Tsai and Shah applied the discrete approximation of the gradientrst, then employed the linear approximation of the reflectance function in terms of the depth directly. Their algorithm recovered the depth at each point using a Jacobi iterative scheme. ( P. Tsai and M. Shah. Shape from shading using linear approximation. Image and Vision Computing, 12(8):487–498, 1994.)

# reflectance function

# Shape from shading Pentland approach

Pentland used the linear approximation of the reflectance function in terms of the surface gradient, and applied a Fourier transform to the linear function to get a closed form solution for the depth at each point. (Pentland takes the Fourier transform of both sides of the equation). (Pentland, A., “Shape Information From Shading: A Theory About Human Perception,” Computer Vision., Second International Conference on , vol., no., pp.404-413, 5-8 Dec 1988.)