Nonlinear Least Squares Supervised Optimization Method And Its Preliminary Application

In a certain mathematical model,Newton method often converges to local extreme value,or even isn't able to solve the problem,because of the dependence on the initial value,the derivable requirement of object function and the ill-condition of coefficient matrix of the normal equation.Given the current estimated value,the essence of the nonlinear least squares iterative method can be describled as the process of solving the descent direction.It updates the current estimated value and makes the algorithm converge by using appropriate methods.From this point,by using supervised learning,the supervised optimization method for solving nonlinear squares problem was proposed in this paper.The average opitimation direction,which is an approximation of the direction from the current estimated value to the true value,was learned from a lot of samples in the train.The solution of problem can be seen as the approximation of the train sample.In order to test and verify the effectiveness of supervised optimization method,the preliminary application researches about single image space resection and face feature points' detection are made in this paper.Solving space resection of single image can be describled as a nonlinear least squares problem.But compared with the Euler angle method in photogrammetry,it is more difficult to converge due to the ill-condition of coefficient matrix of the normal equation caused by the ill-condition of model and the decrease of numerical stability.For this reason,according to the characteristics of supervised optimization method,the problem that how to solve space resection of single image by using supervised optimization method is researched in this paper.Experimental result show that the method in this paper can overcome the ill-condition caused by the coplanarity of ground control points in a manner.And it shows weak dependence of the initial value than the current method.It is difficult to sovle the facial feature points' detection in real time by using the normal methods for solving nonlinear least squares,because of the non-derivative of image feature function,the unkown of real shape when detecting facial feature points and the time consumption to calculate Hessian matrix and gradient matrix.For this reason,the problem that how to solve the facial feature point's detection by using supervised optimization method is researched in this paper.Experimental results show that the method in this paper has the same precision as the ASM method,but it is faster than ASM method and more eariser used in facial feature points' detection in real time.