Description of complex optical surface based on sparse radial basis function approximation with spatially variable shape parameters

It is necessary to fit the discrete sampling value of the optical element surface obtained by measuring equipment, because the results of fitting are useful for manufacturing and optical design. The commonly used fitting methods are X-Y polynomial approximation, Zernike polynomial approximation and radial basis function (RBF) approximation. Compared with others, radial basis function is more suitable to fit the complex optical surface. However, the further improvement of fitting accuracy and cost are limited by the fixed shape parameter of the classic RBF approximation. In this paper, we propose the sparse radial basis function approximation with spatially variable shape parameters to fit discretely sampled optical surfaces. Our main purpose is to improve fitting accuracy and to reduce computational cost. Then, we analyze the impact of the spatial distribution of RBF nodes on fitting. Finally, we compare the accuracy and cost between the classic RBF approximation and the sparse RBF approximation with spatially variable shape parameters by fitting various complex surface.