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Natural signals and images are well-known to be approximately sparse intransform domains such as Wavelets and DCT. This property has been heavilyexploited in various applications in image processing and medical imaging.Compressed sensing exploits the sparsity of images or image patches in atransform domain or synthesis dictionary to reconstruct images fromundersampled measurements. In this work, we focus on blind compressed sensing,where the underlying sparsifying transform is a priori unknown, and propose aframework to simultaneously reconstruct the underlying image as well as thesparsifying transform from highly undersampled measurements. The proposed blockcoordinate descent type algorithms involve highly efficient optimal updates.Importantly, we prove that although the proposed blind compressed sensingformulations are highly nonconvex, our algorithms are globally convergent(i.e., they converge from any initialization) to the set of critical points ofthe objectives defining the formulations. These critical points are guaranteedto be at least partial global and partial local minimizers. The exact point(s)of convergence may depend on initialization. We illustrate the usefulness ofthe proposed framework for magnetic resonance image reconstruction from highlyundersampled k-space measurements. As compared to previous methods involvingthe synthesis dictionary model, our approach is much faster, while alsoproviding promising reconstruction quality.

Efficient Blind Compressed Sensing Using Sparsifying Transforms with Convergence Guarantees and Application to MRI