A Closed-Form Analytical Conversion between Zernike and Gatinel–Malet Basis Polynomials to Present Relevant Aberrations in Ophthalmology and Refractive Surgery

Abstract

The Zernike representation of wavefronts interlinks low- and high-order aberrations, which may result in imprecise clinical estimates. Recently, the Gatinel–Malet wavefront representation has been introduced to resolve this problem by deriving a new, unlinked basis originating from Zernike polynomials. This new basis preserves the classical low and high aberration subgroups’ structure, as well as the orthogonality within each subgroup, but not the orthogonality between low and high aberrations. This feature has led to conversions relying on separate wavefront reconstructions for each subgroup, which may increase the associated numerical errors. This study proposes a robust, minimised-error (lossless) analytical approach for conversion between the Zernike and Gatinel–Malet spaces. This method analytically reformulates the conversion as a nonhomogeneous system of linear equations and computationally solves it using matrix factorisation and decomposition techniques with high-level accuracy. This work fundamentally demonstrates the lossless expression of complex wavefronts in a format that is more clinically interpretable, with potential applications in various areas of ophthalmology, such as refractive surgery.