A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns.

A new method of estimating the phase-shift between interferograms is introduced. The method is based on a recently introduced two-dimensional Fourier-Hilbert demodulation technique. Three or more interferogram frames in an arbitrary sequence are required. The first stage of the algorithm calculates frame differences to remove the fringe pattern offset; allowing increased fringe modulation. The second stage is spatial demodulation to estimate the analytic image for each frame difference. The third stage robustly estimates the inter-frame phase-shifts and then uses the generalised phase-shifting algorithm of Lai and Yatagai to extract the offset, the modulation and the phase exactly. Initial simulations of the method indicate that high accuracy phase estimates are obtainable even in the presence of closed or discontinuous fringe patterns.

[1]  Paul Taylor,et al.  Multiphase fringe analysis with unknown phase shifts , 1994 .

[2]  H H Barrett,et al.  Phase-Shifting Interferometry and Maximum-Likelihood Estimation Theory. II. A Generalized Solution. , 1998, Applied optics.

[3]  Dennis Gabor,et al.  Theory of communication , 1946 .

[4]  B. Oreb,et al.  Design and assessment of symmetrical phase-shifting algorithms , 1992 .

[5]  K. Larkin,et al.  Topics in Multi dimensional Signal Demodulation , 2000 .

[6]  Thomas Kreis,et al.  Digital holographic interference-phase measurement using the Fourier-transform method , 1986 .

[7]  Kenneth H. Womack,et al.  Interferometric Phase Measurement Using Spatial Synchronous Detection , 1983 .

[8]  Miguel A. Alonso,et al.  Measures of spread for periodic distributions and the associated uncertainty relations , 2001 .

[9]  Mitsuo Takeda,et al.  Spatiotemporal frequency multiplex heterodyne interferometry , 1992 .

[10]  J C Wyant,et al.  Phase shifter calibration in phase-shifting interferometry. , 1985, Applied optics.

[11]  K. Larkin.  Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform. , 2001, Journal of the Optical Society of America. A, Optics, image science, and vision.

[12]  T. Yatagai,et al.  Generalized phase-shifting interferometry , 1991 .

[13]  Kieran G. Larkin,et al.  Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function , 1993, Optics & Photonics.

[14]  Bing Zhao.  A statistical method for fringe intensity-correlated error in phase-shifting measurement : The effect of quantization error on the N-bucket algorithm , 1997 .

[15]  T. Kreis.  Holographic Interferometry: Principles and Methods , 1996 .

[16]  P L Ransom,et al.  Interferogram analysis by a modified sinusoid fitting technique. , 1986, Applied optics.

[17]  Katsuyuki Okada,et al.  Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry , 1991 .

[18]  Kieran G. Larkin,et al.  Design and assessment of symmetrical phase-shifting algorithms , 1992 .

[19]  Hedser H. van Brug.  Phase-step calibration for phase-stepped interferometry , 1999 .

[20]  Andrew W. Fitzgibbon,et al.  Direct least squares fitting of ellipses , 1996, Proceedings of 13th International Conference on Pattern Recognition.

[21]  Colin J. R. Sheppard,et al.  Axial behavior of pupil-plane filters , 1988 .

[22]  D J Bone,et al.  Fringe-pattern analysis using a 2-D Fourier transform. , 1986, Applied optics.

[23]  Andrew W. Fitzgibbon,et al.  Direct Least Square Fitting of Ellipses , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[24]  C. J. Morgan.  Least-squares estimation in phase-measurement interferometry. , 1982, Optics letters.

[25]  T. Eiju,et al.  Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm. , 1987, Applied optics.

[26]  H J Tiziani,et al.  Semispatial, robust, and accurate phase evaluation algorithm. , 1995, Applied optics.

[27]  Alan C. Bovik,et al.  The analytic image , 1997, Proceedings of International Conference on Image Processing.

[28]  C. Vest.  Holographic Interferometry , 1979 .

[29]  K A Nugent,et al.  Interferogram analysis using an accurate fully automatic algorithm. , 1985, Applied optics.

[30]  W. Macy,et al.  Two-dimensional fringe-pattern analysis. , 1983, Applied optics.

[31]  B Ströbel,et al.  Processing of interferometric phase maps as complex-valued phasor images. , 1996, Applied optics.

[32]  K. Hibino,et al.  Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts , 1997 .

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

[34]  Kenneth H. Womack.  Frequency Domain Description Of Interferogram Analysis , 1984 .

[35]  H van Brug.  Phase-step calibration for phase-stepped interferometry. , 1999, Applied optics.

[36]  J. McKelvie,et al.  Reference phase shift determination in phase shifting interferometry , 1995 .

[37]  L. Mertz,et al.  Real-time fringe-pattern analysis. , 1983, Applied optics.

[38]  In-Bok Kong,et al.  General algorithm of phase-shifting interferometry by iterative least-squares fitting , 1995 .

[39]  J. Bokor,et al.  Fourier-transform method of phase-shift determination. , 2001, Applied optics.

[40]  Michael A. Player,et al.  Phase step measurement and variable step algorithms in phase-shifting interferometry , 1992 .

[41]  J Schwider,et al.  Some considerations of reduction of reference phase error in phase-stepping interferometry. , 1999, Applied optics.

[42]  M. A. Oldfield,et al.  Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform. , 2001, Journal of the Optical Society of America. A, Optics, image science, and vision.

[43]  M A Player,et al.  Phase-step insensitive algorithms for phase-shifting interferometry , 1994 .

[44]  X. Su,et al.  Improved Fourier transform profilometry for the automatic measurement of 3D object shapes , 1990 .

[45]  K. Hibino,et al.  Phase shifting for nonsinusoidal waveforms with phase-shift errors , 1995 .

[46]  F Roddier,et al.  Interferogram analysis using Fourier transform techniques. , 1987, Applied optics.

[47]  P. Wizinowich,et al.  Phase shifting interferometry in the presence of vibration: a new algorithm and system. , 1990, Applied optics.

[48]  Y. Surrel.  Phase stepping: a new self-calibrating algorithm. , 1993, Applied optics.

[49]  Chris P. Brophy,et al.  Effect of intensity error correlation on the computed phase of phase-shifting interferometry , 1990 .

[50]  Thomas Bülow,et al.  A Novel Approach to the 2D Analytic Signal , 1999, CAIP.

[51]  Y. Ishii,et al.  Phase-extraction algorithm in laser-diode phase-shifting interferometry. , 1995, Optics letters.

[52]  P. Carré.  Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures , 1966 .

[53]  Hans Knutsson,et al.  Signal processing for computer vision , 1994 .

[54]  J. Schwider,et al.  New compensating four-phase algorithm for phase-shift interferometry , 1993 .

[55]  S. W. Kim,et al.  Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting. , 1994, Applied optics.

[56]  M. Takeda,et al.  Fourier transform profilometry for the automatic measurement of 3-D object shapes. , 1983, Applied optics.

[57]  X Chen,et al.  Phase-shifting interferometry with uncalibrated phase shifts. , 2000, Applied optics.