透视投影下混合表面3D重建的快速SFS算法

Fast Shape-From-Shading Algorithm for 3D Reconstruction of Hybrid Surfaces Under Perspective Projection

针对现有的透视投影下混合表面3D重建从明暗恢复形状(SFS)算法中辐照度方程求解耗时较长、误差较大的问题,提出了一种基于牛顿-拉弗森法的混合表面3D重建快速SFS算法。首先,采用Blinn-Phong模型表征物体表面的混合反射特性,结合摄像机的针孔透视投影模型,建立了与物体表面3D形貌信息相对应的图像偏微分辐照度方程;其次,将辐照度方程转换为关于物体表面高度梯度函数的方程,使用牛顿-拉弗森法迭代求解该方程的解;接着,将此解转换为哈密顿-雅可比偏微分辐照度方程,利用勒让德变换获得偏微分方程对应的哈密顿函数;最后,运用最优控制原理和迭代fast marching策略构建哈密顿函数的逼近方案并得到哈密顿-雅可比方程的黏性解,此黏性解代表了混合反射表面的高度值。实验结果表明:与现有的基于upwind格式的混合表面3D重建SFS算法相比,所提算法的运行时间得到大幅度的降低,获得的物体表面高度的平均绝对误差与均方根误差亦有较大程度的减小。

To address the problems associated with excessive time consumption and large errors in irradiance equation solutions using existing shape-from-shading(SFS)algorithms for three-dimensional(3 D)reconstruction of hybrid surfaces under perspective projection,a fast SFS algorithm based on the Newton-Raphson method is proposed.First,the Blinn-Phong model is adopted to characterize the hybrid reflection property of object surfaces.Then,an image partial differential irradiance equation that corresponds to the 3 D shape information of the surfaces is established by considering the pinhole camera perspective projection model.Second,the irradiance equation is transformed into a functional equation in the height gradient of object surfaces,which is iteratively solved using the Newton-Raphson method.Then,the solution is converted into a Hamilton-Jacobi partial differential irradiance equation whose Hamiltonian function can be obtained through Legendre transformation.Finally,a scheme to approximate the Hamiltonian function based on optimal control theory and iterative fast marching strategy is applied to obtain the viscosity solution of the Hamilton-Jacobi equation,which represents the heights of hybrid reflection surfaces.Experimental results demonstrate that the mean absolute error and root mean square error of the heights of the object surface decrease and the run time of the proposed algorithm is greatly reduced compared with the existing SFS algorithm for 3 D reconstruction of hybrid surfaces based on the upwind scheme.