Application of Computer Algebra to the Reconstruction of Surface from Its Photometric Images

Abstract

This paper addresses the problem of reconstructing the shape of an unknown Lambertian surface given in the 3D space by a continuously differentiable function \(z = u(x,y)\). The surface is reconstructed from its photometric images obtained by its successive illumination with three different remote light sources. Using computer algebra methods, we show that the unique solution of the problem, which exists in the domain \(\Omega \) of all three images, can be continued beyond this domain based on the solutions obtained for any pair of the three images. To disambiguate the reconstruction of the surface from its two images, we compute the corresponding value of the parameter \(\varepsilon \) at the boundary of the domain \(\Omega \). Soundness of the theoretical results is confirmed by simulating photometric images of various surfaces.