A Fast Algorithm for the Decomposition of Functions Into Series of Zernike Polynomials

The Zernike transformation finds application in a range of scientific and technical fields, including mathematical modeling in Hartmann surface analysis of mirrors, diffraction theory of aberrations, optical design and testing, ophthalmology and adaptive optics, image analysis, aerodynamics, and radio astronomy. The process of decomposing an original function into a Fourier series using Zernike polynomials is highly computationally intensive. This complexity arises primarily from the repeated calculation of high-order factorials when determining the radial components of the polynomials, as well as the need for multiple computations of vector-matrix multiplications involving dense high-order matrices. This paper introduces an efficient method for compressing transformation matrices for Zernike functions, leveraging the fast discrete Fourier transform algorithm and the extra-component method. The proposed approach allows the efficient multiplication of the original transformation matrix by a vector in the trigonometric basis, thereby reducing the computational complexity of the numerical method. Computational experiments have demonstrated that the proposed procedures can reduce computation time by several orders of magnitude compared to direct matrix-vector multiplication in the transformation.