%0 Journal Article %J Pattern Analysis and Machine Intelligence, IEEE Transactions on %D 2003 %T Lambertian reflectance and linear subspaces %A Basri,R. %A Jacobs, David W. %K 2D %K 4D %K 9D %K analog; %K analytic %K characterization; %K convex %K convolution %K distant %K functions; %K harmonics; %K image %K image; %K intensities; %K Lambertian %K light %K lighting %K linear %K methods; %K nonnegative %K normals; %K object %K optimization; %K programming; %K query %K recognition; %K reflectance; %K reflectivity; %K set; %K sources; %K space; %K spherical %K subspace; %K subspaces; %K surface %X We prove that the set of all Lambertian reflectance functions (the mapping from surface normals to intensities) obtained with arbitrary distant light sources lies close to a 9D linear subspace. This implies that, in general, the set of images of a convex Lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low-dimensional linear subspace, explaining prior empirical results. We also provide a simple analytic characterization of this linear space. We obtain these results by representing lighting using spherical harmonics and describing the effects of Lambertian materials as the analog of a convolution. These results allow us to construct algorithms for object recognition based on linear methods as well as algorithms that use convex optimization to enforce nonnegative lighting functions. We also show a simple way to enforce nonnegative lighting when the images of an object lie near a 4D linear space. We apply these algorithms to perform face recognition by finding the 3D model that best matches a 2D query image. %B Pattern Analysis and Machine Intelligence, IEEE Transactions on %V 25 %P 218 - 233 %8 2003/02// %@ 0162-8828 %G eng %N 2 %R 10.1109/TPAMI.2003.1177153