## Abstract

In eye aberrometry it is often necessary to transform the aberration coefficients in order to express them in a scaled, rotated, and/or displaced pupil. This is usually done by applying to the original coefficients vector a set of matrices accounting for each elementary transformation. We describe an equivalent algebraic approach that allows us to perform this conversion in a single step and in a straightforward way. This approach can be applied to any particular definition, normalization, and ordering of the Zernike polynomials, and can handle a wide range of pupil transformations, including, but not restricted to, anisotropic scalings. It may also be used to transform the aberration coefficients between different polynomial basis sets.

© 2006 Optical Society of America

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### Equations (10)

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(1)
$${\mathbf{r}}^{\prime}=\mathbf{M}\left(\alpha \right)(\mathbf{r}-\mathbf{d}),$$
(2)
$${\mathbf{\rho}}^{\prime}=L\left(\mathbf{\rho}\right)=\beta \mathbf{M}\left(\alpha \right)(\mathbf{\rho}-\mathbf{\delta}),$$
(3)
$$W\left(\mathbf{\rho}\left(P\right)\right)=\sum _{j=0}^{M}{a}_{j}{Z}_{j}\left(\mathbf{\rho}\right),$$
(4)
$$W\left(\mathbf{\rho}\left(P\right)\right)={W}^{\prime}\left({\mathbf{\rho}}^{\prime}\left(P\right)\right)=\sum _{j=0}^{{M}^{\prime}}{a}_{j}^{\prime}{Z}_{j}\left({\mathbf{\rho}}^{\prime}\right),$$
(5)
$${\mathbf{a}}^{\prime}=\mathbf{T}\mathbf{a}.$$
(6)
$$\sum _{j=0}^{M}{a}_{j}{Z}_{j}\left(\mathbf{\rho}\right)=\sum _{j=0}^{M}{a}_{j}^{\prime}{Z}_{j}\left({\mathbf{\rho}}^{\prime}\right).$$
(7)
$$\mathbf{Z}\mathbf{a}={\mathbf{Z}}^{\prime}{\mathbf{a}}^{\prime}.$$
(8)
$${\mathbf{a}}^{\prime}={\left({\mathbf{Z}}^{\prime T}{\mathbf{Z}}^{\prime}\right)}^{-1}{\mathbf{Z}}^{\prime T}\mathbf{Z}\mathbf{a},$$
(9)
$$\mathbf{T}={\left({\mathbf{Z}}^{\prime T}{\mathbf{Z}}^{\prime}\right)}^{-1}{\mathbf{Z}}^{\prime T}\mathbf{Z}.$$
(10)
$$\mathbf{\rho}={L}^{\prime}\left({\mathbf{\rho}}^{\prime}\right)={\beta}^{\prime}\mathbf{M}\left({\alpha}^{\prime}\right)({\mathbf{\rho}}^{\prime}-{\mathbf{\delta}}^{\prime}),$$