We present parallel algorithms for computing cycle orders and cycle perimeters in relative neighborhood graphs. This parallel algorithm has wide-ranging applications from microscopic to macroscopic domains, e.g., in histopathological image analysis and wireless network routing. Our algorithm consists of the following steps (sub-algorithms): (1) Uniform partitioning of the graph vertices across processes, (2) Parallel Delaunay triangulation and (3) Parallel computation of the relative neighborhood graph and the cycle orders and perimeters. We evaluated our algorithm on a large dataset with 6.5 Million points and demonstrate excellent fixed-size scalability. We also demonstrate excellent isogranular scalability up to 131K processes. Our largest run was on a dataset with 13 billion points on 131K processes on ORNL's Cray XK7 Titan supercomputer.

Detection of multiple lesions in images is a medically important task and free-response receiver operating characteristic (FROC) analyses and its variants, such as alternative FROC (AFROC) analyses, are commonly used to quantify performance in such tasks. However, ideal observers that optimize FROC or AFROC performance metrics have not yet been formulated in the general case. If available, such ideal observers may turn out to be valuable for imaging system optimization and in the design of computer aided diagnosis techniques for lesion detection in medical images. In this paper, we derive ideal AFROC and FROC observers. They are ideal in that they maximize, amongst all decision strategies, the area, or any partial area, under the associated AFROC or FROC curve. Calculation of observer performance for these ideal observers is computationally quite complex. We can reduce this complexity by considering forms of these observers that use false positive reports derived from signal-absent images only. We also consider a Bayes risk analysis for the multiple-signal detection task with an appropriate definition of costs. A general decision strategy that minimizes Bayes risk is derived. With particular cost constraints, this general decision strategy reduces to the decision strategy associated with the ideal AFROC or FROC observer.

Diffusion tensor magnetic resonance imaging (DT-MRI) provides a comprehensive characterization of white matter (WM) in the brain and therefore, plays a crucial role in the investigation of diseases in which WM is suspected to be compromised such as multiple sclerosis and neuropsychiatric disorders like schizophrenia. However changes induced by pathology may be subtle and affected regions of the brain can only be revealed by a group-based analysis of patients in comparison with healthy controls. This in turn requires voxel-based statistical analysis of spatially normalized brain DT images, as in the case of conventional MR images. However this process is rendered extremely challenging in DT-MRI due to the high dimensionality of the data and its inherent non-linearity that causes linear component analysis methods to be inapplicable. We therefore propose a novel framework for the statistical analysis of DT-MRI data using manifold-based techniques such as isomap and kernel PCA that determine the underlying manifold structure of the data, embed it to a manifold and help perform high dimensional statistics on the manifold to determine regions of difference between the groups of patients and controls. The framework has been successfully applied to DT-MRI data from patients with schizophrenia, as well as to study developmental changes in small animals, both of which identify regional changes, indicating the need for manifold-based methods for the statistical analysis of DTI.

This paper addresses the problem of statistical analysis of diffusion tensor magnetic resonance images (DT-MRI). DT-MRI cannot be analyzed by commonly used linear methods, due to the inherent nonlinearity of tensors, which are restricted to lie on a nonlinear submanifold of the space in which they are defined, namely R6. We estimate this submanifold using the Isomap manifold learning technique and perform tensor calculations using geodesic distances along this manifold. Multivariate statistics used in group analyses also use geodesic distances between tensors, thereby warranting that proper estimates of means and covariances are obtained via calculations restricted to the proper subspace of R6. Experimental results on data with known ground truth show that the proposed statistical analysis method properly captures statistical relationships among tensor image data, and it identifies group differences. Comparisons with standard statistical analyses that rely on Euclidean, rather than geodesic distances, are also discussed.

For the 2-class detection problem (signal absent/present), the likelihood ratio is an ideal observer in that it minimizes Bayes risk for arbitrary costs and it maximizes the area under the receiver operating characteristic (ROC) curve [AUC]. The AUC-optimizing property makes it a valuable tool in imaging system optimization. If one considered a different task, namely, joint detection and localization of the signal, then it would be similarly valuable to have a decision strategy that optimized a relevant scalar figure of merit. We are interested in quantifying performance on decision tasks involving location uncertainty using the localization ROC (LROC) methodology. Therefore, we derive decision strategies that maximize the area under the LROC curve, A(LROC). We show that these decision strategies minimize Bayes risk under certain reasonable cost constraints. The detection-localization task is modeled as a decision problem in three increasingly realistic ways. In the first two models, we treat location as a discrete parameter having finitely many values resulting in an (L + 1) class classification problem. In our first simple model, we do not include search tolerance effects and in the second, more general, model, we do. In the third and most general model, we treat location as a continuous parameter and also include search tolerance effects. In all cases, the essential proof that the observer maximizes A(LROC) is obtained with a modified version of the Neyman-Pearson lemma. A separate form of proof is used to show that in all three cases, the decision strategy minimizes the Bayes risk under certain reasonable cost constraints.