Mathematical Challenges in Magnetic Resonance Imaging (MRI)
Magnetic resonance imaging (MRI) requires solutions to (at least) two inverse problems. The first problem is the design of radio frequency (RF) pulses that excite the spins in a certain slice or volume according to a prescribed pattern. The second problem is the reconstruction of 2D, 3D, or 4D images from the measured data. Both of these problems can be solved with simple Fourier transform methods in some cases, but there are interesting applications, including functional MRI (fMRI), where more advanced methods based on more accurate physical models are beneficial. These usually require iterative optimization methods. This talk will first review the problems and traditional solutions, and then will describe some more recent advanced methods. Along the way I will identify some open mathematical challenges in this field.
Jeffrey Fessler, University of Michigan, Ann Arbor