Orthogonal tensor dictionary learning for accelerated dynamic MRI.

A direct application of the compressed sensing (CS) theory to dynamic magnetic resonance imaging (MRI) reconstruction needs vectorization or matricization of the dynamic MRI data, which is composed of a stack of 2D images and can be naturally regarded as a tensor. This 1D/2D model may destroy the inherent spatial structure property of the data. An alternative way to exploit the multidimensional structure in dynamic MRI is to employ tensor decomposition for dictionary learning, that is, learning multiple dictionaries along each dimension (mode) and sparsely representing the multidimensional data with respect to the Kronecker product of these dictionaries. In this work, we introduce a novel tensor dictionary learning method under an orthonormal constraint on the elementary matrix of the tensor dictionary for dynamic MRI reconstruction. The proposed algorithm alternates sparse coding, tensor dictionary learning, and updating reconstruction, and each corresponding subproblem is efficiently solved by a closed-form solution. Numerical experiments on phantom and synthetic data show significant improvements in reconstruction accuracy and computational efficiency obtained by the proposed scheme over the existing method that uses the 1D/2D model with overcomplete dictionary learning. Graphical abstract Fig. 1 Comparison between (a) the traditional method and (b) the proposed method based on dictionary learning for dynamic MRI reconstruction.