Iterative root-multiple signal classification algorithm for eliminating parasitic reflections of transparent planar elements

The figure measurement of transparent planar elements is a critical challenge in modern high-accuracy optical systems. The Fourier-based multi-frequency fringe algorithm is promising but still struggles to separate closely spaced frequencies, especially with short data records, causing spectral leakage and overlap that undermine surface-reconstruction accuracy. To address this, this paper proposes a phase measuring deflectometry method combining the root-multiple signal classification (Root-MUSIC) algorithm with the non-dominated sorting genetic algorithm II (NSGA-II). First, the covariance matrix of intensity sequences from each pixel is decomposed via eigenvalue analysis to estimate the noise subspace of the transparent planar, providing initial screen coordinates of front and rear surface reflections. A nonlinear optimization framework based on NSGA-II then minimizes the mean square error (MSE) between actual sequence values and model estimated values, iterating until convergence. Numerical simulations on a 10 mm-thick glass element show a root mean square (RMS) reconstruction error of 5 nm, while experimental tests on a 55 mm-diameter window glass achieve an RMS error of 59 nm compared to interferometry. Both results demonstrate that Root-MUSIC enhances frequency resolution than the multi-frequency method, while parallel computation improves optimization efficiency. This method enables non-destructive, high-precision in-situ measurement of the transparent planar element.