ZernikeLegendreFit: Fast, precise, high-capacity fitting

Rapid and accurate surface fitting via orthogonality and Legendre quadrature, supporting thousands of Zernike terms.

Function for fitting a surface with Zernike polynomials

Author: Yiwen Fan ([e-mail])

Institute of Optics, University of Rochester

Usage:

[output_coeff, z_recon_map] = ZernikeLegendreFit(z_map, index_type, coeff_max,

J, K, center_x, center_y)

04/30/2026 Ver 1.3.0 update - Recursively calculate {jacobi(m, n) * u^m} as a whole for better numerical stability

12/01/2025 Ver 1.2.1 update - Clarify the X, Y coordinate mapping of the input z_map. No functional update

01/20/2025 Ver 1.2 update - Support rectangular input matrices z_map (raw and column number does not have to be equal)

10/26/2024 Ver 1.1 update - Support Fringe Zernike index

Description:

This function fits a given surface to Zernike polynomials, supporting up to hundreds of terms. It returns the coefficients of the fit and the reconstructed map.

Inputs:

z_map - The sag table of the freeform surface. Please note that z_map(1, 1) corresponding to normalized x = -1, y = -1 position: first row corresponding to y = -1 horizontal line (the bottom line in Cartesian coordinate system), first column corresponding to x = -1 vertical line.

index_type - A string presenting the Zernike type. Currently support "2indices" (default) and "fringe".

coeff_max - The highest fitting order.

1) if index_type == "2indices", this should be a 1x2 cell where ~{1} = M, ~{2} = N.

2) if index_tyoe == "fringe", this shoule an integer J.

J - The sampling number on azimuthal direction, default: 2*M+1

K - The sampling number on radial direction, default: 2*(N+1)

center_x, center_y - The center of the freeform surface, default: will be calculated by FindCenter function.

Outputs:

output_coeff - The fitted coefficients,

1) if index_type == "2indices", this should be a 1x2 cell, where ~{1} = amn, ~{2} = bmn.

2) if index_type == "fringe", this should be a 1x1 cell, where ~{1} = cj.

z_recon_map - The reconstructed surface

Acknowledgments:

This work was inspired by the research of Greg Forbes and funded by the National Science Foundation I/UCRC Center for Freeform Optics (IIP-1822026, IIP-1822049).

Part of this work was contributed by Ilhan Kaya in 2010, who wrote the initial jacobiZernike function. This function was later revised by Yiwen Fan and renamed jacobiZernike_table.