Demonstration of Zernike Polynomial Fitting

Zernike polynomials are essential mathematical tools in optics for characterizing wavefront aberrations, providing a complete set of orthogonal basis functions defined over a unit circle to describe wavefront distortions. Implementing Zernike polynomial fitting in MATLAB R2006b typically involves the following key steps:

Data Preparation Phase Acquire wavefront phase data for fitting, typically obtained from interferometer measurements or simulation-generated data matrices. Data should be distributed in polar coordinates within the unit circle, with non-circular domain data requiring mask processing. Code implementation involves reading data files and applying circular masks using logical indexing.

Polynomial Order Selection Determine the Zernike polynomial order (e.g., 36 terms) based on aberration complexity. Lower-order terms correspond to basic aberrations like tilt and defocus, while higher-order terms describe more complex wavefront distortions. The selection can be automated using variance analysis or manual configuration through input parameters.

Core Coefficient Solving Solve for Zernike coefficients using least squares method. When constructing the design matrix: - Calculate corresponding Zernike basis function values for each data point - Use pseudoinverse (pinv) to handle potential rank deficiency issues - Consider weighted fitting to improve edge region accuracy MATLAB implementation typically involves creating a design matrix with zernfun() calculations and applying pinv() or backslash operator for coefficient estimation.

Reconstruction and Evaluation Reconstruct the wavefront using obtained coefficients and evaluate fitting quality through residual analysis. Common evaluation metrics include RMS error and PV value, with polar coordinate contour plots recommended for visualization. Implementation requires calculating reconstructed wavefronts and generating comparative plots using contourf() function.

Key Development Considerations: - Ensure numerical stability in polar-to-Cartesian coordinate conversions - Account for unit circle normalization effects on high-order terms - Implement memory management strategies (block processing for large datasets) Code should include safeguards against numerical instability and optimized memory allocation for large-scale computations.

This implementation can be extended to applications like adaptive optics system correction and corneal topography analysis. For more complex requirements, consider integrating GPU acceleration using Parallel Computing Toolbox or incorporating regularization methods to improve fitting performance through techniques like Tikhonov regularization.