Arizona State University

Most current methods of Magnetic Resonance Imaging (MRI) reconstruction interpret raw signal values as samples of the Fourier transform of the object. Although this is computationally convenient, it neglects relaxation and off–resonance evolution in phase, both of which can occur to significant extent during a typical MRI signal. A more accurate model, known as Parameter Assessment by Recovery from Signal Encoding (PARSE), takes the time evolution of the signal into consideration. This model uses three parameters that depend on tissue properties: transverse magnetization, signal decay rate, and frequency offset from resonance. Two difficulties in recovering an image using this model are the low SNR for long acquisition times in single-shot MRI, and the nonlinear dependence of the signal on the decay rate and frequency offset. In this talk, we address the latter issue by using a second order approximation of the original PARSE model. The linearized model can be solved using convex optimization augmented with well-stablished regularization techniques such as total variation. The sensitivity of the parameters to noise and computational challenges associated with this approximation will be discussed.

In this talk I review the use of the spectral decomposition for understanding the solution of ill-posed inverse problems. It is immediate to see that regularization is needed in order to find stable solutions. These solutions, however, do not typically allow reconstruction of signal features such as edges. Generalized regularization assists but is still insufficient and methods of total variation are commonly suggested as an alternative. In the talk I consider application of standard approaches from Tikhonov regularization for finding appropriate regularization parameters in the total variation augmented Lagrangian implementations. Areas for future research will be considered.