Modeling anisotropic undersampling of magnetic resonance angiographies and reconstruction of a high-resolution isotropic volume using half-quadratic regularization techniques

In this paper we address the problem of reconstructing a high resolution volumic image from several low resolution data sets. A solution to this problem is proposed in the particular framework of magnetic resonance angiography (MRA), where the resolution is limited by a trade-off between the spatial resolution and the acquisition time, both being proportional to the number of samples acquired in k-space. For this purpose only the meaningful spatial frequencies of the 3D k-space of the vessel are acquired, which is achieved using successive acquisitions with decreased spatial resolution, leading to highly anisotropic data sets in one or two specific directions. The reconstruction of the MRA volume from these data sets relies on an edge-preserving regularization method and leads to two different implementations: the first one is based on a conjugate gradient algorithm, and the second one on half-quadratic developments. The hyper parameters of the method were experimentally determined using a set of simulated data, and promising results were obtained on aorta and carotid artery acquisitions, where on the one hand a good fidelity to the acquired data is maintained, and on the other hand homogeneous areas are smooth and edges are well preserved. Half-quadratic regularization proved to be particularly well adapted to the MRA problem and leads to a fast iterative algorithm requiring only scalar and FFT computations.