Methods for providing practical magnetic resonance imaging systems that utilize non-homogeneous background fields, B.sub.0, as well as, possibly non-linear, gradient fields G.sub.1, G.sub.2 to make non-invasive measurements to determine, among other things, a spin density function. These methods are time and SAR efficient, using one or two refocusing pulses for each line measured in k-space. Two types of non-homogeneous background fields are considered: background fields B.sub.0 in which the function |B.sub.0| does not have a critical point within the field of view, and background fields B.sub.0 such that the function |B.sub.0| has a single critical point within the field of view. In the first case, an MR-imaging device may be constructed by using the permanent gradient in the background field, B.sub.0, as a slice select gradient, so long as particular criteria are met. The methods demonstrate that there are many practical circumstances where these criteria can all be met. An apparatus meeting such criteria can allow one to make non-invasive measurements that allow a reconstruction of the spin density function determined by a 3-dimensional object. The measurement process using such an apparatus includes the steps of placing the object to be imaged in an inhomogeneous background field B.sub.0 for a sufficient time for the nuclear spins of a desired species to be polarized, selectively exciting the polarized nuclear spins using standard selective excitation RF-pulse sequences from apparatus that generates substantially homogeneous RF-fields within a field of view of the apparatus, and spatially encoding phases of the excited nuclear spins using gradient fields G.sub.1, G.sub.2, where the functions |B.sub.0|, , define coordinates within the field of view of the apparatus. Also, under these same assumptions on the magnetic fields, one can use a combination of the fields B.sub.0, G.sub.1, and G.sub.2 to define a slice select direction not parallel to the permanent gradient in B.sub.0 In this case one recovers a 2D-imaging protocol, using an inhomogeneous background field, with most of the features of standard spin-warp imaging. In the second case, magnets may be constructed so that |B.sub.0| has an isolated non-zero local minimum. Using selective excitation, one can excite only the spins lying in a small neighborhood of this local minimum. In this way one can do spatially localized, high SNR, spectroscopic measurements, without any need for further spatial encoding.