Windowed Fourier transform for fringe pattern analysis: theoretical analyses.

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 sigma(x)=sigma(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.

[1]  M. Takeda,et al.  Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry , 1982 .

[2]  Guillermo H. Kaufmann,et al.  Speckle noise reduction in television holography fringes using wavelet thresholding , 1996 .

[3]  Richard Kronland-Martinet,et al.  Characterization of acoustic signals through continuous linear time-frequency representations , 1996, Proc. IEEE.

[4]  E Trouvé,et al.  Fringe detection in noisy complex interferograms. , 1996, Applied optics.

[5]  Manuel Servin,et al.  Adaptive quadrature filters and the recovery of phase from fringe pattern images , 1997 .

[6]  Otmar Loffeld,et al.  Presentation of an improved phase unwrapping algorithm based on Kalman filters combined with local slope estimation , 1997 .

[7]  Henri Maître,et al.  Improving phase unwrapping techniques by the use of local frequency estimates , 1998, IEEE Trans. Geosci. Remote. Sens..

[8]  Stephan Waldner,et al.  A simple and effective method for filtering speckle-interferometric phase fringe patterns , 1999 .

[9]  M Servin,et al.  Phase unwrapping through demodulation by use of the regularized phase-tracking technique. , 1999, Applied optics.

[10]  M Servin,et al.  Regularization methods for processing fringe-pattern images. , 1999, Applied optics.

[11]  Alejandro Federico,et al.  Comparative study of wavelet thresholding methods for denoising electronic speckle pattern interferometry fringes , 2001 .

[12]  Chung Ki Hong,et al.  Least-squares phase estimation with multiple parameters in phase-shifting electronic speckle pattern interferometry. , 2003, Journal of the Optical Society of America. A, Optics, image science, and vision.

[13]  Kemao Qian.  Windowed Fourier transform method for demodulation of carrier fringes , 2004 .

[14]  Qian Kemao,et al.  Windowed Fourier transform for fringe pattern analysis: addendum. , 2004, Applied optics.

[15]  Qian Kemao,et al.  Windowed Fourier transform for fringe pattern analysis. , 2004, Applied optics.

[16]  U. Schnars,et al.  Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques , 2004 .

[17]  Seah Hock Soon,et al.  Two-dimensional windowed Fourier frames for noise reduction in fringe pattern analysis , 2005 .

[18]  Jingang Zhong,et al.  Multiscale windowed Fourier transform for phase extraction of fringe patterns. , 2007, Applied optics.

[19]  Qian Kemao,et al.  Sequential demodulation of a single fringe pattern guided by local frequencies. , 2007, Optics letters.

[20]  Qian Kemao,et al.  Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations , 2007 .

[21]  Qian Kemao.  On window size selection in the windowed Fourier ridges algorithm: Addendum , 2007 .

[22]  Wolfgang Osten,et al.  Kinematic and deformation parameter measurement by spatiotemporal analysis of an interferogram sequence. , 2007, Applied optics.

[23]  S. H. Soon,et al.  Comparative analysis on some filters for wrapped phase maps. , 2007, Applied optics.

[24]  Qian Kemao,et al.  On window size selection in the windowed Fourier ridges algorithm , 2007 .

[25]  Qian Kemao.  A simple phase unwrapping approach based on filtering by windowed Fourier transform: A note on the threshold selection , 2008 .

[26]  J. Goodman.  Speckle Phenomena in Optics: Theory and Applications , 2020 .