This paper presents a comprehensive theory of photometric surface reconstruction from image derivatives. For unknown isotropic BRDFs, we show that two measurements of spatial and temporal image derivatives, under unknown light sources on a circle, suffice to determine the surface. This result is the culmination of a series of fundamental observations. First, we discover a photometric invariant that relates image derivatives to the surface geometry, regardless of the form of isotropic BRDF. Next, we show that just two pairs of differential images from unknown light directions suffice to recover surface information from the photometric invariant. This is shown to be equivalent to determining isocontours of constant magnitude of the surface gradient, as well as isocontours of constant depth. Further, we prove that specification of the surface normal at a single point completely determines the surface depth from these isocontours. In addition, we propose practical algorithms that require additional initial or boundary information, but recover depth from lower order derivatives. Our theoretical results are illustrated with several examples on synthetic and real data. 1.

Radiometric image analysis methods heavily rely on reflectance models. Due to the complexity of real materials, methods based on simple models such as the Lambertian model often suffer from inaccuracy. On the other hand, more advanced models such as the Cook-Torrance model severely complicate the analysis problem. We tackle this dilemma by focusing on the low-frequency component of the reflectance. We propose a compact biquadratic reflectance model to represent the reflectance of a broad class of materials precisely in the low-frequency domain. We validate our model by fitting to both existing parametric models and non-parametric measured data, and show that our model outperforms existing parametric diffuse models. We show applications of reflectometry using general diffuse surfaces and photometric stereo for general isotropic materials. Experimental results show the effectiveness of our biquadratic model and its usefulness in radiometric image analysis. 1.

Abstract. This paper exploits the monotonicity of general isotropic reflectances for estimating elevation angles of surface normal given the azimuth angles. With an assumption that the reflectance includes at least one lobe that is a monotonic function of the angle between the surface normal and half-vector (bisector of lighting and viewing directions), we prove that elevation angles can be uniquely determined when the surface is observed under varying directional lights densely and uniformly distributed over the hemisphere. We evaluate our method by experiments using synthetic and real data to show its wide applicability, even when the assumption does not strictly hold. By combining an existing method for azimuth angle estimation, our method derives complete surface normal estimates for general isotropic reflectances. 1