Estimating phase shifts from three fringe patterns by use of cross spectrum

Haifei Ding; Hongwei Guo

doi:10.1364/AO.56.000916

Applied Optics
Vol. 56,
Issue 4,
pp. 916-927
(2017)
•https://doi.org/10.1364/AO.56.000916

Estimating phase shifts from three fringe patterns by use of cross spectrum

Haifei Ding and Hongwei Guo

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Haifei Ding and Hongwei Guo, "Estimating phase shifts from three fringe patterns by use of cross spectrum," Appl. Opt. 56, 916-927 (2017)

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Abstract

In phase-shifting technique, using self-calibrating algorithms allows us to determine phases and phase shifts simultaneously, thus eliminating the errors caused by miscalibrations of phase shifters. However, it is difficult to estimate phase shifts when only three fringe patterns are available, because in this case the problem becomes underdetermined. In this paper, we analyze the effects of phase-shift errors on the calculated phases, and find that the phase-shift errors introduce correlations between different frequency components of the calculated phases. We measure these correlations by calculating the cross spectrum of cis functions between the calculated phases and their trebles, and further define a single-valued objective function. A gradient-guided search strategy is used for minimizing this objective function, so that the phase shifts are estimated from three fringe patterns. The simulation and experimental results demonstrate that the newly proposed algorithm, in comparison with the existing correlation-based algorithms, has several advantages, such as being insensitive to the nonuniformities of the background intensities and the modulations, having a high stability, and offering improved computational efficiency.

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	Fringe Patterns Not Smoothed			Fringe Patterns Smoothed Using a Gaussian Filter
Noise SDs	Estimates of $δ_{1}$ and Their Errors Enclosed in Brackets (rad)	Estimates of $δ_{2}$ and Their Errors Enclosed in Brackets (rad)	RMS Errors of Phases (rad)	Estimates of $δ_{1}$ and Their Errors Enclosed in Brackets (rad)	Estimates of $δ_{2}$ and Their Errors Enclosed in Brackets (rad)	RMS Errors of Phases (rad)
0	0.7923 (0.0069)	1.5495 (−0.0213)	0.0308	0.7938 (0.0084)	1.5563 (−0.0145)	0.0330
0.005	0.7897 (0.0043)	1.5436 (−0.0271)	0.0585	0.7947 (0.0093)	1.5567 (−0.0139)	0.0362
0.01	0.7855 (0.0001)	1.5279 (−0.0429)	0.1510	0.7942 (0.0088)	1.5526 (−0.0182)	0.0417
0.02	0.8057 (0.0203)	1.5697 (−0.0011)	0.2128	0.7974 (0.0120)	1.5537 (−0.0171)	0.0582
0.05	1.0287 (0.2433)	2.0167 (0.4459)	0.6135	0.8126 (0.0272)	1.5661 (−0.0047)	0.1223
0.1	1.4694 (0.6840)	2.8855 (1.3147)	1.0986	0.8635 (0.0781)	1.6804 (0.1096)	0.2312

	Fringe Patterns Not Smoothed			Fringe Patterns Smoothed Using a Gaussian Filter
Noise SDs	Estimates of $δ_{1}$ and Their Errors Enclosed in Brackets (rad)	Estimates of $δ_{2}$ and Their Errors Enclosed in Brackets (rad)	RMS Errors of Phases (rad)	Estimates of $δ_{1}$ and Their Errors Enclosed in Brackets (rad)	Estimates of $δ_{2}$ and Their Errors Enclosed in Brackets (rad)	RMS Errors of Phases (rad)
0	0.7923 (0.0069)	1.5495 (−0.0213)	0.0308	0.7938 (0.0084)	1.5563 (−0.0145)	0.0330
0.005	0.7913 (0.0059)	1.5480 (−0.0228)	0.0365	0.7941 (0.0087)	1.5563 (−0.0144)	0.0346
0.01	0.7896 (0.0042)	1.5447 (−0.0261)	0.0512	0.7932 (0.0078)	1.5542 (−0.0166)	0.0350
0.02	0.7829 (−0.0025)	1.5359 (−0.0349)	0.0887	0.7975 (0.0121)	1.5594 (−0.0114)	0.0422
0.05	0.7853 (−0.0001)	1.5511 (−0.0197)	0.2305	0.8007 (0.0153)	1.5608 (−0.0100)	0.0616
0.1	0.9606 (0.1752)	1.9139 (0.3431)	0.5114	0.7962 (0.0108)	1.5550 (−0.0158)	0.0963
0.2	1.3046 (0.5193)	2.6613 (1.0905)	0.8900	0.7945 (0.0091)	1.5608 (−0.0100)	0.1902

	Nominal	Cross Bispectrum- Based [25]	Proposed
$δ_{0}$ (rad)	0	0	0
$δ_{1}$ (rad)	$2 π / 3$	2.0593	2.0943
$δ_{2}$ (rad)	$4 π / 3$	4.4251	4.4276
RMS of phase differences (rad)	0.0814	0.0108	0.0222
Computational time ( $s$ )	0.038	208 (for processing 10 cross sections of fringe patterns)	128.31 (for processing full fringe patterns with $512 pixels \times 512 pixels$ )

	Fringe Patterns Not Smoothed			Fringe Patterns Smoothed Using a Gaussian Filter
Noise SDs	Estimates of $δ_{1}$ and Their Errors Enclosed in Brackets (rad)	Estimates of $δ_{2}$ and Their Errors Enclosed in Brackets (rad)	RMS Errors of Phases (rad)	Estimates of $δ_{1}$ and Their Errors Enclosed in Brackets (rad)	Estimates of $δ_{2}$ and Their Errors Enclosed in Brackets (rad)	RMS Errors of Phases (rad)
0	0.7923 (0.0069)	1.5495 (−0.0213)	0.0308	0.7938 (0.0084)	1.5563 (−0.0145)	0.0330
0.005	0.7897 (0.0043)	1.5436 (−0.0271)	0.0585	0.7947 (0.0093)	1.5567 (−0.0139)	0.0362
0.01	0.7855 (0.0001)	1.5279 (−0.0429)	0.1510	0.7942 (0.0088)	1.5526 (−0.0182)	0.0417
0.02	0.8057 (0.0203)	1.5697 (−0.0011)	0.2128	0.7974 (0.0120)	1.5537 (−0.0171)	0.0582
0.05	1.0287 (0.2433)	2.0167 (0.4459)	0.6135	0.8126 (0.0272)	1.5661 (−0.0047)	0.1223
0.1	1.4694 (0.6840)	2.8855 (1.3147)	1.0986	0.8635 (0.0781)	1.6804 (0.1096)	0.2312

	Fringe Patterns Not Smoothed			Fringe Patterns Smoothed Using a Gaussian Filter
Noise SDs	Estimates of $δ_{1}$ and Their Errors Enclosed in Brackets (rad)	Estimates of $δ_{2}$ and Their Errors Enclosed in Brackets (rad)	RMS Errors of Phases (rad)	Estimates of $δ_{1}$ and Their Errors Enclosed in Brackets (rad)	Estimates of $δ_{2}$ and Their Errors Enclosed in Brackets (rad)	RMS Errors of Phases (rad)
0	0.7923 (0.0069)	1.5495 (−0.0213)	0.0308	0.7938 (0.0084)	1.5563 (−0.0145)	0.0330
0.005	0.7913 (0.0059)	1.5480 (−0.0228)	0.0365	0.7941 (0.0087)	1.5563 (−0.0144)	0.0346
0.01	0.7896 (0.0042)	1.5447 (−0.0261)	0.0512	0.7932 (0.0078)	1.5542 (−0.0166)	0.0350
0.02	0.7829 (−0.0025)	1.5359 (−0.0349)	0.0887	0.7975 (0.0121)	1.5594 (−0.0114)	0.0422
0.05	0.7853 (−0.0001)	1.5511 (−0.0197)	0.2305	0.8007 (0.0153)	1.5608 (−0.0100)	0.0616
0.1	0.9606 (0.1752)	1.9139 (0.3431)	0.5114	0.7962 (0.0108)	1.5550 (−0.0158)	0.0963
0.2	1.3046 (0.5193)	2.6613 (1.0905)	0.8900	0.7945 (0.0091)	1.5608 (−0.0100)	0.1902

Abstract

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Applied Optics