Algebraic reconstruction for magnetic resonance imaging under B0 inhomogeneity

In magnetic resonance imaging, spatial localization is usually achieved using Fourier encoding which is realized by applying a magnetic field gradient along the dimension of interest to create a linear correspondence between the resonance frequency and spatial location following the Larmor equation. In the presence of B0 inhomogeneities along this dimension, the linear mapping does not hold and spatial distortions arise in the acquired images. In this paper, the problem of image reconstruction under an inhomogeneous field is formulated as an inverse problem of a linear Fredholm equation of the first kind. The operators in these problems are estimated using field mapping and the k-space trajectory of the imaging sequence. Since such inverse problems are known to be ill-posed in general, robust solvers, singular value decomposition and conjugate gradient method, are employed to obtain corrected images that are optimal in the Frobenius norm sense. Based on this formulation, the choice of the imaging sequence for well-conditioned matrix operators is discussed, and it is shown that nonlinear k-space trajectories provide better results. The reconstruction technique is applied to sequences where the distortion is more severe along one of the image dimensions and the two-dimensional reconstruction problem becomes equivalent to a set of independent one-dimensional problems. Experimental results demonstrate the performance and stability of the algebraic reconstruction methods.