Impossible Shaded Images

Abmact-In this correspondence, we show that images that could not have arisen from shading on a smooth surface with uniform reflecting properties and lighting exist. Much work has been done on recovering surface shape from images, and there has been some attention paid to the question of uniqueness. It has been shown, for example, that singular points curtail ambiguity. However, little has been said about the existence of solutions, perhaps because in practice, the given image is assumed to have arisen from a uniform, smoothly curved surface, and therefore, one b o w s that there must be a t least one solution. What if, however, the reflecting properties of the surface vary from place to place? What if the actual surface does not reflect light the way one has assumed or that the light source is not where it was thought to be? Will the solution only be warped by these departures from the ideal model, or may there in fact be situations where there is no smooth surface that could have given rise to the given shading pattern? Can the fact that a shaded image of some surface with spatially varying surface reflectance is impossible in this sense be used to detect surface albedo variations?