Weighted Zernike expansion with applications to the optical aberration of the human eye

Coefficients for $G_n^m$ , $\gamma =0$ , 0.5, $0⩽n⩽4$ ^a

^a  The numbers enclosed by parentheses are the coefficients for the weighted Zernike polynomials when $\gamma =0.5$ .

$(V_n^m,V_{n^{\prime }}^{m^{\prime }})$ : $n=3$ , $m=3$ ^a

^a  The integration for $(V_n^m,V_{n^{\prime }}^{m^{\prime }})$ is performed with the trapezoid rule. For ${(V_n^m,V_{n^{\prime }}^{m^{\prime }})}_{\mathit{\text{new}}}$ , the integration is done with weights that take into account the expected circular shape of the aperture.

Table 1

Coefficients for $G_n^m$ , $\gamma =0$ , 0.5, $0⩽n⩽4$ ^a

^a  The numbers enclosed by parentheses are the coefficients for the weighted Zernike polynomials when $\gamma =0.5$ .

Table 2

$(V_n^m,V_{n^{\prime }}^{m^{\prime }})$ : $n=3$ , $m=3$ ^a

^a  The integration for $(V_n^m,V_{n^{\prime }}^{m^{\prime }})$ is performed with the trapezoid rule. For ${(V_n^m,V_{n^{\prime }}^{m^{\prime }})}_{\mathit{\text{new}}}$ , the integration is done with weights that take into account the expected circular shape of the aperture.