Fast Evaluation of Equal-spaced Zernike Polynomial Expansion Samples
Abstract
<H4>PURPOSE</H4><p>To develop a method to quickly calculate equal-spaced Zernike polynomial expansion samples on a rectangular or polar grid for analysis or display. </P> <H4>METHODS</H4><p>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 1000×1000 sample grid. </P> <H4>RESULTS</H4><p>The direct calculation of the 10th order Zernike polynomial expansion required over 400 times more processing time than the difference equation technique for a 1000×1000 sample grid. The largest difference in calculated values between the two techniques was negligible, indicating 11 digits of accuracy when using double precision variables. </P> <H4>CONCLUSIONS</H4><p>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. [<CITE>J Refract Surg</CITE>. 2010;26:61-65.] </P> <P>doi:10.3928/1081597X-20101215-10 </P> <H4>AUTHORS</H4> <P>From Sarver and Associates Inc, Carbondale, Ill (Sarver); and Solutions-4-C, Corona, Calif (Hall). </P> <P>The authors have no proprietary interest in the materials presented herein. </P> <P>This paper was presented at the Lens, Refractive & Wavefront Summit ARI/WFC 2009; March 5-7, 2009; Alicante, Spain. </P> <P>Correspondence: Edwin J. Sarver, PhD, 131 Phillips Rd, Carbondale, IL 62902. Tel: 618.529.4225; Fax: 618.457.5600; E-mail: <A HREF="mailto:[email protected]">[email protected]</A> </P>
