Algebraic Reconstruction Method For Off-Resonance And Eddy-Current Correction In Functional And Diffusion Weighted Magnetic Resonance Imaging

A method of providing corrected magnetic resonance imaging (MRI), the method comprising:

• acquiring raw MRI data that is fully sampled in a first direction (x), undersampled in a second direction (y) and undersampled in a third direction (z), wherein the raw MRI data includes first raw MRI data and second raw MRI data having equal and opposite regular k-space encoding;
• performing a 1-D Fourier transformation of the raw MRI data in the first direction, thereby providing hybrid MRI data, wherein the hybrid MRI data includes first hybrid data (sup) corresponding to the first raw MRI data and second hybrid data (sdown) corresponding to the second raw MRI data;
• determining a first effective encoding matrix (Fup) that includes at least the effects of regular k-space encoding and known phase variations due to system non-ideality for the first hybrid data;
• determining a second effective encoding matrix (Fdown) that includes at least the effects of regular k-space encoding and known phase variations due to system non-ideality for the second hybrid data;
• determining a joint least-squares solution for magnetization (m) in a matrix equation (sup, sdown)−(Fup, Fdown)m using a singular value decomposition of an effective joint encoding matrix (Fup, Fdown);
• providing the magnetization (m) as an output.