TY - JOUR T1 - Lambertian reflectance and linear subspaces JF - Pattern Analysis and Machine Intelligence, IEEE Transactions on Y1 - 2003 A1 - Basri,R. A1 - Jacobs, David W. KW - 2D KW - 4D KW - 9D KW - analog; KW - analytic KW - characterization; KW - convex KW - convolution KW - distant KW - functions; KW - harmonics; KW - image KW - image; KW - intensities; KW - Lambertian KW - light KW - lighting KW - linear KW - methods; KW - nonnegative KW - normals; KW - object KW - optimization; KW - programming; KW - query KW - recognition; KW - reflectance; KW - reflectivity; KW - set; KW - sources; KW - space; KW - spherical KW - subspace; KW - subspaces; KW - surface AB - 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. VL - 25 SN - 0162-8828 CP - 2 M3 - 10.1109/TPAMI.2003.1177153 ER -