Analysis of third order spherical aberration with the continuous wavelet transform

There are several numerical techniques to solve the value of aberration coefficients. One classical technique is the Gaussian elimination method, which has been described in most standard numerical analysis textbooks, such as Ralston's text, the conventional direct inversion method is numerically unstable. To obtain the Zernike coefficients form a samples wavefront with inherent measurement noise, the classical least-squares matrix inversion method and the Gram-Schmidt orthogonalization method would become ill- conditioned due to an improper data sampling. In this paper, we present the continuous wavelet transform (CWT) technique to find the defocus aberration and 3rd order spherical aberration coefficients. The technique we proposed is superior to the conventional methods in two ways. (1) Our method is much faster than the conventional methods, especially in applications with a few sampling points. (2) Our method is also more accurate in fitting aberration coefficients than the conventional methods, particularly in applications involving noise. Furthermore, the aberration coefficients determined through the CWT are independent of the order of the polynomial expansion. So we can find a true value from the datum of fitting.