Impossible shaded images

Abstract: In this paper, we show that there exist images that could not have arisen from shading on a smooth surface with uniform reflecting properties and lighting. 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 and occluding boundaries often curtail ambiguity. But 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 so one knows that there must be at 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 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?