| 1. | Levenberg - marquardt method is one of the most important methods for solving systems of nonlinear equations
Levenberg - marquardt方法是求解非线性方程组的最重要的方法之一。 |
| 2. | It is well - known that when solving a system of nonlinear equations , the damped newton method is globally and superlinearly / quadratically convergent
求解非线性方程组的阻尼newton法不仅具有快速收敛的特点，而且有全局收敛性。 |
| 3. | Numerical computations show that the approach has global searching capability and can give satisfactory solutions , especially for the system of nonlinear equations , which is sensitive to the variables
数值计算结果表明此方法具有全局搜索性，特别是，它能够以满意的精度求出对未知数具有敏感性的非线性方程组的解。 |
| 4. | With the basis of the system of nonlinear equations which is established by minimizing the error quadratic sum of theoretical and actual shaded value of the points on the typical surface , the parameters of the illumination models can be firstly determined by means of the least - square procedure
首先，以典型曲面上各点的理论灰度值与实测灰度值的误差平方和最小为目标建立非线性方程组，以非线性最小二乘理论为基础，通过解非线性方程组确定光照模型各个参数值。 |
| 5. | Then 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 can be obtained from gauss - newton and levenberg - marquardt algorithms , so that the normal vector at that point of the surface can be defined
再根据多幅图像上固定位置一点的灰度值变化列多个非线性方程，以gauss - newton算法和levenberg - marquardt算法为基础求非线性方程组的最优近似解，将选取点的法向量确定下来。 |
| 6. | A complex particle swarm optimization ( cpso ) algorithm , which combines the advantages of method of complex ( mc ) and particle swarm optimization ( pso ) , is put forward to solve systems of nonlinear equations , and it can be used to overcome the difficulty in selecting good initial guess for newton ' s method and the inaccuracy of mc and pso due to being easily trapped into local minima for solving systems of nonlinear equations
摘要结合复形法与粒子群算法的优点，提出粒子群复形法，用于求解非线性方程组，以克服牛顿法初始点不易选择的问题，同时克服复形法与粒子群算法由于易陷入局部极值而导致方程组的解的精度不够的不足。 |
| 7. | 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
以非线性最小二乘理论为基础，以典型曲面上各点的理论灰度值与实测灰度值的误差平方和最小为目标建立非线性方程组，通过求解非线性方程组来确定光照模型的各个光照参数。 |
| 8. | 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 can be 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
根据多幅图像上固定位置一点灰度值的变化列多个非线性方程，通过求解该非线性方程组，确定出各选取点的法向量；然后通过复化积分确定选取点的高度值，并利用变分和有限差分思想对所得表面进行进一步的迭代和修正，以减小重构误差。 |
