Estimating the illumination and the reflectance properties of an object surface from a sparse set of images is an important but inherently ill-posed problem. The problem becomes even harder if we wish to account for the spatial variation of material properties on the surface. In this paper, we derive a novel method for estimating the spatially varying specular reflectance properties, of a surface of known geometry, as well as the illumination distribution from a specular-only image, for instance, captured using polarization to separate reflection components. Unlike previous work, we do not assume the illumination to be a single point light source. We model specular reflection with a spherical statistical distribution and encode the spatial variation with radial basis functions of its parameters. This allows us to formulate the simultaneous estimation of spatially varying specular reflectance and illumination as a sound probabilistic inference problem, in particular, using Csisźar's I-divergence measure. To solve it, we derive an iterative algorithm similar to expectation maximization. We demonstrate the effectiveness of the method on synthetic and real-world scenes.