The direct computation of height from shading

We present a method of recovering shape from shading that solves directly for the surface height. By using a discrete formulation of the problem, we are able to achieve good convergence behavior by employing numerical solution techniques more powerful than gradient descent methods derived from variational calculus. Because we directly solve for height, we avoid the problem of nding an integrable surface maximally consistent with surface orientation. Furthermore, since we do not need additional constraints to make the problem well posed, we use a smoothness constraint only to drive the system towards a good solution; the weight of the smoothness term is eventually reduced to near zero. Also, by solving directly for height, we can use stereo processing to provide initial and boundary conditions. Our shape from shading technique, as well as its relation to stereo, is demonstrated on both synthetic and real imagery.