Optimal real-time Q-ball imaging using regularized Kalman filtering with incremental orientation sets

Diffusion MRI has become an established research tool for the investigation of tissue structure and orientation. Since its inception, Diffusion MRI has expanded considerably to include 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. In this article, we first revisit 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 offline 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. This work enables a clinician to start an acquisition with just the minimum number of gradient directions and an initial estimate of the orientation distribution functions (ODF) and then the next gradient directions and ODF estimates can be recursively and optimally determined, allowing the acquisition to be stopped as soon as desired or at any iteration with the optimal ODF estimates. 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.