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

Atlases are normalized representations of anatomy that can provide a standard coordinate system for in vivo
imaging studies. For Optical Bioluminescence Tomography (OBT) in small animals, the animal's surface topography
can be reconstructed from structured light measurements, but internal anatomy is unavailable unless
additional CT or MR images are acquired. We present a novel method for estimating the internal organ structure
of a mouse by warping a labeled 3D volumetric mouse atlas with the constraint that the surfaces of the two
should match. Surface-constrained harmonic maps used for this bijective warping are computed by minimizing
the covariant harmonic energy. We demonstrate the application of this warping scheme in OBT, where scattering
and absorption coefficients of tissue are functions of the internal anatomy and hence, better estimates
of the organ structures can lead to a more accurate forward model resulting in improved source localization.
We first estimated the subject's internal geometry using the atlas-based warping scheme. Then the mouse was
tessellated and optical properties were assigned based on the estimated organ structure. Bioluminescent sources
were simulated, an optical forward model was computed using a finite-element solver, and multispectral data
were simulated. We evaluate the accuracy of the forward model computed using the warped atlas against that
assuming a homogeneous mouse model. This is done by comparing each model against a 'true' optical forward
model where the anatomy of the mouse is assumed known. We also evaluate the impact of anatomical alignment
on bioluminescence source localization.

A method for atlas-based volumetric registration with surface constraints for Optical Bioluminescence Tomography in small animal imaging

Simple threshold and gradient based segmenta- tion algorithms are unable to provide reliable soft tissue clas- sification in MR images. To obtain improved estimates, we have developed a new unsupervised hierarchical segmentation algorithm upon which our tissue classification scheme is based. The MR images are modeled as a first order Gauss-Markov ran- dom field (GMRF) with unknown parameters. Through test- ing hypotheses of homogeneity for either labeled or unlabeled data, the segmentation algorithm seeks to group the pixels in- to connected regions which are homogeneous GMRFs. This is achieved in three steps. The image is represented by a quadtree and a split-and-merge procedure is applied to find large homo- geneous regions within the image. This is followed by a segmen- tation refinement step involving connected component labeling and region growing. Here, the use of a hierarchical approach allows the algorithm to learn about the image by first process- ing the easy-to-classify pixels while delaying decisions on the hard pixels until more information has been gathered. Final- ly, a novel graph theoretic clustering algorithm is developed to reduce the number of resulting connected regions. This clus- tering method is globally optimal for a properly defined family of cost functions.

Tissue Classification In MR Images Using Hierarchical Segmentation

Despite being routinely required in medical applications, deformable surface registration is notoriously difficult due to large intersubject variability and complex geometry of most medical datasets.We present a general and flexible deformable matching framework based on generalized surface flows that efficiently tackles these issues through tailored deformation priors and multiresolution computations. The value of our approach over existing methods is demonstrated for automatic and user-guided cortical registration.

/pdf/generalized-surface-flows-for-deformable-registration-and-140nsp9io1.pdf

Generalized surface flows for deformable registration and cortical matching

Volumetric registration of brain MR images presents a challenging problem due to the wide variety of sulcal folding patterns. We present a novel volumetric registration method based on an intermediate parameter space in which the shape differences are normalized. First, we generate a 3D harmonic map of each brain volume to unit ball which is used as an intermediate space. Cortical surface features and volumetric intensity are then used to find a simultaneous surface and volume registration. We present a finite element method for the registration by using a tetrahedral volumetric mesh for registering the interior volumetric information and the corresponding triangulated mesh at the surface points. This framework aligns the convoluted sulcal folding patterns as well as the subcortical structures by allowing simultaneous flow of surface and volumes for registration. We describe the methodology and FEM implementation and then evaluate the method in terms of the overlap between segmented structures in coregistered brains.

/pdf/a-framework-for-brain-registration-via-simultaneous-surface-4zsqwkp9tq.pdf

A Framework for Brain Registration via Simultaneous Surface and Volume Flow

Several different techniques have appeared in the magnetoencephalography (MEG) literature for solving the ambiguous inverse problem. We show that if the same second order statistics are uniformly applied in each case, then the resulting solutions are the same. Additionally, for the common assumption of independent sources, then the "noise significance " measure emphasises the need to select the data and noise covariance statistics, not the specific values of source statistics. Thus such algorithms as "dynamic SPM" and "synthetic aperture magnetoencephalography " are theoretically equivalent and differ in practice by practical matters of specification or estimation of data covariances.

Richard M. Leahy

Papers

A method for atlas-based volumetric registration with surface constraints for Optical Bioluminescence Tomography in small animal imaging

Tissue Classification In MR Images Using Hierarchical Segmentation

Generalized surface flows for deformable registration and cortical matching

A Framework for Brain Registration via Simultaneous Surface and Volume Flow

Equivalence of linear approaches in bioelectromagnetic inverse solutions