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Applied Optics

Applied Optics


  • Editor: James C. Wyant
  • Vol. 47, Iss. 29 — Oct. 10, 2008
  • pp: 5408–5419

Windowed Fourier transform for fringe pattern analysis: theoretical analyses

Qian Kemao, Haixia Wang, and Wenjing Gao  »View Author Affiliations

Applied Optics, Vol. 47, Issue 29, pp. 5408-5419 (2008)

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A windowed Fourier ridges (WFR) algorithm and a windowed Fourier filtering (WFF) algorithm have been proposed for fringe pattern analysis and have been demonstrated to be versatile and effective. Theoretical analyses of their performances are of interest. Local frequency and phase extraction errors by the WFR and WFF algorithms are analyzed in this paper. Effectiveness of the WFR and WFF algorithms will thus be theoretically proven. Consider four phase-shifted fringe patterns with local quadric phase [ c 20 = c 02 = 0.005   rad / ( pixel ) 2 ], and assume that the noise in these fringe patterns have mean values of zero and standard deviations the same as the fringe amplitude. If the phase is directly obtained using the four-step phase-shifting algorithm, the phase error has a mean of zero and a standard deviation of 0.7 rad . However, when using the WFR algorithm with a window size of σ x = σ y = 10 pixels, the local frequency extraction error has a mean of zero and a standard deviation of less than 0.01   rad / pixel and the phase extraction error in the WFR algorithm has a mean of zero and a standard deviation of about 0.02 rad . When using the WFF algorithm with the same window size, the phase extraction error has a mean of zero and a standard deviation of less than 0.04 rad and the local frequency extraction error also has a mean of zero and a standard deviation of less than 0.01   rad / pixel . Thus, an unbiased estimation with very low standard deviation is achievable for local frequencies and phase distributions through windowed Fourier transforms. Algorithms applied to different fringe patterns, different noise models, and different dimensions are discussed. The theoretical analyses are verified by numerical simulations.

© 2008 Optical Society of America

OCIS Codes
(100.2650) Image processing : Fringe analysis
(100.5070) Image processing : Phase retrieval
(120.2650) Instrumentation, measurement, and metrology : Fringe analysis
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(110.7410) Imaging systems : Wavelets

ToC Category:
Image Processing

Original Manuscript: June 27, 2008
Revised Manuscript: September 2, 2008
Manuscript Accepted: September 11, 2008
Published: October 8, 2008

Qian Kemao, Haixia Wang, and Wenjing Gao, "Windowed Fourier transform for fringe pattern analysis: theoretical analyses," Appl. Opt. 47, 5408-5419 (2008)

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