A FAST ALGORITHM FOR STRUCTURED LOW-RANK MATRIX RECOVERY WITH APPLICATIONS TO UNDERSAMPLED MRI RECONSTRUCTION.

Area of Science:

• Medical Imaging
• Computational Imaging
• Applied Mathematics

Background:

• Traditional image recovery methods like total variation (TV) and wavelet regularization face limitations in large-scale applications.
• Structured low-rank matrix priors are a promising alternative but often suffer from high computational complexity and memory demands.
• These challenges stem from the need to transform images into high-dimensional dense matrices.

Purpose of the Study:

• To develop a computationally efficient and memory-saving algorithm for image recovery using structured low-rank matrix priors.
• To address the limitations of existing methods in handling large-scale image reconstruction problems.
• To improve the quality of reconstructed magnetic resonance (MR) images from undersampled data.

Main Methods:

• Introduced a novel algorithm that leverages the convolutional structure of the lifted matrix.
• Developed a method to operate in the original non-lifted image domain, avoiding high-dimensional matrix transformations.
• Applied the algorithm to the problem of MR image reconstruction from undersampled measurements.

Main Results:

• The proposed algorithm significantly reduces computational complexity and memory requirements.
• Experimental results demonstrate improved image reconstructions compared to traditional total variation (TV) regularization.
• Achieved comparable computation times to existing TV regularization methods.

Conclusions:

• The developed algorithm offers a practical and efficient approach for image recovery using structured low-rank matrix priors.
• This method provides superior MR image reconstruction quality from undersampled data.
• The approach effectively overcomes the computational and memory challenges associated with high-dimensional matrix lifting.