Shadows, shading, and projective ambiguity

In a scene observed from a fixed viewpoint, the set of shadow curves in an image changes as a point light source (nearby or at infinity) assumes diffe rent locations. We show that for any finite set of point light sources illu minating an object viewed under either orthographic or perspective projecti on, there is an equivalence class of object shapes having the same set of s hadows. Members of this equivalence class differ by a four parameter family of projective transformations, and the shadows of a transformed object are identical when the same transformation is applied to the light source loca tions. Under orthographic projection, this family is the generalized bas-re lief (GBR) transformation, and we show that the GBR transformation is the o nly family of transformations of an object's shape for which the complete s et of imaged shadows is identical. Furthermore, for objects with Lambertian surfaces illuminated by distant light sources, the equivalence class of ob ject shapes which preserves shadows also preserves surface shading. Finally , we show that given multiple images under differing and unknown light sour ce directions, it is possible to reconstruct an object's shape up to these transformations from the shadows alone.