Fast Trigonometric Polynomial Approach for Computing the Shape from Shading Mohamed Eisa, Y. M. Fouda and Gamal. F. Elhadi Mathematics Department, Faculty of science, Mansoura University Mansoura 35516, Egypt. eisa@informatik.uni-freiburg.de , gamalfaruk@yahoo.com Abstract Several recently developed techniques for recons- truction surface shape from shading information estimate surface slopes with out ensuring that they integrable. This paper presents a new approach for estimating the shape of three dimension 3-D obje- ct from its two dimension 2-D shade image in ter-ms of a approximating the height map by a second order trigonometric polynomial. The proposed approach satisfies the integrability condition and does not need any boundary condition assumptio-ns, we have designed a stand-alone, flexible Matlab implementation that enables to evaluation the proposed approach. The experiments on real images show the approaches ability to reconstruc- tion the surface from SFS. Keywords: Computer vision, shape from shading, trigonometric polynomial approximate, gray sca- le, integrability. 1. Introduction Shape-from-Shading (SFS) is a fundamental prob-lem in Computer Vision. The common way to obtain shape information is to solve the image irradiance equation, which relates the reflectance map to image intensity. As this task is nontrivial, most of the works in the field employ simplifying assumptions, and in particular the assumption that projection of scene points during a photographic process is orthographic [5,17,1,13,16,21]. This resulted in low stability of reconstruction algorit- thms. The SFS problem consists of computing the three dimensional shape of a surface from the brightness variations in a black and white image of that surface. Pioneered by Horn [6], this problem has been central in the field of computer vision since the early days, because of several reasons, the interest in this problem has slightly decreased at the end of the 90s. First, due to the difficulty of the problem, progress in SFS research is very slow. Second, until recently, the results obtained on real images have been very disappointing. For example, in [21], Zhang et al. acknowledge failure. Third, the various con constraints imposed by the existing solutions to the SFS problem limit its applications. The SFS problem consists of recovering the shape of a scene from a single grey-level image, by means of the analysis of the shading. The craze for SFS in the past seems to have subsided, probably because of rather disappointing results on real images [21]. Nevertheless, several recent works [2,3,9,10,11] have (independently) attempted to modelize SFS in a more realistic way, in particular by considering perspective projection. The present work fits into this scheme; we outline a new modeling of the SFS problem and validate it through a practical application. Our final purpose is to design a system that “unwarps” the image, taken by a digital camera and we attempt to change this situation completely and hope to revive the interest of the community for this old problem and its applications. This article is organized as follows. We first review the reflectance map in section 2, and the problem formulation is given in section 3. Our trigonometric polynomial approach is presented in section 4. Section 5 presents our results and discussions. Finally, the paper summarizes and the future directions in section 6. 2. The Reflectance Map Let Z(x,y) be the unknown surface height of the 3-D object above the (x,y) plane, E(x,y) is defined GVIP Journal, Volume 5, Issue 9, Dec. 2005 21