Efficient blind compressed sensing using sparsifying transforms with convergence guarantees and application to magnetic resonance imaging

Saiprasad Ravishankar, Yoram Bresler

Research output: Contribution to journalArticle

Abstract

Natural signals and images are well known to be approximately sparse in transform domains such as wavelets and discrete cosine transform. This property has been heavily exploited in various applications in image processing and medical imaging. 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 the underlying image as well as the sparsifying transform from highly undersampled measurements. The proposed block coordinate descent-type algorithms involve highly efficient optimal updates. Importantly, we prove that although the proposed blind compressed sensing formulations are highly nonconvex, our algorithms are globally convergent (i.e, they converge from any initialization) to the set of critical points of the objectives defining the formulations. These critical points are guaranteed to be at least partial global and partial local minimizers. The exact point(s) of convergence may depend on initialization. We illustrate the usefulness of the proposed framework for magnetic resonance image reconstruction from highly undersampled k-space measurements. As compared to previous methods involving the synthesis dictionary model, our approach is much faster, while also providing promising reconstruction quality.

Original languageEnglish (US)
Pages (from-to)2519-2557
Number of pages39
JournalSIAM Journal on Imaging Sciences
Volume8
Issue number4
DOIs
StatePublished - Nov 10 2015

Keywords

  • Compressed sensing
  • Dictionary learning
  • Inverse problems
  • Magnetic resonance imaging
  • Medical imaging
  • Sparse representation
  • Sparsifying transforms

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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