# Fourier spectra for nonuniform phase-shifting algorithms based on principal component analysis

## TL;DR

This paper introduces a mathematically rigorous, error-corrected nonuniform phase-shifting algorithm based on PCA, with detailed analysis of its frequency transfer function, harmonic robustness, and SNR, improving phase demodulation accuracy.

## Contribution

It presents a novel PCA-based nPSA with a comprehensive mathematical framework, including formulas for its FTF, correction methods, and performance metrics, addressing previous limitations.

## Key findings

- PCA-nPSA can be modeled as a linear quadrature filter.
- The FTF explains why plain PCA fails with nonuniform phase shifts.
- The proposed formulas enable unbiased performance evaluation.

## Abstract

We develop an error-free, nonuniform phase-stepping algorithm (nPSA) based on principal component analysis (PCA). PCA-based algorithms typically give phase-demodulation errors when applied to nonuniform phase-shifted interferograms. We present a straightforward way to correct those PCA phase-demodulation errors. We give mathematical formulas to fully analyze PCA-based nPSA (PCA-nPSA). These formulas give a) the PCA-nPSA frequency transfer function (FTF), b) its corrected Lissajous figure, c) the corrected PCA-nPSA formula, d) its harmonic robustness, and e) its signal-to-noise-ratio (SNR). We show that the PCA-nPSA can be seen as a linear quadrature filter, and as consequence, one can find its FTF. Using the FTF, we show why plain PCA often fails to demodulate nonuniform phase-shifted fringes. Previous works on PCA-nPSA (without FTF), give specific numerical/experimental fringe data to "visually demonstrate" that their new nPSA works better than competitors. This often leads to biased/favorable fringe pattern selections which "visually demonstrate" the superior performance of their new nPSA. This biasing is herein totally avoided because we provide figures-of-merit formulas based on linear systems and stochastic process theories. However, and for illustrative purposes only, we provide specific fringe data phase-demodulation, including comprehensive analysis and comparisons.

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Source: https://tomesphere.com/paper/1904.01071