Shape and Source from Shading

Abstract

Well-known methods for solving the shape-from-shading problern require knowledge of the reflectance map . Herewe show how the shape-from-shading problern can be solved wh en th e 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 which 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 Stereographie coordinates, simplifies the analysis . Our approach also Ieads to an iterative scheme for computing shape from shading that adjusts the current estimates of the local 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 co nstrained forms of reflectance maps.