysis.- Spatiotemporal Analysis of Functional Images Using the Fixed Effect Model.- Spatio-Temporal Covariance Model for Medical Images Sequences: Application to Functional MRI Data.- Microvascular Dynamics in the Nailfolds of Scleroderma Patients Studied Using Na-Fluorescein dye.- Time Curve Analysis Techniques for Dynamic Contrast MRI Studies.- Detecting Functionally Coherent Networks in fMRI Data of the Human Brain Using Replicator Dynamics.- Smoothness Prior Information in Principal Component Analysis of Dynamic Image Data.- Estimation of Baseline Drifts in fMRI.- Analyzing the Neocortical Fine-Structure.- fMRI/EEG/MEG.- Motion Correction Algorithms of the Brain Mapping Community Create Spurious Functional Activations.- Estimability of Spatio-Temporal Activation in fMRI.- A New Approach to the MEG/EEG Inverse Problem for the Recovery of Cortical Phase-Synchrony.- Neural Field Dynamics on the Folded Three-Dimensional Cortical Sheet and Its Forward EEG and MEG.- Depormable Registration.- A Unified Feature Registration Method for Brain Mapping.- Cooperation between Local and Global Approaches to Register Brain Images.- Landmark and Intensity-Based, Consistent Thin-Plate Spline Image Registration.- Validation of Non-rigid Registration Using Finite Element Methods.- Poster Session II:Shape Analysis.- A Linear Time Algorithm for Computing the Euclidean Distance Transform in Arbitrary Dimensions.- An Elliptic Operator for Constructing Conformal Metrics in Geometric Deformable Models.- Using a Linear Diagnostic Function and Non-rigid Registration to Search for Morphological Differences Between Populations: An Example Involving the Male and Female Corpus Callosum.- Shape Constrained Deformable Models for 3D Medical Image Segmentation.- Stenosis Detection Using a New Shape Space for Second Order 3D-Variations.- Graph-Based Topology Correction for Brain Cortex Segmentation.- Intuitive, Localized Analysis of Shape Variability.- A Sequential 3D Thinning Algorithm and Its Medical Applications.- Poster Session II: Functional Image Analysis.- An Adaptive Level Set Method for Medical Image Segmentation.- Partial Volume Segmentation of Cerebral MRI Scans with Mixture Model Clustering.- Nonlinear Edge Preserving Smoothing and Segmentation of 4-D Medical Images via Scale-Space Fingerprint Analysis.- Spatio-Temporal Segmentation of Active Multiple Sclerosis Lesions in Serial MRI Data.- Time-Continuous Segmentation of Cardiac Image Sequences Using Active Appearance Motion Models.- Feature Enhancement in Low Quality Images with Application to Echocardiography.- 3D Vascular Segmentation Using MRA Statistics and Velocity Field Information in PC-MRA.- Markov Random Field Models for Segmentation of PET Images.- Analysis of Brain Structure.- Statistical Study on Cortical Sulci of Human Brains.- Detecting Sisease-Specific Patterns of Brain Structure Using Cortical Pattern Matching and a Population-Based Probabilistic Brain Atlas.- Medial Models Incorporating Object Variability for 3D Shape Analysis.- Deformation Analysis for Shape Based Classification.

Information processing in medical imaging : 17th International Conference, IPMI 2001, Davis, CA, USA, June 18-22, 2001 : proceedings

Due to the high noise levels in frame-by-frame reconstructed dynamic PET images, voxel-wise estimation of kinetic parameters remains a challenge Most traditional denoising schemes compute the time activity curve for a given voxel by averaging voxels in its immediate local neighborhood We recognize the fact that similarities between time activity curves are not necessarily local due to the distributed nature of the tracer uptake properties of similar tissue types We, therefore, present a denoising scheme based on the nonlocal mean (NLM) that allows us to compute voxel-wise estimates of kinetic parameters We validate our method using a simulation study and compare the results of our approach with that of the traditional Gaussian filtering technique The results demonstrate that nonlocal averaging generates lower bias as well as lower variance in the denoised time series for different regions of interest We show that this method can thereby be used to generate better estimates of kinetic parameters than those obtained using a Gaussian filter To enable application to real data sets, we have accelerated the developed NLM denoising framework using a prior clustering step The compound approach was applied to preclinical [18F] FDG PET data from a mouse study Significant qualitative improvement was observed in Patlak parametric images computed after NLM denoising

Richard M. Leahy

Papers

Autoregression and Structured Low-Rank Modeling of Sinograms

Dynamic MEG imaging of focal neuronal sources

Investigations of dipole localization accuracy in MEG using the bootstrap

Information processing in medical imaging : 17th International Conference, IPMI 2001, Davis, CA, USA, June 18-22, 2001 : proceedings

A nonlocal averaging technique for kinetic parameter estimation from dynamic PET data