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We derive precise conditions under which material reflectance properties may be estimated from a single image of a homogeneous curved surface (canonically a sphere), lit by a directional source. Based on the observation that light is reflected along certain (a priori unknown) preferred directions such as the half-angle, we propose a semiparametric BRDF abstraction that lies between purely parametric and purely data-driven models. Formulating BRDF estimation as a particular type of semiparametric regression, both the preferred directions and the form of BRDF variation along them can be estimated from data. Our approach has significant theoretical, algorithmic and empirical benefits, lends insights into material behavior and enables novel applications. While it is well-known that fitting multi-lobe BRDFs may be ill-posed under certain conditions, prior to this work, precise results for the well-posedness of BRDF estimation had remained elusive. Since our BRDF representation is derived from physical intuition, but relies on data, we avoid pitfalls of both parametric (low generalizability) and non-parametric regression (low interpretability, curse of dimensionality). Finally, we discuss several applications such as single-image relighting, light source estimation and physically meaningful BRDF editing.