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Both inhomogeneity in the main magnetic field and non-linearity in the gradient fields introduce errors in NMR images reconstructed from projections. It is found that the image artefacts caused by these two sources are separable. The inhomogeneous main field blurs the image while the nonlinear gradients distort the image. Firstly, blurring is removed by reconstructing the image curvilinearly in a coordinate system defined by the gradient fields. The distortions in the image are then corrected by taking an inverse coordinate transformation from the gradient-field coordinates to the spatial coordinates. Experiments performed by computer simulations are presented to demonstrate that, for the first time, accurate images can be obtained under large inhomogeneity and non-linearity in the fields. Mathematical derivations and computational techniques are also described and discussed.