Fast evaluation of equal-spaced Zernike polynomial expansion samples.

PURPOSE: To develop a method to quickly calculate equal-spaced Zernike polynomial expansion samples on a rectangular or polar grid for analysis or display. METHODS: It is well known that a Zernike polynomial expansion can be converted into an equivalent rectangular or polar two-dimensional Taylor polynomial expansion. It is also known how to quickly calculate equal-spaced polynomial samples using difference equations. Using these two techniques, a software class was developed that provides fast evaluation of Zernike polynomial expansion samples on a rectangular or polar grid. To test the method, the time for the direct calculation of 10th order Zernike polynomial expansion was compared to the difference equation approach for a 1000x1000 sample grid. RESULTS: The direct calculation of the 10th order Zernike polynomial expansion required over 400 times more processing time than the difference equation technique for a 1000x1000 sample grid. The largest difference in calculated values between the two techniques was negligible, indicating 11 digits of accuracy when using double precision variables. CONCLUSIONS: The difference equation approach proves to be a fast and accurate method to calculate equal-spaced Zernike polynomial expansion samples on a rectangular or polar grid. This algorithm has application in both the analysis of optical systems and display of results.