Q2dChebyshevFit: Fit MSF surface to 2dQ polynomials

Rapid and accurate surface fitting that supporting thousands of Q terms

Function for fitting a surface to 2DQ polynomials

Usage:

• [amn, bmn, z_recon_map] = Q2dChebyshevFit(z_map, M, N, J, K)

Description:

• This function takes as input a freeform surface and fits it to 2D-Q polynomials, returning the coefficients of the fit as well as the reconstructed map.

Inputs:

• z_map - The sag table of the freeform surface.
• M - The highest azimutal fitting order.
• N - The highest radial fitting order.
• J - The sampling number on azimuthal direction, default: 2M+1
• K - The sampling number on radial direction, default: 2(N+1)

Outputs:

• amn, bmn - The fitted coefficients z_recon_map
• The surface calculated by amn, bmn coefficients

Author: Yiwen Fan ([e-mail])

Date: 05/22/2025

Acknowledgments:

• This fitting algorithm was completely developed by Professor Greg Forbes. For detailed deduction please refer to (1) https://doi.org/10.1364/OE.18.013851 (2) https://doi.org/10.1364/OE.18.019700 (3) https://doi.org/10.1364/OE.20.002483 (4) https://doi.org/10.1364/OE.21.019061 (5) https://doi.org/10.1117/12.2191135
• This work was partially contributed by Ilhan Kaya and Nick Takaki, who wrote all the supporting functions attached below. I (Yiwen) have made some modifications to handle the numerical accuracy when comes to high order.
• I (Yiwen) use traditional least-square fitting for getting the rotationally-symmetric component (Qbfs) for a stabler performance. If you know how to properly handle the Qbfs fitting with decent numerical accuracy, please let me know ♥
• There is a Python package for 2d-Q fitting: https://github.com/npdata/Scikit-qfit/tree/master

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There is another function I developed that fits MSF surfaces to Zernike polynomials https://www.mathworks.com/matlabcentral/fileexchange/132713-zernikelegendrefit-fast-precise-high-capacity-fitting