Data-Driven Learning of a Union of Sparsifying Transforms Model for Blind Compressed Sensing

TL;DR

This paper introduces a data-driven framework for blind compressed sensing that learns a union of sparsifying transforms to improve image reconstruction quality, especially in MRI, from highly undersampled data.

Contribution

It proposes a novel union of transforms model for blind compressed sensing and efficient algorithms with convergence guarantees, outperforming existing methods.

Findings

Better MRI image reconstruction quality compared to recent methods

Learning a union of transforms outperforms single transform models

Algorithms converge to partial global and local minimizers

Abstract

Compressed sensing is a powerful tool in applications such as magnetic resonance imaging (MRI). It enables accurate recovery of images from highly undersampled 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 (BCS), 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 image(s) are approximately sparse in a transform domain. We also extend this model to a union of transforms model that better captures the diversity of features in natural images. The proposed block coordinate descent type algorithms for blind compressed sensing are highly efficient, and are guaranteed to converge to at least the partial global and partial local minimizers of the highly non-convex BCS problems. Our numerical experiments show that the proposed framework usually leads to better quality of image reconstructions in 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 adaptive transform.