Subspace Joint Parameter Estimation Algorithm Based on Hilbert Transform for Constructing Analytical Signals in High-Precision Multi-Surface Phase-Shifting Interferometry

n advanced manufacturing and precision metrology, accurate measurement of the surface topography of each surface within transparent parallel optical components is crucial for controlling device performance. Multi-surface wavelength-tuned interferometry often suffers from spectral crowding and severe crosstalk due to the superposition of harmonic signals. Conventional FFT-based analysis is prone to spectral leakage under conditions of low signal-to-noise ratio (SNR) and non-integer period sampling, making it difficult to ensure measurement accuracy. To address these challenges, we propose a Hilbert-subspace Joint Parameter Estimation for Multi-surface Phase-shifting Interferometry (HJPE-MPI) algorithm. This algorithm utilizes the Hilbert transform to construct analytic signals, which are then used to form a Hankel matrix for subspace decomposition, enabling high-resolution frequency estimation of sampled interference signals. The estimated frequency offsets are subsequently used to correct the amplitude and phase of the harmonic components corresponding to each surface, thereby enabling joint and precise parameter localization. Simulation and experimental results demonstrate that the proposed method significantly outperforms conventional algorithms in the estimation accuracy of frequency, phase, and amplitude for multi-harmonic interference signals, and exhibits excellent repeatability and stability in measurements of a 30 mm-thick transparent plate.