We introduce a method for calculating the local 3D shape of a smooth object from its 2D shaded image. We assume a Lambertian shading model and orthogonal projection. Unlike previous work, we make no assumptions on the surface albedo or curvature nor on the position of the light sources. Rather, we develop the problem in the biologically inspired representation of orientation filters. This leads us to represent the shading patterns as covariant derivatives of differential forms and vector fields. We are able to solve implicitly for all possible surfaces and light source pairs that are possible given a local shaded image. However, this leads to a nonlinear system of partial differential equations (PDEs).

We analyze solutions to this PDE system in special cases (both restricted domain and restricted function classes). We consider a second order assumption and the restriction of this PDE system to critical points. We use some approximations to arrive at a global reconstruction method based on the Euler Lagrange equations and consider some topological constraints.