Field Inhomogeneity Correction based on Gridding Reconstruction

i i 2 1 N N t e e m s ρ ω ρ κ ρ κ κ ρ r k or − = ≈ 1 0 i - i - M t e e s d m κ ω κ κ ρ ρ κ κ ρ r k , where s denotes the estimated signal at position k at time t , m the magnetiza- tion and the angular off-resonance frequency at position r , N1N2 the number of pixels, M the number of samples in the k-space domain, and d an optional sam- pling density compensation. In principle, and t may be considered as extra di- mensions in image and k-space, respectively. Evaluating one of the sums then amounts to calculating a trivariate Fourier transform with non-equispaced sam- pling in both domains. To improve accuracy and efficiency, we suggest to directly apply the above approximation to the field inhomogeneity-induced exponential. For this purpose, we define