Windowed Fourier transform for fringe pattern analysis: theoretical analyses.

Area of Science:

• Optics and Photonics
• Signal Processing
• Image Analysis

Background:

• Fringe pattern analysis is crucial in various optical metrology applications.
• Existing methods like phase-shifting algorithms can be sensitive to noise and local frequency variations.
• Windowed Fourier ridges (WFR) and windowed Fourier filtering (WFF) algorithms have shown promise for improved fringe analysis.

Purpose of the Study:

• To theoretically analyze the local frequency and phase extraction errors of WFR and WFF algorithms.
• To demonstrate the effectiveness and accuracy of WFR and WFF algorithms in fringe pattern analysis.
• To compare the performance of WFR and WFF algorithms against traditional phase-shifting methods.

Main Methods:

• Theoretical analysis of local frequency and phase extraction errors.
• Simulation of four phase-shifted fringe patterns with local quadric phase and Gaussian noise.
• Implementation and error evaluation of WFR and WFF algorithms with specified window sizes (sigma(x)=sigma(y)=10 pixels).
• Comparison of error metrics (mean and standard deviation) with the standard four-step phase-shifting algorithm.

Main Results:

• The standard phase-shifting algorithm resulted in a phase error standard deviation of 0.7 rad.
• The WFR algorithm achieved a phase error standard deviation of approximately 0.02 rad and a local frequency error standard deviation below 0.01 rad/pixel.
• The WFF algorithm achieved a phase error standard deviation below 0.04 rad and a local frequency error standard deviation below 0.01 rad/pixel.
• Both WFR and WFF algorithms demonstrated unbiased estimation with low standard deviations for local frequencies and phase distributions.

Conclusions:

• Windowed Fourier transform-based algorithms (WFR and WFF) provide highly accurate local frequency and phase extraction in fringe pattern analysis.
• These algorithms offer significant improvements in reducing phase and frequency errors, especially in the presence of noise.
• The theoretical analysis confirms the effectiveness and versatility of WFR and WFF algorithms for various fringe patterns and noise models.