Study On 3D Surface Reconstruction Based On 2D Gray Image In Reverse Engineering

Measuring method and technique that 3D surface shape is recovered from 2D gray image (shape from shading) can obtain surface shape by natural way and easily realize the information integration of intersystem. Particularly, this technique belongs to non-contact measurement, and it has become a puzzle and hotspot question in image measuring domain based on vision in recent years. In this paper, with high resolution digital camera as experimental means, and based on principle of photometry and theory and numerical solution for nonlinear partial differential equation, research is developed around the theory and technique to reconstruct 3D surface shape from 2D gray image. The parameters of illumination model are determined by optimization method, so as to reduce the error caused inaccurate illumination parameters. It is our purpose to do some initial work in order to develop non-contact measuring method and technique for complex surface (free-form surface) based on shape from shading.For consideration of principle of photometry and experimental research, with 2D gray image of typical surface (spherical surface) taken with digital camera, and by use of nonlinear least square optimizing method, the illumination parameters of four local illumination models, that are Lambert, Phong, Torrance-Sparrow and Cook-Torrance model, are determined. And with statistics analytical principle, regression analyses are made to compare the fitting precision of these illumination models under the current experiment condition. Comparing Torrance-Sparrow model with others, an improved illumination model for measurement of surface is developed.With the basis of the theory and numerical solution of ideal Lambert diffuse model on shape from shading, the image irradiance equation based on improved illumination model is derived, thereby the fact is disclosed that the question of reconstruction from a single gray image is how to solve a nonlinear partial differential equation. In view of ill-posed characteristic of the equation, with variational principle, solving of a nonlinear hyperbolic equation is attributed to minimization of a functional. So the equation has unique solution through regularization theory. Then variables of the functional are discretized and linearized with finite difference and nonlinear least squares techniques, accordingly minimization of the functional is converted into standard least square question.With a view to the difficulty and precision of surface reconstruction from a single image, the reconstruction from monocular multiple images is developed in the paper. With the basis of nonlinear least squares theory, the system of nonlinear equations is established by minimizing the error quadratic sum of theoretical and actual gray level of the points on the typical surface, and the parameters of the illumination models can be determined by means of the least-squares procedure. A new system of nonlinear equations can be formed from the variation gray of same point in multiple images and the optimum solution of the system canbe obtained, so that the normal vector at that point of the surface can be defined. Then we can get the surface height at the point by applying composite numerical integration. According to variational calculus and finite difference method, the fitted surface is further iterated and modified, so the reconstruction error can be reduced.The accuracy of the algorithm is verified with synthetic and real image of spherical surface. The result of experiment and calculation shows that the algorithm is effective and feasible and has stated application value. 3D surface reconstruction software based on 2D gray images has been developed with Visual C++ Programming language on the platform of Windows operation system.