Magnetic resonance image reconstruction from highly undersampled K-Space data using dictionary learning

摘要

Compressed sensing (CS) utilizes the sparsity of MR images to enable accurate reconstruction from undersampled k-space data. Recent CS methods have employed analytical sparsifying transforms such as wavelets, curvelets, and finite differences. In this thesis, we propose a novel framework for adaptively learning the sparsifying transform (dictionary), and reconstructing the image simultaneously from highly undersampled k-space data. The sparsity in this framework is enforced on overlapping image patches emphasizing local structure. Moreover, the dictionary is adapted to the particular image instance, thereby favoring better sparsities and consequently much higher undersampling rates. The proposed alternating reconstruction algorithm learns the sparsifying dictionary, and uses it to remove aliasing and noise in one step, and subsequently restores and fills in the k-space data in the other step. Numerical experiments are conducted on MR images and on real MR data of several anatomies with a variety of sampling schemes. The results demonstrate dramatic improvements on the order of 4-18 dB in reconstruction error and doubling of the acceptable undersampling factor using the proposed adaptive dictionary as compared to previous CS methods. These improvements persist over a wide range of practical data SNRs, without any parameter tuning. As a further enhancement to the proposed dictionary learning scheme for MRI reconstruction, we explore the use of an additive multiscale dictionary formulation. This formulation enforces sparsity of the reconstructed image simultaneously at multiple scales (patch sizes) and combines the results at those scales to obtain superior reconstructions. The multiscale dictionary in the proposed formulation is a collection of several single scale dictionaries that operate separately. The alternating reconstruction algorithm learns the various single scale sparsifying dictionaries and uses them to remove image artifacts in one step, and then restores and fills in k-space in the other step. Experiments conducted on several MR images using simulated k-space undersampling with a variety of sampling schemes show promising improvements of up to 1.4 dB in reconstruction error with the proposed multiscale dictionary as compared to a dictionary learned at only one scale. This improvement is also achieved at a substantially lower computational complexity for the multiscale formulation, thereby demonstrating that (additive) multiscale sparse representations are both better and faster. The final improvement explored in this thesis is a sequential multiscale reconstruction algorithm that starts with the lowest scale and adds in the higher scales sequentially over iterations. This approach is shown to be faster than the one where all scales are used for all the iterations, while achieving the same PSNR in the reconstructed image.