Shape Fitting for the Shape Control System of Silicon Single Crystal Growth

Abstract

Shape fitting, including straight line and ellipse fitting, plays an important role in the (cylinder-) shape control system of silicon single crystal growth, because the straight lines and ellipse in the crystal image contain the important horizontal circle center and diameter information. This information can be used as control variables so that the grown crystal approximates to a perfect cylinder, and thus can be used as high-quality source materials. In this paper, we develop new straight line and ellipse fitting algorithms. The key points are as follows. We formulate the two-dimensional (2-D) binary image into a single-snapshot array signal of a virtual sensor array, and casts the angle estimation problem of straight lines into the direction finding one of virtual incoming sources. Based on the virtual array manifold and potential incoming angles, the relevant over-complete dictionary is constructed, and thus a sparse regression problem is formed. To solve such a regression problem, we introduce the weight vector sparsity term into the conventional linear least-squares support vector regression framework to estimate the angles of these straight lines. Based on the estimated angles and potential offsets, another over-complete dictionary is constructed, and thus the image can be looked upon as the sparse representation of these dictionary atoms. Since the constructed dictionary is of the same size as the image, we use the compressed sensing theory to reduce the relevant dimensionality and then apply the aforementioned sparse regression method to obtain the relevant offsets of these straight lines. We derive a new second-order polynomial of ellipse equation to obtain the ellipse parameters to avoid the trival solution from the conventional polynomial model. Some simulation and experimental examples are given to illustrate the effectiveness of the proposed algorithms.