Improved MRI Restoration by Integrating Bayesian formalism and Support Vector Machines in a Time Delayed priors Framework

Abstract

A data acquisition process is needed to form the MR images. Such data acquisition occurs in the spatial frequency (kspace) domain, where sampling theory determines resolution and field of view, and it results in the formation of the kspace matrix. Strategies for reducing image artifacts are often best developed in this domain. After obtaining such a k-space matrix, image reconstruction involves fast multi-dimensional Inverse Fourier transforms, often preceded by data interpolation and re-sampling. Sampling the k-space matrix occurs along suitable trajectories [1,2,3]. Ideally, these trajectories are chosen to completely cover the k-space according to the Nyquist sampling criterion. The measurement time of a single trajectory can be made short. However, prior to initiating a trajectory, return to thermal equilibrium of the nuclear spins needs to be awaited. The latter is governed by an often slow natural relaxation process that is beyond control of the scanner and impedes fast scanning. Therefore, the only way to shorten scan time in MRI when needed, as for instance in functional MRI, is to reduce the overall waiting time by using fewer trajectories, which in turn should individually cover more of k-space through added curvatures. Although, however, such trajectory omissions achieve the primary goal, i.e. more rapid measurements, they entail under-sampling and violations of the Nyquist criterion thus, leading to concomitant problems for image reconstruction. The above mentioned rapid scanning in MRI problem is highly related with two other ones. The first is the selection of the optimal scanning scheme in k-space that is the problem of finding the shape of sampling trajectories that more fully cover the k-space using fewer trajectories. Mainly three such alternative shapes have been considered in the literature and are used in actual scanners, namely, Cartesian, radial and spiral [1], associated with different reconstruction techniques. More specifically, the Cartesian scheme uses the inverse 2D FFT, while the radial and spiral scanning involve the Projection Reconstruction, the linogram or the SRS-FT approaches [1,2,3]. The second one is associated with image estimation from fewer samples in k-space that is the problem of omitting as many trajectories as possible without attaining worse reconstruction results. The main result of such scan trajectories omissions is that we have fewer samples in kspace than needed for estimating all pixel intensities in image space. Therefore, there is infinity of MRI images satisfying the sparse k-space data and thus, the image restoration problem becomes ill-posed. Additionally, omissions usually cause violation of the Nyquist sampling condition. Despite the fact that solutions are urgently needed, in functional MRI for instance, very few research efforts exist in the literature. –The goal of this paper is to present the development of a new image restoration methodology for extracting Magnetic Resonance Images (MRI) from reduced scans in k-space. The proposed approach considers the combined use of Support Vector Machine (SVM) models in a time delayed Bayesian priors framework and Bayesian restoration, in the problem of MR image reconstruction from sparsely sampled k-space, following several different sampling schemes, including spiral and radial. Effective solutions to this problem are indispensable especially when dealing with MRI of dynamic phenomena since then, rapid sampling in k-space is required. The goal in such a case is to make measurement time smaller by reducing scanning trajectories as much as possible. In this way, however, underdetermined equations are introduced and poor image extraction follows. It is suggested here that significant improvements could be achieved, concerning quality of the extracted image, by judiciously applying time delayed SVM priors and Bayesian estimation methods to the k-space data. More specifically, it is demonstrated that SVM neural network techniques could construct efficient time delayed Bayesian priors and introduce them in the procedure of Bayesian restoration. These time delayed Bayesian Priors are independent of specific image properties and probability distributions. They are based on training SVM neural filters to estimate the missing samples of complex k-space and thus, to improve k-space information capacity. Such a neural filter based time delayed Bayesian prior is integrated to the maximum likelihood procedure involved in the Bayesian reconstruction. It is found that the proposed methodology leads to enhanced image extraction results favorably compared to the ones obtained by the Integrated Bayesian MRI reconstruction approach involving simple and non time delayed SVM models priors [7], by the traditional Bayesian MRI reconstruction approach [1] as well as by the pure Neural Network (NN) filter based reconstruction approach [6].