InternationalJournal of ComputerVision, 14, 227-251 (1995) @ 1995 Kluwer AcademicPublishers, Boston. Manufacturedin The Netherlands. Generalization of the Lambertian Model and Implications for Machine Vision MICHAEL OREN AND SHREE K. NAYAR Department of Computer Science, Columbia University, New York, N.Y. 10027 Received March 22, 1993; Revised March 4, 1994. Abstract. Lambert's model for diffuse reflection is extensively used in computational vision. It is used explicitly by methods such as shape from shading and photometric stereo, and implicitly by methods such as binocular stereo and motion detection. For several real-world objects, the Lambertian model can prove m be a very inaccurate approximation to the diffuse component. While the brightness of a Lambertian surface is independent of viewing direction, the brightness of a rough diffuse surface increases as the viewer approaches the source direction. A comprehensive model is developed that predicts reflectance from rough diffuse surfaces. The model accounts for complex geometric and radiometric phenomena such as masking, shadowing, and interreflections between points on the surface. Experiments have been conducted on real samples, such as, plaster, clay, sand, and cloth. All these surfaces demonstrate significant deviation from Lambertian behavior. The reflectance measurements obtained are in strong agreement with the reflectance predicted by the proposed model. The paper is concluded with a discussion on the implications of these results for machine vision. Keywords: diffuse reflection, Lambertian model, surface roughness, masking, shadowing, interreflections, diffuse reflectance model, qualitative model, radiance measurements, image rendering, reflectance maps, lunar reflectance, shape from shading, photometric stereo, binocul~tr stereo and motion 1 Introduction One of the primary goals of a machine vision system is to recover physical properties of a scene from im- ages. Image brightness values are closely related to the reflectance properties of points in the scene. Hence, accurate reflectance models are fundamental to the ad- vancement of machine vision. Several mechanisms in- volved in the reflection process are reviewed in Nayar et al. (1991b) and Tagare and deFigueiredo (1991). These mechanisms, or components, can be classified into two broad categories; diffuse and specular. 1 A surface that obeys Lambert's Law (Lambert 1760) ap- pears equally bright from all viewing directions. This model for diffuse reflection was advanced by Lambert over 200 years ago and remains one of the most widely used models in machine vision. It is used explicitly by shape recovery techniques such as shape from shad- ing and photometric stereo. It is also invoked by vi- sion techniques such as binocular stereo and motion detection to solve the correspondence problem. In the field of remote sensing, the Lambertian model is of- ten used to apply brightness corrections to images of the same scene obtained under different illumination conditions. The widespread use of the Lambertian model arises from its simplicity and because it does rea- sonably well in approximating reflection from a wide range of matte surfaces. For several real-world objects, however, the Lam- bertian model can prove to be a poor and inadequate approximation to the diffuse component. Figure l(a) shows areal image of a clay vase obtained using a CCD camera. The vase is illuminated by a single distant light source in the same direction as the sensor. Figure 1(b) shows a rendered image of a vase with the same shape as the one shown in Figure l(a). This image is rendered using the Lambertian model, and the same illumination direction as in the case of the real vase. As expected, the Lambertian model predicts that the brightness of the cylindrical vase will decrease as we approach the occluding boundaries on both sides. However, the real vase is very flat in appearance, image brightness re- maining almost constant over the entire surface. The vase is clearly not Lambertian in reflectance. 2 The deviation from Lambertian behavior illustrated above can be significant for a variety of other real-world ma- terials, such as, concrete, sand, and cloth. An accu- rate model that describes diffuse reflectance from such