90,984 research outputs found
An enumerative formula for the spherical cap discrepancy
The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes. Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy with respect to a sample of given size
Single- and Multiple-Shell Uniform Sampling Schemes for Diffusion MRI Using Spherical Codes
In diffusion MRI (dMRI), a good sampling scheme is important for efficient
acquisition and robust reconstruction. Diffusion weighted signal is normally
acquired on single or multiple shells in q-space. Signal samples are typically
distributed uniformly on different shells to make them invariant to the
orientation of structures within tissue, or the laboratory coordinate frame.
The Electrostatic Energy Minimization (EEM) method, originally proposed for
single shell sampling scheme in dMRI, was recently generalized to multi-shell
schemes, called Generalized EEM (GEEM). GEEM has been successfully used in the
Human Connectome Project (HCP). However, EEM does not directly address the goal
of optimal sampling, i.e., achieving large angular separation between sampling
points. In this paper, we propose a more natural formulation, called Spherical
Code (SC), to directly maximize the minimal angle between different samples in
single or multiple shells. We consider not only continuous problems to design
single or multiple shell sampling schemes, but also discrete problems to
uniformly extract sub-sampled schemes from an existing single or multiple shell
scheme, and to order samples in an existing scheme. We propose five algorithms
to solve the above problems, including an incremental SC (ISC), a sophisticated
greedy algorithm called Iterative Maximum Overlap Construction (IMOC), an 1-Opt
greedy method, a Mixed Integer Linear Programming (MILP) method, and a
Constrained Non-Linear Optimization (CNLO) method. To our knowledge, this is
the first work to use the SC formulation for single or multiple shell sampling
schemes in dMRI. Experimental results indicate that SC methods obtain larger
angular separation and better rotational invariance than the state-of-the-art
EEM and GEEM. The related codes and a tutorial have been released in DMRITool.Comment: Accepted by IEEE transactions on Medical Imaging. Codes have been
released in dmritool
https://diffusionmritool.github.io/tutorial_qspacesampling.htm
Optimal experiment design in a filtering context with application to sampled network data
We examine the problem of optimal design in the context of filtering multiple
random walks. Specifically, we define the steady state E-optimal design
criterion and show that the underlying optimization problem leads to a second
order cone program. The developed methodology is applied to tracking network
flow volumes using sampled data, where the design variable corresponds to
controlling the sampling rate. The optimal design is numerically compared to a
myopic and a naive strategy. Finally, we relate our work to the general problem
of steady state optimal design for state space models.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS283 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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