Fast evaluation of equal-spaced Zernike polynomial expansion samples

To develop a method to quickly calculate equal-spaced Zernike polynomial expansion samples on a rectangular or polar grid for analysis or display.

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.

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.

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.