Wiener phase-shifting algorithms for maximum signal-to-noise ratio phase demodulation

In this work, we introduce a new class, to our knowledge, of temporal phase-shifting algorithms (PSAs), called Wiener PSAs (WPSAs). These algorithms combine the optimal Wiener least-squares noise-reduction criterion with the spectral constraints required for any valid PSA in phase-shifting interferometry. Importantly, WPSAs are not just a straightforward extension of the Wiener filter; rather, they constitute an entirely new class of quadrature filters that share with Wiener filtering only the principle of optimal mean-square error minimization. Unlike the conventional Wiener filter design, WPSAs incorporate, via Lagrange multipliers, explicit frequency transfer function (FTF) quadrature conditions: full suppression of the DC and the conjugate signals at frequencies ω∈[-ω₀,0], while keeping the desired quadrature at ω=+ω₀. The WPSA synthesis yields a complex quadrature filter that minimizes output noise power while guaranteeing exact phase demodulation from temporal fringes corrupted by additive white Gaussian noise (AWGN). The resulting WPSAs achieve the highest attainable signal-to-noise ratio (SNR) for any given number of interferograms N and phase-step ω₀, even when the standard least-squares condition ω₀=2π/N is not satisfied. Practical examples with common phase-step values confirm that WPSAs consistently outperform existing PSAs in noise robustness. This work not only extends the optimal Wiener criterion to quadrature filtering but also highlights the essential role of the FTF formalism in formulating and solving quadrature-constrained optimization problems for high-precision, low-noise phase retrieval.