Exploring Structural Sparsity of Coil Images from 3-Dimensional Directional Tight Framelets for SENSE Reconstruction^\ast

Each coil image in a parallel magnetic resonance imaging (pMRI) system is an imaging slice modulated by the corresponding coil sensitivity. These coil images, structurally similar to each other, are stacked together as 3-dimensional (3D) image data, and their sparsity property can be explored via 3D directional Haar tight framelets. The features of the 3D image data from the 3D framelet systems are utilized to regularize sensitivity encoding (SENSE) pMRI reconstruction. Accordingly, a so-called SENSE3d algorithm is proposed to reconstruct images of high quality from the sampled K-space data with a high acceleration rate by decoupling effects of the desired image (slice) and sensitivity maps. Since both the imaging slice and sensitivity maps are unknown, this algorithm repeatedly performs a slice step followed by a sensitivity step by using updated estimations of the desired image and the sensitivity maps. In the slice step, for the given sensitivity maps, the estimation of the desired image is viewed as the solution to a convex optimization problem regularized by the sparsity of its 3D framelet coefficients of coil images. This optimization problem, involving data from the complex field, is solved by a primal-dual three-operator splitting (PD3O) method. In the sensitivity step, the estimation of sensitivity maps is modeled as the solution to a Tikhonovtype optimization problem that favors the smoothness of the sensitivity maps. This corresponding problem is nonconvex and could be solved by a forward-backward splitting method. Experiments on real phantoms and in vivo data show that the proposed SENSE3d algorithm can explore the sparsity property of the imaging slices and efficiently produce reconstructed images of high quality with reduced aliasing artifacts caused by high acceleration rate, additive noise, and the inaccurate estimation of each coil sensitivity. To provide a comprehensive picture of the overall performance of our SENSE3d model, we provide the quantitative index (HaarPSI) and comparisons to some deep learning methods such as VarNet and fastMRI-UNet.