Optimal Real-Time QBI using Regularized Kalman Filtering with Incremental Orientation Sets

Diffusion MRI has become an established research tool for the investigation of tissue structure and orientation from which has stemmed a number of variations, such as Diffusion Tensor Imaging (DTI), Diffusion Spectrum Imaging (DSI) and Q-Ball Imaging (QBI). The acquisition and analysis of such data is very challenging due to its complexity. Recently, an exciting new Kalman filtering framework has been proposed for DTI and QBI reconstructions in real time during the repetition time (TR) of the acquisition sequence \cite{Miccai:2007,Med. Image Analysis -Vol 12, Issue 5, June 2008}. In this article, we first revisite and thoroughly analyze this approach and show it is actually sub-optimal and not recursively minimizing the intended criterion due to the Laplace-Beltrami regularization term. Then, we propose a new approach that implements the QBI reconstruction algorithm in real-time using a fast and robust Laplace-Beltrami regularization without sacrificing the optimality of the Kalman filter. We demonstrate that our method solves the correct minimization problem at each iteration and recursively provides the optimal QBI solution. We validate with real QBI data that our proposed real-time method is equivalent in terms of QBI estimation accuracy to the standard off-line processing techniques and outperforms the existing solution. Last, we propose a fast algorithm to recursively compute gradient orientation sets whose partial subsets are almost uniform and show that it can also be applied to the problem of efficiently ordering an existing point-set of any size. Our work allows to start an acquisition just with the minimum number of gradient directions and an initial estimate of the q-ball and then all the rest, including the next gradient directions and the q-ball estimates, are recursively and optimally determined, allowing the acquisition to be stopped as soon as desired or at any iteration with the optimal q-ball estimate. This opens new and interesting opportunities for real-time feedback for clinicians during an acquisition and also for researchers investigating into optimal diffusion orientation sets and, real-time fiber tracking and connectivity mapping

Online orientation distribution function reconstruction in constant solid angle and its application to motion detection in HARDI

International audienceThe diffusion orientation distribution function (ODF) can be reconstructed from q-ball imaging (QBI) to map the complex intravoxel structure of water diffusion. As acquisition time is particularly large for high angular resolution diffusion imaging (HARDI), fast estimation algorithms have recently been proposed, as an on-line feedback on the reconstruction accuracy. Thus the acquisition could be stopped or continued on demand. We adapt these real-time algorithms to the mathematically correct definition of ODF in constant solid angle (CSA), and develop a motion detection algorithm upon this reconstruction. Results of improved fiber crossing detection by CSA ODF are shown, and motion detection was implemented and tested in vivo

Incremental gradient table for multiple Q-shells diffusion MRI

International audienceMost studies on sampling optimality for diffusion MRI deal with single Q-shell acquisition. For single Q-shell acquisition, incremental gradient table has proved useful in clinical setup, where the subject is likely to move, or for online reconstruction. In this article, we propose a generalization of the electrostatic repulsion to generate gradient tables for multiple Q-shells acquisitions, designed for incremental reconstruction or processing of data prematurely aborted

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

A filtered approach to neural tractography using the Watson directional function

We propose a technique to simultaneously estimate the local fiber orientations and perform multifiber tractography. Existing techniques estimate the local fiber orientation at each voxel independently so there is no running knowledge of confidence in the measured signal or estimated fiber orientation. Further, to overcome noise, many algorithms use a filter as a post-processing step to obtain a smooth trajectory. We formulate fiber tracking as causal estimation: at each step of tracing the fiber, the current estimate of the signal is guided by the previous. To do this, we model the signal as a discrete mixture of Watson directional functions and perform tractography within a filtering framework. Starting from a seed point, each fiber is traced to its termination using an unscented Kalman filter to simultaneously fit the signal and propagate in the most consistent direction. Despite the presence of noise and uncertainty, this provides an accurate estimate of the local structure at each point along the fiber. We choose the Watson function since it provides a compact representation of the signal parameterized by the principal diffusion direction and a scaling parameter describing anisotropy, and also allows analytic reconstruction of the oriented diffusion function from those parameters. Using a mixture of two and three components (corresponding to two-fiber and three-fiber models) we demonstrate in synthetic experiments that this approach reduces signal reconstruction error and significantly improves the angular resolution at crossings and branchings. In vivo experiments examine the corpus callosum and internal capsule and confirm the ability to trace through regions known to contain such crossing and branching while providing inherent path regularization

Adaptive Design of Sampling Directions in Diffusion Tensor MRI and Validation on Human Brain Images

International audienceDiffusion tensor reconstruction is made possible through the acquisition of several diffusion weighted images, each corresponding to a given sampling direction in the Q-space. In this study, we address the question of sampling efficiency, and show that in case we have some prior knowledge on the diffusion characteristics, we may be able to adapt the sampling directions for better reconstruction of the diffusion tensor. The prior is a tensor distribution function, estimated over a given region of interest, possibly on several subjects. We formulate an energy related to error on tensor reconstruction, and calculate analytical gradient expression for efficient minimization. We validate our approach on a set of 5199 tensors taken within the corpus callosum of the human brain, and show improvement by an order of 10% on the MSE of the reconstructed tensor

Optimal Design of Multiple Q-shells experiments for Diffusion MRI

diffusion mri; experimental design; sampling scheme; diffusion propagator; optimal samplingdiffusion mri; experimental design; sampling scheme; diffusion propagator; optimal samplingInternational audienceRecent advances in diffusion MRI make use of the diffusion signal sampled on the whole Q-space, rather than on a single sphere. While much effort has been done to design uniform sampling schemes for single shell experiment, it is yet not clear how to build a strategy to sample the diffusion signal in the whole Fourier domain. In this article, we propose a method to generate acquisition schemes for multiple Q-shells experiment in diffusion MRI. The acquisition protocols we design are incremental, which means they remain approximately optimal when truncated before the acquisition is complete. Our method is fast, incremental, and we can generate diffusion gradients schemes for any number of acquisitions, any number of shells, and any number of points per shell. The samples arranged on different shells do not share the same directions. The method is tested for Spherical Polar Fourier reconstruction of the diffusion signal, and based on Monte-Carlo simulations, several preferred acquisition parameters are identified

Large Scale Inverse Problems

This book is thesecond volume of a three volume series recording the "Radon Special Semester 2011 on Multiscale Simulation &amp Analysis in Energy and the Environment" that took placein Linz, Austria, October 3-7, 2011. This volume addresses the common ground in the mathematical and computational procedures required for large-scale inverse problems and data assimilation in forefront applications. The solution of inverse problems is fundamental to a wide variety of applications such as weather forecasting, medical tomography, and oil exploration. Regularisation techniques are needed to ensure solutions of sufficient quality to be useful, and soundly theoretically based. This book addresses the common techniques required for all the applications, and is thus truly interdisciplinary. This collection of survey articles focusses on the large inverse problems commonly arising in simulation and forecasting in the earth sciences