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

We describe lossless methods for compression of 3D PET data. Static list-mode data is re-sorted into sinograms and compressed using a combination of run-length and entropic coding. Dynamic list-mode data is compressed using a sinogram/timogram format in which the arrival times of each photon pair are stored in spatial order, indexed by the sinogram. Compression of the timogram is performed using entropic coding of the differential arrival times in each sinograin bin. We describe these compression methods and present results for static and dynamic data. In the dynamic case, we compare compression of multiple frame sinograin data with sinogram/timogram formats. Both formats produce substantial reductions in data size compared to the raw data, with higher compression factors achieved using the sinogram/timogram format when high temporal resolution is required.

/pdf/lossless-compression-of-list-mode-3d-pet-data-56pgmveule.pdf

Lossless compression of list-mode 3D PET data

This paper describes a new family of linear transforms for data restricted to the surface of a 2-sphere in three-dimensional Fourier space. These transforms generalize the existing Funk-Radon Transform, which has previously been used with great success to extract microstructural tissue orientation information from high angular resolution magnetic resonance diffusion imaging data. Several properties of the new transforms are described, and computationally efficient implementations are derived using spherical harmonic basis functions. A special case from this family, called the FunkRadon and Cosine Transform, is introduced and evaluated. The method is illustrated with simulated and real diffusion weighted MRI data.

New linear transforms for data on a Fourier 2-sphere with application to diffusion MRI

PURPOSE In many MRI scenarios, magnetization is often excited from spatial regions that are not of immediate interest. Excitation of uninteresting magnetization can complicate the design of efficient imaging methods, leading to either artifacts or acquisitions that are longer than necessary. While there are many hardware- and sequence-based approaches for suppressing unwanted magnetization, this paper approaches this longstanding problem from a different and complementary angle, using beamforming to suppress signals from unwanted regions without modifying the acquisition hardware or pulse sequence. Unlike existing beamforming approaches, we use a spatially invariant sensor-domain approach that can be applied directly to raw data to facilitate image reconstruction. THEORY AND METHODS We use beamforming to linearly mix a set of original coils into a set of region-optimized virtual (ROVir) coils. ROVir coils optimize a signal-to-interference ratio metric, are easily calculated using simple generalized eigenvalue decomposition methods, and provide coil compression. RESULTS ROVir coils were compared against existing coil-compression methods, and were demonstrated to have substantially better signal suppression capabilities. In addition, examples were provided in a variety of different application contexts (including brain MRI, vocal tract MRI, and cardiac MRI; accelerated Cartesian and non-Cartesian imaging; and outer volume suppression) that demonstrate the strong potential of this kind of approach. CONCLUSION The beamforming-based ROVir framework is simple to implement, has promising capabilities to suppress unwanted MRI signal, and is compatible with and complementary to existing signal suppression methods. We believe that this general approach could prove useful across a wide range of different MRI applications.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/mrm.28706

Region-optimized virtual (ROVir) coils: Localization and/or suppression of spatial regions using sensor-domain beamforming.

The amount of manually labeled data is limited in medical applications, so semi-supervised learning and automatic labeling strategies can be an asset for training deep neural networks. However, the quality of the automatically generated labels can be uneven and inferior to manual labels. In this paper, we suggest a semi-supervised training strategy for leveraging both manually labeled data and extra unlabeled data. In contrast to the existing approaches, we apply robust loss for the automated labeled data to automatically compensate for the uneven data quality using a teacher-student framework. First, we generate pseudo-labels for unlabeled data using a teacher model pre-trained on labeled data. These pseudo-labels are noisy, and using them along with labeled data for training a deep neural network can severely degrade learned feature representations and the generalization of the network. Here we mitigate the effect of these pseudo-labels by using robust loss functions. Specifically, we use three robust loss functions, namely beta cross-entropy, symmetric cross-entropy, and generalized cross-entropy. We show that our proposed strategy improves the model performance by compensating for the uneven quality of labels in image classification as well as segmentation applications.

Semi-supervised Learning using Robust Loss

Quantifying PET tracer uptake in a region of interest (ROI) is an important task in a variety of applications including brain imaging, myocardial imaging, and tumor activity assessment. Many post-reconstruction correction methods that compensate for spatial resolution or partial volume effects have been proposed for ROI quantification. Here our goal is to optimize the image reconstruction methods themselves for this task through the use of spatially variant regularization. We investigate and analyze a selective regularization strategy where reduced smoothing is imposed across the boundary of a pre-specified ROI which can be drawn, for example, from a coregistered CT image. Preliminary simulation results show that this strategy leads to better bias/variance trade-offs than spatially uniform regularization. However, the penalty function for selective smoothing is space-variant and therefore it is not straightforward to predict the bias and variance of ROI uptake estimators in a computationally efficient way without expensive Monte Carlo simulation. Here we develop a computationally efficient method to predict bias and variance through use of the Sherman-Morrison-Woodbury matrix identity and local Fourier approximations. Simulation results show that the prediction is reasonably accurate.

Richard M. Leahy

Papers

Lossless compression of list-mode 3D PET data

New linear transforms for data on a Fourier 2-sphere with application to diffusion MRI

Region-optimized virtual (ROVir) coils: Localization and/or suppression of spatial regions using sensor-domain beamforming.

Semi-supervised Learning using Robust Loss

Analysis of Region of Interest Quantification for PET Image Reconstruction with Selective Regularization