Exploiting Spatial Information to Enhance DTI Segmentations via Spatial Fuzzy c-Means with Covariance Matrix Data and Non-Euclidean Metrics

A diffusion tensor models the covariance of the Brownian motion of water at a voxel and is required to be symmetric and positive semi-definite. Therefore, image processing approaches, designed for linear entities, are not effective for diffusion tensor data manipulation, and the existence of artefacts in diffusion tensor imaging acquisition makes diffusion tensor data segmentation even more challenging. In this study, we develop a spatial fuzzy c-means clustering method for diffusion tensor data that effectively segments diffusion tensor images by accounting for the noise, partial voluming, magnetic field inhomogeneity, and other imaging artefacts. To retain the symmetry and positive semi-definiteness of diffusion tensors, the log and root Euclidean metrics are used to estimate the mean diffusion tensor for each cluster. The method exploits spatial contextual information and provides uncertainty information in segmentation decisions by calculating the membership values for assigning a diffusion tensor at one voxel to different clusters. A regularisation model that allows the user to integrate their prior knowledge into the segmentation scheme or to highlight and segment local structures is also proposed. Experiments on simulated images and real brain datasets from healthy and Spinocerebellar ataxia 2 subjects showed that the new method was more effective than conventional segmentation methods

Statistical analysis of diffusion tensor imaging

This thesis considers the statistical analysis of diffusion tensor imaging (DTI). DTI is an advanced magnetic resonance imaging (MRI) method that provides a unique insight into biological microstructure \textit{in vivo} by directionally describing the water molecular diffusion. We firstly develop a Bayesian multi-tensor model with reparameterisation for capturing water diffusion at voxels with one or more distinct fibre orientations. Our model substantially alleviates the non-identifiability issue present in the standard multi-tensor model. A Markov chain Monte Carlo (MCMC) algorithm is then developed to study the uncertainty of the model parameters based on the posterior distribution. We apply the Bayesian method to Monte Carlo (MC) simulated datasets as well as a healthy human brain dataset. A region containing crossing fibre bundles is investigated using our multi-tensor model with automatic model selection. A diffusion tensor, a covariance matrix related to the molecular displacement at a particular voxel in the brain, is in the non-Euclidean space of 3x3 positive semidefinite symmetric matrices. We define the sample mean of tensor data to be the Fréchet mean. We carry out the non-Euclidean statistical analysis of diffusion tensor data. The primary focus is on the use of Procrustes size-and-shape space. Comparisons are made with other non-Euclidean techniques, including the log-Euclidean, Riemannian, Cholesky, root Euclidean and power Euclidean methods. The weighted generalised Procrustes analysis has been developed to efficiently interpolate and smooth an arbitrary number of tensors with the flexibility of controlling individual contributions. A new anisotropy measure, Procrustes Anisotropy is defined and compared with other widely used anisotropy measures. All methods are illustrated through synthetic examples as well as white matter tractography of a healthy human brain. Finally, we use Giné’s statistic to design uniformly distributed diffusion gradient direction schemes with different numbers of directions. MC simulation studies are carried out to compare effects of Giné’s and widely used Jones' schemes on tensor estimation. We conclude by discussing potential areas for further research

Competitive analysis of interrelated price online inventory problems with demands

This paper investigates the interrelated price online inventory problems in which decisions as to when and how much to replenish must be made in an online fashion to meet some demand even without concrete knowledge of future prices. The objective of the decision maker is to minimize the total cost with the demands met. Two different types of demand are considered carefully, which are linearly related demand to price and exponentially related demand to price. In this paper, the prices are online with only the price range variation known in advance, which are interrelated with the preceding price. Two models of price correlations are investigated. Namely an exponential model and a logarithmic model. The corresponding algorithms of the problems are developed and the competitive ratio of the algorithms are also derived by the solutions of linear programming

Cluster Analysis of Diffusion Tensor Fields with Application to the Segmentation of the Corpus Callosum

AbstractAccurate segmentation of the Corpus Callosum (CC) is an important aspect of clinical medicine and is used in the diagnosis of various brain disorders. In this paper, we propose an automated method for two and three dimensional segmentation of the CC using diffusion tensor imaging. It has been demonstrated that Hartigan's K-means is more efficient than the traditional Lloyd algorithm for clustering. We adapt Hartigan's K-means to be applicable for use with the metrics that have a f -mean (e.g. Cholesky, root Euclidean and log Euclidean). Then we applied the adapted Hartigan's K-means, using Euclidean, Cholesky, root Euclidean and log Euclidean metrics along with Procrustes and Riemannian metrics (which need numerical solutions for mean computation), to diffusion tensor images of the brain to provide a segmentation of the CC. The log Euclidean and Riemannian metrics provide more accurate segmentations of the CC than the other metrics as they present the least variation of the shape and size of the tensors in the CC for 2D segmentation. They also yield a full shape of the splenium for the 3D segmentation

Competitive analysis of interrelated price online inventory problems with demands

This paper investigates the interrelated price online inventory problems in which decisions as to when and how much to replenish must be made in an online fashion to meet some demand even without concrete knowledge of future prices. The objective of the decision maker is to minimize the total cost with the demands met. Two different types of demand are considered carefully, which are linearly related demand to price and exponentially related demand to price. In this paper, the prices are online with only the price range variation known in advance, which are interrelated with the preceding price. Two models of price correla- tions are investigated. Namely an exponential model and a logarithmic model. The corresponding algorithms of the problems are developed and the competitive ratio of the algorithms are also derived by the solutions of linear programming

Non-Euclidean statistics for covariance matrices with applications to diffusion tensor imaging

The statistical analysis of covariance matrix data is considered and, in particular, methodology is discussed which takes into account the nonEuclidean nature of the space of positive semi-definite symmetric matrices. The main motivation for the work is the analysis of diffusion tensors in medical image analysis. The primary focus is on estimation of a mean covariance matrix and, in particular, on the use of Procrustes size-and-shape space. Comparisons are made with other estimation techniques, including using the matrix logarithm, matrix square root and Cholesky decomposition. Applications to diffusion tensor imaging are considered and, in particular, a new measure of fractional anisotropy called Procrustes Anisotropy is discussed

Effect of football boot upper padding on shooting accuracy and velocity performance

Football boots are marketed with a specific performance feature focus, for example, power boots are marketed for optimal shooting performance. However, little evidence exists on the impact of boot design on shooting performance. This study assessed the effect of upper padding on shooting velocity and accuracy using a test–retest reliable test setup. Nine university level football players performed a protocol of shooting to: (1) maximise velocity; and (2) maximise accuracy in football boots with and without upper padding (Poron Memory foam). The protocol was completed twice; the non-padded boot results were used for test–retest validation, while the non-padded versus padding results were used to investigate the effect of padding. Velocity was assessed through actual ball velocity, percentage of maximum velocity and perceived velocity. Accuracy was assessed through radial offset, vertical offset, horizontal offset, success (goal/no goal), zonal offset and perceived accuracy. No significant differences between boots were observed in the velocity measures for either velocity or accuracy focused shots. Significant differences between boots were observed in vertical offset for both accuracy (without padding mean ± standard deviation − 0.02 ± 1.05 m, with padding 0.28 ± 0.87 m, P = 0.029) and velocity (without padding 0.04 ± 1.33 m, with padding 0.38 ± 0.86 m, P = 0.042) focused shots resulting in more missed shots above the goal for the padded boot (without padding 41–43% missed, with padding 56–72% missed). These findings suggest the addition of upper padding has a negative impact on shooting accuracy while not impacting shooting velocity.</div

Procrustes analysis for diffusion tensor image processing

There is an increasing need to develop processing tools for diffusion tensor image data with the consideration of the non-Euclidean nature of the tensor space. In this paper Procrustes analysis, a non-Euclidean shape analysis tool under similarity transformations (rotation, scaling and translation), is proposed to redefine sample statistics of diffusion tensors. A new anisotropy measure Procrustes Anisotropy (PA) is defined with the full ordinary Procrustes analysis. Comparisons are made with other anisotropy measures including Fractional Anisotropy and Geodesic Anisotropy. The partial generalized Procrustes analysis is extended to a weighted generalized Procrustes framework for averaging sample tensors with different fractions of contributions to the mean tensor. Applications of Procrustes methods to diffusion tensor interpolation and smoothing are compared with Euclidean, Log-Euclidean and Riemannian methods

Regularisation, interpolation and visualisation of diffusion tensor images using non-Euclidean statistics.

Practical statistical analysis of diffusion tensor images is considered, and we focus primarily on methods that use metrics based on Euclidean distances between powers of diffusion tensors. First we describe a family of anisotropy measures based on a scale invariant power-Euclidean metric, which are useful for visualisation. Some properties of the measures are derived and practical considerations are discussed, with some examples. Second we discuss weighted Procrustes methods for diffusion tensor interpolation and smoothing, and we compare methods based on different metrics on a set of examples as well as analytically. We establish a key relationship between the principal-square-root-Euclidean metric and the size-and-shape Procrustes metric on the space of symmetric positive semi-definite tensors. We explain, both analytically and by experiments, why the size-and-shape Procrustes metric may be preferred in practical tasks of interpolation, extrapolation, and smoothing, especially when observed tensors are degenerate or when a moderate degree of tensor swelling is desirable. Third we introduce regularisation methodology, which is demonstrated to be useful for highlighting features of prior interest and potentially for segmentation. Finally, we compare several metrics in a dataset of human brain diffusion-weighted MRI, and point out similarities between several of the non-Euclidean metrics but important differences with the commonly used Euclidean metric