X-ray backlit radiographs of dense plasma shells can be significantly altered by refraction of x rays that would otherwise travel straight-ray paths, and this effect can be a powerful tool for diagnosing the spatial structure of the plasma being radiographed. We explore the conditions under which refraction effects may be observed, and we use analytical and numerical approaches to quantify these effects for one-dimensional radial opacity and density profiles characteristic of inertial-confinement fusion (ICF) implosions. We also show how analytical and numerical approaches allow approximate radial plasma opacity and density profiles to be inferred from point-projection refraction-enhanced radiography data. This imaging technique can provide unique data on electron density profiles in ICF plasmas that cannot be obtained using other techniques, and the uniform illumination provided by point-like x-ray backlighters eliminates a significant source of uncertainty in inferences of plasma opacity profiles from area-backlit pinhole imaging data when the backlight spatial profile cannot be independently characterized. The technique is particularly suited to in-flight radiography of imploding low-opacity shells surrounding hydrogen ice, because refraction is sensitive to the electron density of the hydrogen plasma even when it is invisible to absorption radiography. It may also provide an alternative approach to timing shockwaves created by the implosion drive, that are currently invisible to absorption radiography.