Research Of Ellipse Fitting Problem Based On Second-order ADMM Algorithm

Quick and accurate ellipse detection has important applications in computer vision,intelligent manufacturing and other fields.In the Czochralski silicon single crystal control system,the diameter of silicon single crystal is measured mainly by fitting the high brightness halo at the solid-liquid interface.In order to further estimate the current aperture,it is necessary to design a fast and accurate ellipse fitting method.With the continuous improvement of the technical of crystal growth,the diameter measurement of Czochralski silicon single crystal growth with water cold jacket is worthy of attention.How to design the method that can ensure the fitting accuracy and speed is one of the urgent problems in the case of a small amount of data points.The main research contents and work of this paper are as follows:(1)For the ellipse fitting problem,the paper simplify the general formula of the ellipse to reduce the number of parameters fitted into four to extremely improve the computational efficiency.Subsequently,the objective function and constraints are established by comparing with the least squares support vector regression method,so that the ellipse fitting problem is converted into the optimization problem,which provides convenience for subsequent research.(2)In order to fit the ellipse quickly and obtain more accurate ellipse parameters,the paper determines the steps of ADMM algorithm to update variables alternately,and the subproblems that need to be optimized are summarized as unconstrained optimization problems.Subsequently,a hybrid BFGS algorithm is proposed by improving the BFGS algorithm,and combined with ADMM algorithm to further solve the ellipse fitting problem.Finally,the rapidity and accuracy of the proposed algorithm are verified by numerical simulations and experiments based on the actual data of crystal growth images captured by CCD camera.(3)For the diameter measurement problem of Czochralski silicon single crystal growth with water cold jacket,the paper proposes an improved geometric method to determine the coordinates of the center point of the ellipse based on the idea of data-driven and clustering algorithms.Firstly,the significance of adding water cold jacket and its influence on diameter measurement are introduced,and the problem is summarized as an ellipse fitting problem with fewer data points.Secondly,the proposed hybrid BFGS algorithm is used to solve the problem and analyze the application scope of the proposed method.Then,based on the parallel chord midline theorem an improved geometric method is proposed to determine the ellipse center by using the idea of clustering,neighborhood and dichotomy.In the end,the effectiveness and superiority of the proposed method is verified by simulation experiments.(4)In order to solve the long and short semi-axes of the ellipse under the condition of few data points,we supplement the data for the low experimental data points situation according to the geometric properties of symmetry in the center of the ellipse which is based on the center point of the ellipse obtained in(3).The coordinate of the center point of the ellipse is substituted into the objective function to further reduce the variable dimensions that need to be optimized.Then,ADMM algorithm and hybrid BFGS algorithm are used to solve the optimization problem.Finally,the feasibility and accuracy of the above method are verified by simulation experiments.