Shape and Source from Shading
Well-known methods for solving the shape-from-shading problem require knowledge of the reflectance map. Here we show how the shape-from-shading problem can be solved when the reflectance map is not available, but is known to have a given form with some unknown parameters. This happens, for example, when the surface is known to be Lambertian, but the direction to the light source is not known. We give an iterative algorithm that alternately estimates the surface shape and the light source direction. Use of the unit normal in parameterizing the reflectance map, rather than the gradient or stereographic coordinates, simpliflies the analysis. Our approach also leads to an iterative scheme for computing shape from shading that adjusts the current estimates of the focal normals toward or away from the direction of the light source. The amount of adjustment is proportional to the current difference between the predicted and the observed brightness. We also develop generalizations to less constrained forms of reflectance maps.