Abstract
A volume NMR image can be reconstructed by backprojecting a set of filtered projection profiles onto a 3D matrix. It has been demonstrated previously that the image can be obtained by either a direct 3D backprojection or by an indirect, two-step, 2D backprojection. In this paper, the mathematical equivalence of these different procedures is explained from the geometrical nature of the 3D backprojection. The two-step technique is simply a fast algorithm which groups the summations properly and efficiently in computing the backprojection. It is however applicable only when the main magnetic field is uniform. If the field is inhomogeneous, then direct backprojection must be used in order to obtain an accurate image by a curvilinear reconstruction technique. The linearity of the gradient fields is not a factor determining the applicability of the fast backprojection.