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 affine 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.

翻译：波前重构是自适应光学、干涉测量和相衬成像等多种光学系统中的关键环节。传统重构方法通常采用笛卡尔（像素）基或泽尼克多项式基。笛卡尔基擅长捕捉高频特征，但由于自由度数量庞大，容易出现过拟合且效率低下。泽尼克基能有效表征常见光学像差，但对于光学涡旋、贝塞尔光束或具有尖锐不连续性的波前等复杂或非标准波前则难以处理。本文提出一种利用过完备相位字典结合稀疏表示技术的波前重构新方法。通过构建包含多样化基函数（从泽尼克多项式到表征光学涡旋及其他复杂模式的专用函数）的字典，我们实现了对复杂波前更灵活高效的表征。此外，该方法引入了可训练的仿射变换以校正对准误差。基于压缩感知与稀疏编码原理，我们在系数空间中施加稀疏性约束以避免过拟合，并增强对噪声的鲁棒性。