一种非朗伯表面SFS的快速粘性解算法

A fast viscosity solution algorithm of shape from shading for non-Lambertian surfaces

针对传统的非朗伯表面明暗恢复形状(SFS)算法存在运行时间长、精度不高的问题,提出了一种粘性意义下的明暗恢复形状快速算法。首先,非朗伯物体表面采用Oren-Nayar模型描述其表面反射属性,摄像机镜头使用适合于实际成像过程的透视投影模式,同时假设光源置于镜头像方主点附近,建立了上述情况下的图像辐照度方程,此方程蕴含着非朗伯表面的高度信息。其次,将辐照度方程转化为Hamilton-Jacobi(H-J)类偏微分方程,运用Legendre变换和最优控制理论得到H-J方程对应的Hamilton函数的控制形式。接着,建立了Hamilton函数的逼近算法,使用非线性规划原理构建Hamilton函数最优问题的等价约束问题,利用得到的最优控制并通过Newton法最终得到了H-J方程的粘性解,该粘性解即是非朗伯物体表面的高度。实验结果表明,提出的算法与典型的基于Lax-Friedrichs方法的算法相比,所需要的运行时间大幅度减少,重构的物体表面的高度平均误差与均方根误差也有较大幅度降低。

To address the much time consuming and high error of traditional algorithms of shape from shading （SFS） for non-Lambertian surfaces, a fast SFS method in the viscosity sense is proposed. The Oren-Nayar model is used to characterize the reflectance property of the non-Lambertian surfaces, and the perspective projection model which is suitable for the actual imaging process is adopted to model the camera lens. Assuming that the light source is closed to the second principal point of the lens, the image irradiance equation which contains the height information of the surfaces is established. The equation is transformed into a Hamihon-Jacobi （H-J） partial differenti- al equation and the control formulation of its Hamihonian is conducted by using the Legendre transform and optimal control theory. The approximation algorithm of the Hamiltonian is setup and the equivalent constrained problem of the optimal problem of the Hamiltonian is formulated with the nonlinear programming principle. Finally, the viscosity solution of the H-J equation that is the height of the non-Lam- bertian surfaces is solved by adopting the got optimal control and Newton method. Compared with the typical algorithm based on Lax- Friedrichs method, experimental results demonstrate that the running time of the proposed algorithm is decreased greatly, and the mean absolute and root mean square errors of the height are also reduced.