Closed Form Solution of Local Shape from Shading at Critical Points

In the theory of shape from shading, behaviours of the local solution around a critical point of the image play an important role. This paper shows that the second derivatives of the object surface can be locally determined at these image critical points. Closed form expressions of the surface second derivatives in terms of the second derivatives of the image brightness and of the reflectance map are shown. They are derived as follows: By differentiating the image irradiance equation twice at an image critical point, a set of polynomial equations is obtained that contains the second derivatives of the surface, of the image brightness and of the reflectance map. Regarding these equations as simultaneous equations for unknown surface second derivatives, they are algebraically solved and their explicit expressions are derived. Such a derivation is possible only at image critical points and is impossible at any other image point. The applicability of the derived expressions to noisy images is tested using synthetic images.