Blind compressed sensing using sparsifying transforms

Compressed sensing exploits the sparsity of images or image patches in a transform domain or synthesis dictionary to reconstruct images from undersampled measurements. In this work, we focus on blind compressed sensing, where the underlying sparsifying transform is a priori unknown, and propose a framework to simultaneously reconstruct both the image and the transform from highly undersampled measurements. The proposed block coordinate descent type algorithm involves efficient updates. Importantly, we prove that although the proposed formulation is highly nonconvex, our algorithm converges to the set of critical points of the objective defining the formulation. We illustrate the promise of the proposed framework for magnetic resonance image reconstruction from highly undersampled k-space measurements. As compared to previous methods involving fixed sparsifying transforms, or adaptive synthesis dictionaries, our approach is much faster, while also providing promising image reconstructions.