Distributions on the circle group

In this paper, we extend the definition of a random angle and the definition of a probability distribution of a random angle. We expand P. Lévy’s researches related to wrapping the probability distributions defined on R. We determine a relation between quasi-lattice probability distributions on R and lattice probability distributions on the unit circle S. We use the Bergström identity for comparison of a convolution of probability distributions of random angles. We also prove an inverse formula for lattice probability distributions on S.

The Influence of Radial Undersampling Schemes on Compressed Sensing in Cardiac DTI

Diffusion tensor imaging (DTI) is known to suffer from long acquisition time, which greatly limits its practical and clinical use. Undersampling of k-space data provides an effective way to reduce the amount of data to acquire while maintaining image quality. Radial undersampling is one of the most popular non-Cartesian k-space sampling schemes, since it has relatively lower sensitivity to motion than Cartesian trajectories, and artifacts from linear reconstruction are more noise-like. Therefore, radial imaging is a promising strategy of undersampling to accelerate acquisitions. The purpose of this study is to investigate various radial sampling schemes as well as reconstructions using compressed sensing (CS). In particular, we propose two randomly perturbed radial undersampling schemes: golden-angle and random angle. The proposed methods are compared with existing radial undersampling methods, including uniformity-angle, randomly perturbed uniformity-angle, golden-angle, and random angle. The results on both simulated and real human cardiac diffusion weighted (DW) images show that, for the same amount of k-space data, randomly sampling around a random radial line results in better reconstruction quality for DTI indices, such as fractional anisotropy (FA), mean diffusivities (MD), and that the randomly perturbed golden-angle undersampling yields the best results for cardiac CS-DTI image reconstruction.

Approximating Multivariate Tempered Stable Processes

We give a simple method to approximate multidimensional exponentially tempered stable processes and show that the approximating process converges in the Skorokhod topology to the tempered process. The approximation is based on the generation of a random angle and a random variable with a lower-dimensional Lévy measure. We then show that if an arbitrarily small normal random variable is added, the marginal densities converge uniformly at an almost linear rate, providing a critical tool to assess the performance of existing particle tracking codes.

Approximating Multivariate Tempered Stable Processes

We give a simple method to approximate multidimensional exponentially tempered stable processes and show that the approximating process converges in the Skorokhod topology to the tempered process. The approximation is based on the generation of a random angle and a random variable with a lower-dimensional Lévy measure. We then show that if an arbitrarily small normal random variable is added, the marginal densities converge uniformly at an almost linear rate, providing a critical tool to assess the performance of existing particle tracking codes.