Zernike by One Pascal Triangle - A high performance, low memory cost and flexible computation scheme for Zernike polynomials

This work uncovers two hidden cases of blockwise recurrence in Zernike computations. Based on these findings, a new computation scheme for Zernike polynomials is proposed. It uses one Pascal 's triangle for all internal factors, thus avoiding the computation of factorials, cos/sin functions, and matrix inversions. It offers both a direct transformation method and a blockwise recursive method for calculating Zernike basis functions, thereby fulfilling the requirements for high accuracy, high speed, low memory footprint, and great application flexibility.