Depth and shape from shading using the photometric stereo method

The independent calculation of local position and orientation of the Lambertian surface of an opaque object is proposed using the photometric stereo method. A number of shaded video images are taken using different positions of an ideal point light source which is placed close to the object. Normally, three images are required for a uniform and four for a textured Lambertian surface. By restricting three light sources to lie in a straight line, the depth calculations for an arbitrary surface with textured Lambertian reflection characteristics can be also determined; however, in this case the orientation of the surface cannot be calculated independently. It is shown that for both uniform and textured Lambertian surfaces the equations which are functions of three independent variables, namely, depth (D) and surface normal direction vector (n = [p, q, − 1]), can be reduced to a single nonlinear equation of depth, i.e., the distance between the camera and the point on the surface. Both convergence and a unique solution are ensured because of the simple behavior of the nonlinear equation within a practical range of depth and gradient values. The robustness of the algorithm is demonstrated by synthetic as well as experimental data. The calculation of the approximate positions and orientations of discontinuous surfaces is demonstrated when random noise is added to the synthetically calculated image intensities. Two parallel planes with a gap, two sloped planes, and a spherical surface are used to demonstrate that the algorithms work well. An important feature of calculating both depth and orientation independently is that for smooth surfaces they must obey the partial differential expressions p = δDδxand q = δDδy. If we are certain that the experimental errors are within a known limit then the numerical approximation to these partial derivative expressions can be used to determine discontinuities within the image. On the other hand, if we know that the surfaces are smooth then errors in the numerical evaluation of these differential expressions allow the estimation of experimental errors.