Proton radiography is an important diagnostic method for laser plasma experiments and is particularly important in the analysis of magnetized plasmas. The theory of radiographic image analysis has heretofore only permitted somewhat limited analysis of the radiographs of such plasmas. We furnish here a theory that remedies this deficiency. We show that to linear order in magnetic field gradients, proton radiographs are projection images of the MHD current along the proton trajectories. We demonstrate that in the linear regime (i.e., the small image contrast regime), the full structure of the projected perpendicular magnetic field can be reconstructed by solving a steady-state inhomogeneous 2-dimensional diffusion equation sourced by the radiograph fluence contrast data. We explore the validity of the scheme with increasing image contrast, as well as limitations of the inversion method due to the Poisson noise, discretization errors, radiograph edge effects, and obstruction by laser target structures. We also provide a separate analysis that is suited to the inference of isotropic-homogeneous magnetic turbulence spectra. Finally, we discuss extension of these results to the nonlinear regime (i.e., the order unity image contrast regime).