Sweep MRI is a new technological development to address the difficulty of imaging tissues and organs, such as bone, tendon, and lung, that give weak signals. In Sweep MRI, the RF (radio frequency) excitation is pulsed, while the background magnetic field gradient can be varied in time (see here for a schematic diagram of an MRI machine).  The resulting mathematical problem that needs to be solved is an inverse problem involving two steps [1].  The first step is determining the planar averages of the material properties of the object over a set of planes determined by the gradient of the background magnetic field.  The second step is to recover the spatial distribution of the material properties from these 'projections'.  Once the data are unraveled, the problem of image reconstruction can be viewed as that of Fourier inversion of a 3-d function from samples of its Fourier transform. The goal of this project is to first study the model of the relationship between the measured data and the desired unknown image, and the limitations of this in the case where the measured data is incomplete.  We will start with the case where the magnetic field gradient is assumed to be constant in time.  The time-varying case will also be considered.  Issues such as resolution limit and reconstruction methods will be explored, as well as signal recovery form incomplete data measurements. Below is a picture of a fetal mouse using SWIFT MRI. References: M. Weiger, F. Hennel, and K. Pruessmann, Sweep MRI with Algebraic Reconstruction, Magnetic Resonance in Medicine, 64, 2010, 1685-1695. Z.-P. Liang and P. Lauterbur, Principles of Magnetic Resonance Imaging: A Signal Processing Approach, IEEE Press, 1999. C. Epstein, Introduction to the Mathematics of Medical Imaging, Pearson/Prentice Hall, 2003. Prerequisites: Basic image processing, Fourier analysis, MATLAB programming. Keywords: Magnetic Resonance Imaging, Fourier transforms, inverse problem, tomography.