Closed-form attitude determination under spectrally varying illumination

When a Lambertian surface is illuminated by several chromatic lights the surface normals may be recovered from a single color image. A robust regression is used to find the ellipsoid in color space on which at least half the pixels lie. Then the matrix giving the linear relationship between the color and the surface normal, for non-outlier points is found as a root of the ellipsoid quadratic form. But this root is recovered only up to an arbitrary rotation. An integrability condition can be used to determine the correct rotation. The rotation of recovered surface normals is needed to align partial derivatives p and q with the camera plane and thus establish the object´s attitude. Here a new smoothness condition approximating the integrability condition is introduced that allows one to solve for the rotation matrix in closed form