What is the set of images of an object under all possible lighting conditions?

The appearance of a particular object depends on both the viewpoint from which it is observed and the light sources by which it is illuminated. If the appearance of two objects is never identical for any pose or lighting conditions, then-in theory - the objects can always be distinguished or recognized. The question arises: What is the set of images of an object under all lighting conditions and pose? In this paper, we consider only the set of images of an object under variable illumination (including multiple, extended light sources and attached shadows). We prove that the set of n-pixel images of a convex object with a Lambertian reflectance function, illuminated by an arbitrary number of point light sources at infinity, forms a convex polyhedral cone in IRⁿ and that the dimension of this illumination cone equals the number of distinct surface normals. Furthermore, we show that the cone for a particular object can be constructed from three properly chosen images. Finally, we prove that the set of n-pixel images of an object of any shape and with an arbitrary reflectance function, seen under all possible illumination conditions, still forms a convex cone in IRⁿ. These results immediately suggest certain approaches to object recognition. Throughout this paper, we offer results demonstrating the empirical validity of the illumination cone representation