The joint illumination and motion space of video sequences

It has been proved that the set of all Lambertian reflectance functions obtained with arbitrarily distant light sources lies close to a 9D linear subspace. We extend this result from still images to video sequences. We show that the set of all Lambertian reflectance functions of a moving object at any position, illuminated by arbitrarily distant light sources, lies close to a bilinear subspace consisting of nine illumination variables and six motion variables. This result implies that, when the position and 3D model of an object at one instance of time is known, the reflectance images at future time instances can be estimated using the bilinear subspace. This is based on the fact that, given the illumination direction, the image of a moving surface cannot change suddenly over a short time period. We apply our theory to synthesize video sequences of a 3D face with various combinations of motion and illumination directions.