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We found an error affecting the results presented in Figure 7 of our article “Orthogonal basis with a conicoid first mode for shape specification of optical surfaces". Here we publish the revised Fig. 7.
© 2016 Optical Society of America
In page 5459 of reference [1], third line from bottom, the value c = 0.261082 obtained for the best-fit sphere curvature used in formula (42) is wrong. Consequently, the results presented in Figure 7 (page 5461) are wrong as well.
The correct value for the best-fit sphere is c = 0.964400103. With this correction, Figure 7 of [1] should be replaced by the following new Fig. 1. Moreover, for the sake of clarity, we introduce in this new Figure 7 three minor modifications with respect to the Figure 7 given in [1]:
- - The root mean square error plotted in the vertical axis is now given in a logarithmic scale.
- - The new Figure 7 contains only the error given by three methods: Forbes (Qbfs), the new basis (q0 hyperbola), and Zernike (ZPs). That is, we have eliminated the new basis (q0 sphere) as it performs worse compared with the new basis (q0 hyperbola).
- - Now we applied the criterion of having similar number of terms for the three expansions. The three cases have two indexes: radial n and angular m. Thus, the number of terms is specified by giving values to those indexes. For the orthogonal systems (Zernike and the new basis) we always used complete radial orders. This means that for a given n, m goes from −n to +n. The difference is that for ZPs m increases in steps of 2, whereas in the new basis the step is 1. Then for a given n, we have (n + 1)(n + 2)/2 Zernike polynomials and (n + 1)2 terms of our new basis (that is based on spherical harmonics). In the Forbes system, the number of terms is (n + 1)(2m + 1) and m can take different values. Here we choose the value of m in such a way that the number of terms is close to that of our new basis. This happens for m = [n/2].
Fig. 1 RMSE in the approximation of the function Eq. (41) in [1] provided by the different approximations as a function of the number of terms used in the approximations. The vertical axis is given in a logarithmic scale.
For the same number of terms, Zernike polynomials provide the best fit, but the reason is that, for constant number of terms ZPs reach higher values for the radial order n. In this Figure, the maximum radial order for Forbes and our new basis was n = 6, whereas for Zernike the last point corresponds to n = 8, which explains the difference. Our new basis provides better fit than the Forbes Qbfs.
Funding
Ministerio de Economía y Competitividad (MINECO) award FIS2014-58303.
References and links
1. C. Ferreira, José L. López, R. Navarro, and E. Pérez-Sinusía, “Orthogonal basis with a conicoid first mode for shape specification of optical surfaces,” Opt. Express 24(5), 5448–5462 (2016). [CrossRef]
