Data-driven adaptation of a union of sparsifying transforms for blind compressed sensing MRI reconstruction

Compressed Sensing has been demonstrated to be a powerful tool for magnetic resonance imaging (MRI), where it enables accurate recovery of images from highly undersampled k-space measurements by exploiting the sparsity of the images or image patches in a transform domain or dictionary. In this work, we focus on blind compressed sensing, where the underlying sparse signal model is a priori unknown, and propose a framework to simultaneously reconstruct the underlying image as well as the unknown model from highly undersampled measurements. Specifically, our model is that the patches of the underlying MR image(s) are approximately sparse in a transform domain. We also extend this model to a union of transforms model that is better suited to capture the diversity of features in MR images. The proposed block coordinate descent type algorithms for blind compressed sensing are highly efficient. Our numerical experiments demonstrate the superior performance of the proposed framework for MRI compared to several recent image reconstruction methods. Importantly, the learning of a union of sparsifying transforms leads to better image reconstructions than a single transform.