A piecewise quadratic approach to single image shape from shading

When light from a distant point source illuminates a smooth Lambertian surface, it will generate a pattern of shading. The problem of recovering the original shape from this shading pattern is the so-called shape-from-shading problem for the Lambertian surface. In Brooks et al. (1994), a variational method of shape recovery is presented, which involves integrating from a singular point on the original surface along special curves called base characteristics. There are two major difficulties with this method, which are due to the fact that image data is typically discrete, so that shading information is only known at finitely many points. The first difficulty is that of localising the singular point, since small errors at the beginning of the calculation will result in large errors at the end. The second obstacle is that the base characteristic curves can only be approximated very poorly on the discrete domain. In this paper we present a non-variational method for computing shape from shading, which aims to overcome these problems with the use of continuous quadratic approximations to the discrete data. The method yields a considerable improvement in maximum error over the one presented by Brook et al.