Sparse reconstruction of wavefronts using an over-complete phase dictionary

Wavefront reconstruction is a critical component in various optical systems, including adaptive optics, interferometry, and phase contrast imaging. Traditional reconstruction methods often employ either the Cartesian (pixel) basis or the Zernike polynomial basis. While the Cartesian basis is adept at capturing high-frequency features, it is susceptible to overfitting and inefficiencies due to the high number of degrees of freedom. The Zernike basis efficiently represents common optical aberrations but struggles with complex or non-standard wavefronts such as optical vortices, Bessel beams, or wavefronts with sharp discontinuities. This paper introduces a novel approach to wavefront reconstruction using an over-complete phase dictionary combined with sparse representation techniques. By constructing a dictionary that includes a diverse set of basis functions-ranging from Zernike polynomials to specialized functions representing optical vortices and other complex modes-we enable a more flexible and efficient representation of complex wavefronts. Furthermore, a trainable rigid transform is implemented to account for misalignment. Utilizing principles from compressed sensing and sparse coding, we enforce sparsity in the coefficient space to avoid overfitting and enhance robustness to noise.