A variational approach to magnetic resonance coil sensitivity estimation

摘要

A variational method for estimating a magnetic resonance coil sensitivity from its corresponding non-uniform illumination of magnetic resonance images is proposed and analyzed. The estimated sensitivity can then be used to correct non-uniformities and indeed to facilitate high-speed parallel image acquisition. The data available for estimation include two images of the same field of view but obtained with and without a uniform coil sensitivity. The non-uniformly illuminated image is ideally the product of the uniformly illuminated image with the coil sensitivity. Thus, the desired sensitivity is estimated roughly from a quotient of the two given images over the effective data support. However, the measurements are corrupted by noise, the data are discontinuous at tissue boundaries, and yet the coil sensitivity is very smooth. In the selected estimation procedure, the sum of a residual and a high order penalty is minimized. In this form, the problem is related to the surface estimation problem of early vision. In the present context, higher coil sensitivity decay rates are captured non-parametrically by higher order penalties together with natural boundary conditions. The role of alternative penalties and boundary conditions is investigated both analytically and numerically. Although high order operators are often approximated numerically with a product of low order factors, such factorizations are shown here to lead to conspicuously spurious boundary conditions for surface estimation with sparse data support. Furthermore, while finite elements present a natural numerical approach for solving the optimality system, it is demonstrated here that lumping is required to avoid aberrant consequences in the limit of vanishing regularization corresponding to an ever improving signal-to-noise ratio. Finally, with a proper analytical and numerical formulation, the estimation procedure is demonstrated for a magnetic resonance imaging application involving parallel image acquisition.