Abstract

Here we describe a 2-step temporal phase unwrapping formula that uses 2-sensitivity demodulated phases for measuring static surfaces. The first phase demodulation has at most 1-wavelength sensitivity and the second one is G-times (G>>1.0) more sensitive. Measuring static surfaces with 2-sensitivity fringe patterns is well known and recent published methods combine 2-sensitivities measurements mostly by triangulation. Two important applications for our 2-step unwrapping algorithm is profilometry and synthetic aperture radar (SAR) interferometry. In these two applications the object or surface being analyzed is static and highly discontinuous; so temporal unwrapping is the best strategy to follow. Phase-demodulation in profilometry and SAR interferometry is very similar because both share similar mathematical models.

© 2015 Optical Society of America

1. Introduction

Temporal phase unwrapping was introduced by Huntley and Saldner [1] and it is used to measure optical dynamic wavefronts where one may change the number of interferometry fringes when testing a solid sustaining a dynamic loading [1]. Temporal unwrapping has also been used for measuring a dynamic sequence of holograms [2]. As the name implies the unwrapping is not made in the spatial domain but in the temporal domain over a single spatial pixel. Each spatial pixel in the dynamic fringe pattern is unwrapped independently from the others. This has the advantage that noisy pixels remain isolated and do not spread their noise to less noisy regions ruining the entire spatial unwrapping process. The main drawback of temporal unwrapping is that it needs many wrapped intermediate temporal phase-maps to keep within the limits imposed by the Nyquist temporal sampling rate [1].

Saldner and Huntley [3] also applied temporal phase unwrapping to profilometry of discontinuous three-dimensional (3D) objects. Saldner and Huntley [3] stated that the Nyquist temporal sampling limit, allow them at most 1-wavelength (1λ) sensitivity increase per temporal step. Therefore if one wishes to pass from 1λ to 7λ in phase-sensitivity one would need 7 intermediate temporal phase-maps. Here we are proposing to use only the two extremes wrapped phases to obtain the same result. The Nyquist sampling limit is overcome because the profiling surface is static during the whole temporal fringe-projection profilometry experiment. Another way of seeing this is by noting that the static 2D surface measured at different phase sensitivities always have the same Shannon information entropy. This is because the surface remains static; the relative probability of occurrence of any given point in this surface is the same independently of the measuring sensitivity scale. That is why the Nyquist sampling limit is not a fundamental limit in this case.

Since the publication of reference [3] multi-sensitivity profilometry of 3D discontinuous static objects has been an active research field [412]. Profilometry of highly discontinuous industrial objects is the rule rather than the exception and temporal unwrapping is the best choice for these applications. These researches [312] have demonstrated and applied several algorithms to unwrap by triangulation and other varied techniques wrapped phases with many wavelengths from less sensitive phase-maps.

Another important field of applications of phase unwrapping with variable sensitivity interferometry is Synthetic Aperture Radar (SAR) Interferometry [13]. SAR interferometry is a powerful remote sensing technique for the quantitative measurement geophysical data of the Earth’s surface. By increasing the baseline of the 2 Radar antennas, SAR interferometry enables sub-wavelength phase-measurements of the earth surface with variable sensitivity. Here also we can combine 2-sensitive SAR fringe-patterns to unwrap the highly discontinuous Earth terrain with the accuracy of the highest-baseline phase-measurement. Although 3D fringe projection profilometry and SAR interferometry need widely different experimental set-ups (SAR interferometry normally uses earth-orbiting satellites) both share the same formal or mathematical background.

Here we propose a 2-step temporal phase unwrapping algorithm which uses 2 widely separated sensitive demodulated-wrapped phases of a static surface.

2. Temporal phase unwrapping with sensitivity interferometric measurements

We start by giving the standard mathematical formula for two fringe patterns having different phase modulation sensitivity,

I1(x,y)=a(x,y)+b(x,y)cos[φ(x,y)],φ(x,y)(π,π),I2(x,y)=a(x,y)+b(x,y)cos[Gφ(x,y)],(G1),G.
Here we are assuming thatφ(x,y)is a 1λsensitive phase andGφ(x,y) is G-times more sensitive, i.e. sup|Gφ(x,y)|=Gλ,(G). One may use a large number of phase demodulation algorithms [14] to obtain the 2 demodulated wrapped phase-maps as,
φ1(x,y)=W[φ(x,y)],φ(x,y)(π,π).φ2W(x,y)=W[Gφ(x,y)],(G1),G.
BeingφW=W[φ]=angle[exp(iφ)] the wrapping phase operator. Equation (2) shows the two demodulated phases of the 2 fringe-patterns in Eq. (1). The first demodulation φ1(x,y) is not wrapped because it is less than 1λ; the second phase φ2W(x,y)is highly wrapped because it is scaled-up by G which in practice falls within 6<G<20(G) depending on the quality ofφ1(x,y) and φ2W(x,y).

3. Signal-to-noise ratio gain in 2-sensitivity phase modulation of static surfaces

From the phase-noise perspective there is a good advantage of using variable sensitivity interferometry. Let us assume that the two interferogram are phase modulated by[φ(x,y)+φn(x,y)] and by[Gφ(x,y)+φn(x,y)] as,

I1(x,y)=a(x,y)+b(x,y)cos[φ(x,y)+φn(x,y)],φ(x,y)(π,π),I2(x,y)=a(x,y)+b(x,y)cos[Gφ(x,y)+φn(x,y)],(G1),G.
Being φn(x,y)the phase-noise. The phase-noise is the same in both fringe-patterns because the experimental set-up is the same except for an increase(G1)in phase-sensitivity. In an abstract Hilbert space, the power of a signals(x,y) is given by|s(x,y)|2dxdy; therefore the signal-to-noise power ratios for the modulating phases in Eq. (3) are,
PhaseSignalPowerPhaseNoisePower=(x,y)Ω|Gφ(x,y)|2dxdy(x,y)Ω|φn(x,y)|2dxdy>(x,y)Ω|φ(x,y)|2dxdy(x,y)Ω|φn(x,y)|2dxdy.
Being (x,y)Ωthe 2-dimensional region where the signalφ(x,y)is well-defined. The signal-to-noise power-ratio increases G2times by increasing the phase sensitivity from φ(x,y) to Gφ(x,y) for the same phase-noiseφn(x,y). This is the fundamental reason why it is a good idea to use temporal phase unwrapping in digital 3D profilometry and in SAR interferometry of static surfaces. If we want to have φ(x,y)as noiseless as Gφ(x,y) one would need N (N=G2) phase-shifted fringe-patternsI1(x,y)to obtain the same signal-to-noise than demodulatingI2(x,y)alone [14]

4. Temporal phase unwrapping of static surfaces with 2-measuring sensitivities

No matter which digital phase demodulation method one uses [14] the last step is always phase unwrapping. We are assuming that we end up with 2 wrapped phase measurements as,

φ1(x,y)=W[φ(x,y)],andφ2W(x,y)=W[Gφ(x,y)],(G1),G.
Being W[φ]=angle[exp(iφ)] the wrapping operator. The first phase measurement φ1(x,y)φ(x,y) is less than 1λ (i.e.φ1(x,y)(π,π)) otherwise this method does not work at all. The second phase φ2(x,y) is G-times more sensitive; meaning that φ2(x,y) is scaled-up by G and therefore wrapped several times. The last step however is always to go from phase-radians to actual surface-height in centimeters or meters; this last step is shown in section 6 and it depends on the actual experimental set-up used to obtain the fringe patternsI1(x,y)and I2(x,y).

Let us now display our temporal 2-steps unwrapping formula as,

φ2(x,y)=Gφ1(x,y)+W[φ2W(x,y)Gφ1(x,y)],(radians).
This is the main result of this paper and as far as we know this is a new and useful 2-sensitivity temporal phase-unwrapper. The estimated unwrapped phase φ2(x,y) is the searched continuous phase with the highest sensitivity. The termGφ1(x,y)is the first coarse estimation ofφ2(x,y) and may be represented by,
Gφ1(x,y)=φ2(x,y)+φe(x,y).
Gφ1(x,y)=2πk(x,y)+φ2W(x,y)+φe(x,y).
The coarse estimateGφ1(x,y)equals φ2(x,y) plus an error phase φe(x,y)(Eq. (7a)). The integer 2D-field which unwraps φ2W(x,y) is k(x,y){,2,1,0,1,2,}. Substituting Eq. (7b) into Eq. (6) and rewriting Eq. (6) for the reader’s convenience one obtains,
 φ2(x,y)=Gφ1(x,y)+W[φ2W(x,y)Gφ1(x,y)],φ2(x,y)=Gφ1(x,y)+W[φ2W(x,y)2πk(x,y)φ2W(x,y)φe(x,y)],φ2(x,y)=Gφ1(x,y)+W[2πk(x,y)φe(x,y)].
Given thatφe(x,y)=W[2πk(x,y)+φe(x,y)] and using Eq. (7a) one obtains,
φ2(x,y)=Gφ1(x,y)φe(x,y),φ2(x,y)=φ2(x,y)+φe(x,y)φe(x,y),φ2(x,y)=φ2(x,y).
Equation (9) is valid whenever the phase unwrapped errorφe(x,y)=[φ2(x,y)Gφ1(x,y)]do not exceed 1λ, i.e. φe(x,y)(π,π), that is,
φe(x,y)=[φ2(x,y)Gφ1(x,y)](π,π).
We have mathematically demonstrated that our 2-sensitivity temporal phase-unwrapper obtains the unwrapped phase φ2(x,y) completely untouched by the phase-error φe(x,y) of the first coarse phase estimate Gφ1(x,y)=φ2(x,y)+φe(x,y) whenever the condition in Eq. (10) is satisfied. Figure 1 shows graphically the difference between our 2-steps temporal phase unwrapper and the one reported in [3]. Note that if both φ1(x,y)and φ2W(x,y)were noiseless and distortion-less φe(x,y)0, the factor G could a large number.

 

Fig. 1 At the top we show the 9 temporal samplings needed by standard temporal phase unwrapping [3] to pass from 1-lambda to 9-lambda in sensitivity at the Nyquist sampling rate. In our 2-step temporal phase unwrapper algorithm (Eq. (6) with G = 9) we only need the two extreme sensitive phase-maps to obtain the same results.

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In Fig. 1 we shows that in order to be at the Nyquist temporal sampling rate we need at least 9 phase-maps with phase sensitivities of {λ,2λ,3λ,,9λ} [3] to unwrap this sequence of 9 temporal phase-maps. In contrast using the 2-steps algorithm herein described, only the 2 extreme phase-maps{1λ,9λ} are needed.

5. First computer simulation example with G = 10

As Fig. 2 shows an intrinsically discontinuous surface φ1(x,y)assumed to be 1-wavelenght (1λ) “height”, noisy and distorted.

 

Fig. 2 Panel (a) shows the surfaces and central-cuts of the 1-wavelenght (1λ) phase φ1(x,y)(π,π) containing 3-ring discontinuities. Panel (b) shows the G-times (G = 10) more sensitivity wrapped phase φ2W(x,y)=W[Gφ(x,y)].

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The second surface φ2W(x,y) is wrapped and it is assumed to be noiseless and undistorted. This idealized condition is made to show that our first coarse estimation Gφ1(x,y) errors do not propagate towardsφ2(x,y). Figure 3 shows the unwrapping process where the rounded-plus-sign represents our unwrapping algorithm (Eq. (6)). As Fig. 3 shows, the amplified phase Gφ1(x,y)=10φ1(x,y)is noisy and distorted while φ2W(x,y)and its unwrapped versionφ2(x,y)have neither noise nor distortion.

 

Fig. 3 Here we show the temporal unwrapping of a 3-rings discontinuous surface and their central-cuts; these 3 discontinuous rings are not wrapped-phase discontinuities, they are essential surface discontinuities. Panel (a) shows Gφ1(x,y)=10φ1(x,y). Panel (b) shows the noiseless wrapped-phaseφ2W(x,y)(π,π). Panel (c) showsφ2(x,y)obtained using our 2-step temporal phase unwrapper.

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Next Fig. 4 shows in gray levels, the phases corresponding to 10φ1(x,y) and φ2(x,y). Observe the centered cut-lines in red for Gφ1(x,y) and in blue forφ2(x,y). In panel (b) we have superimposed Gφ1(x,y)andφ2(x,y)to have an intuitive estimation of the phase error φe(x,y)=φ2(x,y)10φ1(x,y).

 

Fig. 4 Here we show the 3-rings discontinuous surface-phase in gray levels. Panel (a) shows the noisy and distorted 1-wavelenght phaseGφ1(x,y)scaled-up by G = 10. The red graph is a central cut of10φ1(x,y). Panel (b) shows in blue a cut-graph of the phaseφ2(x,y). We have superimposed the red and blue graphs to see the error φe(x,y)between them.

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Figure 5 shows that whenever [φ2(x,y)10φ1(x,y)](π,π) holds we end up with the desired high-sensitive unwrapped phaseφ2(x,y) without the phase errors φe(x,y) generated by in our initial coarse approximation10φ1(x,y)=φ2(x,y)+φe(x,y).

 

Fig. 5 This figure shows a crucial fact of the proposed phase unwrapping algorithm with G = 10. The error between the more sensitive estimated unwrapped phaseφ2(x,y)(in radians), and the scaled-up (G = 10) phase Gφ1(x,y)=10φ1(x,y) must fall within φe(x,y)(π,π).

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Finally Fig. 6 shows what happens to the continuous (unwrapped) phase φ2(x,y)when the condition [φ2(x,y)Gφ1(x,y)](π,π) do not hold; spurious phase jumps start to appear inφ2(x,y)(Fig. 6). This is a hallmark indication that we have exceeded the maximum allowed scaling-up G factor. In this case we need to repeat the experiment with a lower G and/or reduce the estimation error ofφ1(x,y) until these spurious phase jumps disappear.

 

Fig. 6 Here have increased the sensitivity gain from G = 10 to G = 12 exceeding the boundaries of the condition [φ2(x,y)Gφ1(x,y)](π,π), generating spurious phase jumps (in radians).

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6. Second example: fringe-projection profilometry of a discontinuous surface

Here we present a profilometry [15] example where the phase amplifying factor G is set to seven (G = 7). The standard fringe-projection profilometry set-up is shown in Fig. 7,

 

Fig. 7 This is the typical configuration to digitize a 3-dimensional (3D) object using fringe-projection profilometry. Theta is the angle between the camera and the fringe-projector.

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In the absence of a 3D object, one obtains two pure carrier-fringe patterns projected over the reference plane,

I1(x,y)ReferencePlane=a(x,y)+b(x,y)cos[ω1x+φn(x,y)],I2(x,y)ReferencePlane=a(x,y)+b(x,y)cos[7ω1x+φn(x,y)],(G=7).
From Eq. (11) the carrier-frequency in I2(x,y)ReferencePlaneis multiplied by G = 7 however the phase-noise φn(x,y)remains the same for both fringe patterns.

Figure 8 shows a computer simulated surfaceh(x,y)having 9 objects with different heights coded in gray levels. These 9 objects are collocated over the reference plane. The two fringe patterns at the CCD camera sensor I1(x,y)and I2(x,y) are:

I1(x,y)=a(x,y)+b(x,y)cos[ω1x+ω1tan(θ)h(x,y)+φn(x,y)],I2(x,y)=a(x,y)+b(x,y)cos[7ω1x+7ω1tan(θ)h(x,y)+φn(x,y)];(G=7).
The phase sensitivity of the fringe pattern I1(x,y) isω1tan(θ), while the phase sensitivity of I2(x,y)is7ω1tan(θ). Phase-demodulating I1(x,y) and I2(x,y), and subtracting their carrier-frequency reference-planes ω1x and 7ω1x one obtains,
φ1(x,y)=angle{exp[iω1tan(θ)h(x,y)+iφn(x,y)]};φ1(π,π),φ2W(x,y)=angle{exp[i7ω1tan(θ)h(x,y)+iφn(x,y)]};(G=7).
The more sensitive wrapped phaseφ2W(x,y)is unwrapped according to Eq. (6) as,
φ2(x,y)=7φ1(x,y)+W[φ2W(x,y)7φ1(x,y)];radians,(G=7).
The last step is the physical calibration from radians to centimeters (cm) as,
h1(x,y)=φ1(x,y)ω1tan(θ)cm;h2(x,y)=φ2(x,y)7ω1tan(θ)cm.
Substituting the demodulated phasesφ1(x,y)and φ2(x,y) in Eq. (12) one obtains,
h1(x,y)=h(x,y)+φn(x,y)ω1tan(θ);h2(x,y)=h(x,y)+φn(x,y)7ω1tan(θ);(G=7).
Equation (16) means that the heighth2(x,y) is 7-times less-noisy than h1(x,y) (Fig. 8). Additionally, Eq. (16) is equivalent to Eq. (4), but now the height-noise reduction is expressed in terms of the noise-amplitude φn(x,y)not in terms of its power|φn(x,y)|2dxdy.

 

Fig. 8 Here a simulated fringe-projection profilometry experiment of a discontinuous surface containing 9 separated objects. The highest object is the whitest triangle. In the blue graphs we can see that the measuring noise in h2(x,0) has decreased in amplitude 7-times with respect to h1(x,0).

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We finally emphasize that this profilometry simulated experiment would have required at least N = 7 phase-maps using standard temporal unwrapping [3]. However, using our 2-steps temporal unwrapping formula, one only needs the 2 extreme phase-maps (see Fig. 1) to achieve exactly the same results in terms of noise and harmonic distortion rejection.

7. Conclusions

We have presented a 2-steps temporal phase-unwrapping algorithm for measuring static surfaces. The first phase φ1(x,y)comes from a fringe-pattern phase-modulated by less than1λ. The wrapped phase φ2W(x,y)is G-times more sensitive. This algorithm was demonstrated mathematically in section 4 and we repeated it here for the reader’s convenience,

φ2(x,y)=Gφ1(x,y)+W[φ2W(x,y)Gφ1(x,y)],(G1,G).
This unwrapping algorithm holds whenever,
φe(x,y)=φ2(x,y)Gφ1(x,y)=W[φ2W(x,y)Gφ1(x,y)].
If the above equality (Eq. (18)) is not fulfilled spurious phase jumps start to appear in the unwrapped phaseφ2(x,y) (Fig. 6). In this case we must repeat the experiment to decrease the amplifying sensitivity factorG, and/or decrease the phase-errors inφ1(x,y). Just for completeness let us show the standard algorithm for temporal phase unwrapping [3,14] applied to N phase-maps{φW0,φW1,,φWk,,φWN1},
φk+1(x,y)=φk(x,y)+W[φk+1(x,y)φWk(x,y)];sup|φW0(x,y)|<λ.
This unwrapping algorithm is valid whenever the Nyquist temporal sampling rate is observed,
sup|φk+1(x,y)φk(x,y)|<λ,k={0,1,,N1}.
We finally list the advantages and limitations of our new 2-steps (2-sensitivities) temporal phase unwrapping algorithm to measure static surfaces with high precision, and compare it against the N-steps standard (Eq. (19) and Eq. (20)) temporal phase unwrapper [3].

  • a) One needs 2-sensitivity fringe-patterns phase-modulated fringe-patterns. One modulated by φ(x,y)(π,π) and another one by Gφ(x,y) (G>>1.0,G) of the static surface under test. The amplifying G parameter in practice typically falls within(8<G<14)(see Fig. 1).
  • b) The demodulated phaseφ1(x,y)φ(x,y)must be less than 1λ sensitivity.
  • c) The wrapped phase φ2W(x,y)is G-times more sensitive thanφ1(x,y).
  • d) The unwrapped phaseφ2(x,y)is roughly approximated by φ2(x,y)Gφ1(x,y)
  • e) SubstitutingGφ1(x,y)and the wrapped phaseφ2W(x,y)into our 2-step phase unwrapping algorithm (Eq. (6) or Eq. (17)) one obtains the desired unwrapped phaseφ2(x,y) without the errors contained in the first estimation ofφ1(x,y).
  • f) If the phase error φe(x,y)(Eq. (10)) lies outside (π,π) spurious phase jumps start to appear. Then we need to repeat the experiment with a lower G, and/or obtain a less noisy and/or less distorted 1λ estimation ofφ1(x,y).
  • g) If (for example) we increase the phase sensitivity from 1λ to 9λ, both the standard temporal phase unwrapper [3] and our 2-steps one, would give the same unwrapping results. But the standard temporal unwrapper [3] would require at least 9 phase-maps while our 2-step unwrapper requires only the 2 extreme phase-maps (see Fig. 1).
  • h) The same applies to the demodulated phase-noiseφn(x,y). For example a 7-step (with 1λsensitivity increase) using the standard unwrapper [3] would increase 49-times the signal-to-noise power ratio (see Eq. (4)) with reference to the estimateφ1(x,y). This is the same amount of signal-to-noise increase obtained using our 2-step temporal unwrapper using only the 1λ and 7λ phase-maps sensitivities (see Fig. 8).

In brief, all anti-noise benefits of N-steps{φW0,φW1,,φWN1} standard temporal phase unwrapping [3] (Eqs. (19) y (20)) are kept unchanged, except that we are using only the 2 extreme phase-maps {φ0,φWN1} (see Fig. 1 and Eq. (6)); this holds whenever the condition in Eq. (10) is maintained.

Acknowledgments

We would like to acknowledge the support of the Mexican Scientific Research Council (CONACyT) for his valuable support through the basic research grant 157044-F.

References and links

1. J. M. Huntley and H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32(17), 3047–3052 (1993). [CrossRef]   [PubMed]  

2. G. Pedrini, I. Alexeenko, W. Osten, and H. J. Tiziani, “Temporal phase unwrapping of digital hologram sequences,” Appl. Opt. 42(29), 5846–5854 (2003). [CrossRef]   [PubMed]  

3. H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36(13), 2770–2775 (1997). [CrossRef]   [PubMed]  

4. J. L. Li, H. J. Su, and X. Y. Su, “Two-frequency grating used in phase-measuring profilometry,” Appl. Opt. 36(1), 277–280 (1997). [CrossRef]   [PubMed]  

5. J. Li, L. G. Hassebrook, and C. Guan, “Optimized two-frequency phase-measuring-profilometry light-sensor temporal-noise sensitivity,” J. Opt. Soc. Am. A 20(1), 106–115 (2003). [CrossRef]   [PubMed]  

6. E. H. Kim, J. Hahn, H. Kim, and B. Lee, “Profilometry without phase unwrapping using multi-frequency and four-step phase-shift sinusoidal fringe projection,” Opt. Express 17(10), 7818–7830 (2009). [CrossRef]   [PubMed]  

7. K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express 18(5), 5229–5244 (2010). [CrossRef]   [PubMed]  

8. Y. Fu, W. Wang, and H. Xiao, “Three-dimensional profile measurement based on modified temporal phase unwrapping algorithm,” Optik (Stuttg.) 124(6), 557–560 (2013). [CrossRef]  

9. Y. Xu, S. Jia, X. Luo, J. Yang, and Y. Zhang, “Multi-frequency projected fringe profilometry for measuring objects with large depth discontinuities,” Opt. Commun. 288, 27–30 (2013). [CrossRef]  

10. T. Liu, C. Zhou, Y. Liu, S. Si, and Z. Lei, “Deflectometry for phase retrieval using a composite fringe,” Opt. Applicata 94, 451–461 (2014).

11. J. Long, J. Xi, M. Zhu, W. Cheng, R. Cheng, Z. Li, and Y. Shi, “Absolute phase map recovery of two fringe patterns with flexible selection of fringe wavelengths,” Appl. Opt. 53(9), 1794–1801 (2014). [CrossRef]   [PubMed]  

12. Y. Xu, S. Jia, Q. Bao, H. Chen, and J. Yang, “Recovery of absolute height from wrapped phase maps for fringe projection profilometry,” Opt. Express 22(14), 16819–16828 (2014). [CrossRef]   [PubMed]  

13. G. Krieger, I. Hajnsek, K. P. Papathanassiou, M. Younis, and A. Moreira, “Interferometric Synthetic Aperture Radar (SAR) Missions Employing Formation Flying,” Proc. IEEE 98(5), 816–843 (2010). [CrossRef]  

14. M. Servin, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology (Wiley-Vch, 2014).

15. K. J. Gasvik, Optical Metrology, 3th ed. (John Wiley & Sons, 2002).

References

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  1. J. M. Huntley and H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32(17), 3047–3052 (1993).
    [Crossref] [PubMed]
  2. G. Pedrini, I. Alexeenko, W. Osten, and H. J. Tiziani, “Temporal phase unwrapping of digital hologram sequences,” Appl. Opt. 42(29), 5846–5854 (2003).
    [Crossref] [PubMed]
  3. H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36(13), 2770–2775 (1997).
    [Crossref] [PubMed]
  4. J. L. Li, H. J. Su, and X. Y. Su, “Two-frequency grating used in phase-measuring profilometry,” Appl. Opt. 36(1), 277–280 (1997).
    [Crossref] [PubMed]
  5. J. Li, L. G. Hassebrook, and C. Guan, “Optimized two-frequency phase-measuring-profilometry light-sensor temporal-noise sensitivity,” J. Opt. Soc. Am. A 20(1), 106–115 (2003).
    [Crossref] [PubMed]
  6. E. H. Kim, J. Hahn, H. Kim, and B. Lee, “Profilometry without phase unwrapping using multi-frequency and four-step phase-shift sinusoidal fringe projection,” Opt. Express 17(10), 7818–7830 (2009).
    [Crossref] [PubMed]
  7. K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express 18(5), 5229–5244 (2010).
    [Crossref] [PubMed]
  8. Y. Fu, W. Wang, and H. Xiao, “Three-dimensional profile measurement based on modified temporal phase unwrapping algorithm,” Optik (Stuttg.) 124(6), 557–560 (2013).
    [Crossref]
  9. Y. Xu, S. Jia, X. Luo, J. Yang, and Y. Zhang, “Multi-frequency projected fringe profilometry for measuring objects with large depth discontinuities,” Opt. Commun. 288, 27–30 (2013).
    [Crossref]
  10. T. Liu, C. Zhou, Y. Liu, S. Si, and Z. Lei, “Deflectometry for phase retrieval using a composite fringe,” Opt. Applicata 94, 451–461 (2014).
  11. J. Long, J. Xi, M. Zhu, W. Cheng, R. Cheng, Z. Li, and Y. Shi, “Absolute phase map recovery of two fringe patterns with flexible selection of fringe wavelengths,” Appl. Opt. 53(9), 1794–1801 (2014).
    [Crossref] [PubMed]
  12. Y. Xu, S. Jia, Q. Bao, H. Chen, and J. Yang, “Recovery of absolute height from wrapped phase maps for fringe projection profilometry,” Opt. Express 22(14), 16819–16828 (2014).
    [Crossref] [PubMed]
  13. G. Krieger, I. Hajnsek, K. P. Papathanassiou, M. Younis, and A. Moreira, “Interferometric Synthetic Aperture Radar (SAR) Missions Employing Formation Flying,” Proc. IEEE 98(5), 816–843 (2010).
    [Crossref]
  14. M. Servin, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology (Wiley-Vch, 2014).
  15. K. J. Gasvik, Optical Metrology, 3th ed. (John Wiley & Sons, 2002).

2014 (3)

2013 (2)

Y. Fu, W. Wang, and H. Xiao, “Three-dimensional profile measurement based on modified temporal phase unwrapping algorithm,” Optik (Stuttg.) 124(6), 557–560 (2013).
[Crossref]

Y. Xu, S. Jia, X. Luo, J. Yang, and Y. Zhang, “Multi-frequency projected fringe profilometry for measuring objects with large depth discontinuities,” Opt. Commun. 288, 27–30 (2013).
[Crossref]

2010 (2)

K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express 18(5), 5229–5244 (2010).
[Crossref] [PubMed]

G. Krieger, I. Hajnsek, K. P. Papathanassiou, M. Younis, and A. Moreira, “Interferometric Synthetic Aperture Radar (SAR) Missions Employing Formation Flying,” Proc. IEEE 98(5), 816–843 (2010).
[Crossref]

2009 (1)

2003 (2)

1997 (2)

1993 (1)

Alexeenko, I.

Bao, Q.

Chen, H.

Cheng, R.

Cheng, W.

Fu, Y.

Y. Fu, W. Wang, and H. Xiao, “Three-dimensional profile measurement based on modified temporal phase unwrapping algorithm,” Optik (Stuttg.) 124(6), 557–560 (2013).
[Crossref]

Guan, C.

Hahn, J.

Hajnsek, I.

G. Krieger, I. Hajnsek, K. P. Papathanassiou, M. Younis, and A. Moreira, “Interferometric Synthetic Aperture Radar (SAR) Missions Employing Formation Flying,” Proc. IEEE 98(5), 816–843 (2010).
[Crossref]

Hao, Q.

Hassebrook, L. G.

Huntley, J. M.

Jia, S.

Y. Xu, S. Jia, Q. Bao, H. Chen, and J. Yang, “Recovery of absolute height from wrapped phase maps for fringe projection profilometry,” Opt. Express 22(14), 16819–16828 (2014).
[Crossref] [PubMed]

Y. Xu, S. Jia, X. Luo, J. Yang, and Y. Zhang, “Multi-frequency projected fringe profilometry for measuring objects with large depth discontinuities,” Opt. Commun. 288, 27–30 (2013).
[Crossref]

Kim, E. H.

Kim, H.

Krieger, G.

G. Krieger, I. Hajnsek, K. P. Papathanassiou, M. Younis, and A. Moreira, “Interferometric Synthetic Aperture Radar (SAR) Missions Employing Formation Flying,” Proc. IEEE 98(5), 816–843 (2010).
[Crossref]

Lau, D. L.

Lee, B.

Lei, Z.

T. Liu, C. Zhou, Y. Liu, S. Si, and Z. Lei, “Deflectometry for phase retrieval using a composite fringe,” Opt. Applicata 94, 451–461 (2014).

Li, J.

Li, J. L.

Li, Z.

Liu, K.

Liu, T.

T. Liu, C. Zhou, Y. Liu, S. Si, and Z. Lei, “Deflectometry for phase retrieval using a composite fringe,” Opt. Applicata 94, 451–461 (2014).

Liu, Y.

T. Liu, C. Zhou, Y. Liu, S. Si, and Z. Lei, “Deflectometry for phase retrieval using a composite fringe,” Opt. Applicata 94, 451–461 (2014).

Long, J.

Luo, X.

Y. Xu, S. Jia, X. Luo, J. Yang, and Y. Zhang, “Multi-frequency projected fringe profilometry for measuring objects with large depth discontinuities,” Opt. Commun. 288, 27–30 (2013).
[Crossref]

Moreira, A.

G. Krieger, I. Hajnsek, K. P. Papathanassiou, M. Younis, and A. Moreira, “Interferometric Synthetic Aperture Radar (SAR) Missions Employing Formation Flying,” Proc. IEEE 98(5), 816–843 (2010).
[Crossref]

Osten, W.

Papathanassiou, K. P.

G. Krieger, I. Hajnsek, K. P. Papathanassiou, M. Younis, and A. Moreira, “Interferometric Synthetic Aperture Radar (SAR) Missions Employing Formation Flying,” Proc. IEEE 98(5), 816–843 (2010).
[Crossref]

Pedrini, G.

Saldner, H.

Saldner, H. O.

Shi, Y.

Si, S.

T. Liu, C. Zhou, Y. Liu, S. Si, and Z. Lei, “Deflectometry for phase retrieval using a composite fringe,” Opt. Applicata 94, 451–461 (2014).

Su, H. J.

Su, X. Y.

Tiziani, H. J.

Wang, W.

Y. Fu, W. Wang, and H. Xiao, “Three-dimensional profile measurement based on modified temporal phase unwrapping algorithm,” Optik (Stuttg.) 124(6), 557–560 (2013).
[Crossref]

Wang, Y.

Xi, J.

Xiao, H.

Y. Fu, W. Wang, and H. Xiao, “Three-dimensional profile measurement based on modified temporal phase unwrapping algorithm,” Optik (Stuttg.) 124(6), 557–560 (2013).
[Crossref]

Xu, Y.

Y. Xu, S. Jia, Q. Bao, H. Chen, and J. Yang, “Recovery of absolute height from wrapped phase maps for fringe projection profilometry,” Opt. Express 22(14), 16819–16828 (2014).
[Crossref] [PubMed]

Y. Xu, S. Jia, X. Luo, J. Yang, and Y. Zhang, “Multi-frequency projected fringe profilometry for measuring objects with large depth discontinuities,” Opt. Commun. 288, 27–30 (2013).
[Crossref]

Yang, J.

Y. Xu, S. Jia, Q. Bao, H. Chen, and J. Yang, “Recovery of absolute height from wrapped phase maps for fringe projection profilometry,” Opt. Express 22(14), 16819–16828 (2014).
[Crossref] [PubMed]

Y. Xu, S. Jia, X. Luo, J. Yang, and Y. Zhang, “Multi-frequency projected fringe profilometry for measuring objects with large depth discontinuities,” Opt. Commun. 288, 27–30 (2013).
[Crossref]

Younis, M.

G. Krieger, I. Hajnsek, K. P. Papathanassiou, M. Younis, and A. Moreira, “Interferometric Synthetic Aperture Radar (SAR) Missions Employing Formation Flying,” Proc. IEEE 98(5), 816–843 (2010).
[Crossref]

Zhang, Y.

Y. Xu, S. Jia, X. Luo, J. Yang, and Y. Zhang, “Multi-frequency projected fringe profilometry for measuring objects with large depth discontinuities,” Opt. Commun. 288, 27–30 (2013).
[Crossref]

Zhou, C.

T. Liu, C. Zhou, Y. Liu, S. Si, and Z. Lei, “Deflectometry for phase retrieval using a composite fringe,” Opt. Applicata 94, 451–461 (2014).

Zhu, M.

Appl. Opt. (5)

J. Opt. Soc. Am. A (1)

Opt. Applicata (1)

T. Liu, C. Zhou, Y. Liu, S. Si, and Z. Lei, “Deflectometry for phase retrieval using a composite fringe,” Opt. Applicata 94, 451–461 (2014).

Opt. Commun. (1)

Y. Xu, S. Jia, X. Luo, J. Yang, and Y. Zhang, “Multi-frequency projected fringe profilometry for measuring objects with large depth discontinuities,” Opt. Commun. 288, 27–30 (2013).
[Crossref]

Opt. Express (3)

Optik (Stuttg.) (1)

Y. Fu, W. Wang, and H. Xiao, “Three-dimensional profile measurement based on modified temporal phase unwrapping algorithm,” Optik (Stuttg.) 124(6), 557–560 (2013).
[Crossref]

Proc. IEEE (1)

G. Krieger, I. Hajnsek, K. P. Papathanassiou, M. Younis, and A. Moreira, “Interferometric Synthetic Aperture Radar (SAR) Missions Employing Formation Flying,” Proc. IEEE 98(5), 816–843 (2010).
[Crossref]

Other (2)

M. Servin, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology (Wiley-Vch, 2014).

K. J. Gasvik, Optical Metrology, 3th ed. (John Wiley & Sons, 2002).

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Figures (8)

Fig. 1
Fig. 1 At the top we show the 9 temporal samplings needed by standard temporal phase unwrapping [3] to pass from 1-lambda to 9-lambda in sensitivity at the Nyquist sampling rate. In our 2-step temporal phase unwrapper algorithm (Eq. (6) with G = 9) we only need the two extreme sensitive phase-maps to obtain the same results.
Fig. 2
Fig. 2 Panel (a) shows the surfaces and central-cuts of the 1-wavelenght ( 1λ ) phase φ1(x,y)(π,π) containing 3-ring discontinuities. Panel (b) shows the G-times (G = 10) more sensitivity wrapped phase φ 2 W (x,y)=W[Gφ(x,y)] .
Fig. 3
Fig. 3 Here we show the temporal unwrapping of a 3-rings discontinuous surface and their central-cuts; these 3 discontinuous rings are not wrapped-phase discontinuities, they are essential surface discontinuities. Panel (a) shows Gφ1(x,y)=10φ1(x,y) . Panel (b) shows the noiseless wrapped-phase φ 2 W (x,y)(π,π) . Panel (c) shows φ2(x,y) obtained using our 2-step temporal phase unwrapper.
Fig. 4
Fig. 4 Here we show the 3-rings discontinuous surface-phase in gray levels. Panel (a) shows the noisy and distorted 1-wavelenght phase Gφ1(x,y) scaled-up by G = 10. The red graph is a central cut of 10φ1(x,y) . Panel (b) shows in blue a cut-graph of the phase φ2(x,y) . We have superimposed the red and blue graphs to see the error φe(x,y) between them.
Fig. 5
Fig. 5 This figure shows a crucial fact of the proposed phase unwrapping algorithm with G = 10. The error between the more sensitive estimated unwrapped phase φ2(x,y) (in radians), and the scaled-up (G = 10) phase Gφ1(x,y)=10φ1(x,y) must fall within φe(x,y)(π,π) .
Fig. 6
Fig. 6 Here have increased the sensitivity gain from G = 10 to G = 12 exceeding the boundaries of the condition [ φ2(x,y)Gφ1(x,y) ](π,π) , generating spurious phase jumps (in radians).
Fig. 7
Fig. 7 This is the typical configuration to digitize a 3-dimensional (3D) object using fringe-projection profilometry. Theta is the angle between the camera and the fringe-projector.
Fig. 8
Fig. 8 Here a simulated fringe-projection profilometry experiment of a discontinuous surface containing 9 separated objects. The highest object is the whitest triangle. In the blue graphs we can see that the measuring noise in h2(x,0) has decreased in amplitude 7-times with respect to h1(x,0).

Equations (21)

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I1(x,y)=a(x,y)+b(x,y)cos[ φ(x,y) ],φ(x,y)(π,π), I2(x,y)=a(x,y)+b(x,y)cos[ Gφ(x,y) ],(G1),G.
φ1(x,y)=W[ φ(x,y) ],φ(x,y)(π,π). φ 2 W (x,y)=W[ Gφ(x,y) ],(G1),G.
I1(x,y)=a(x,y)+b(x,y)cos[ φ(x,y)+φn(x,y) ],φ(x,y)(π,π), I2(x,y)=a(x,y)+b(x,y)cos[ Gφ(x,y)+φn(x,y) ],(G1),G.
PhaseSignalPower PhaseNoisePower = (x,y)Ω | Gφ(x,y) | 2 dxdy (x,y)Ω | φn(x,y) | 2 dxdy > (x,y)Ω | φ(x,y) | 2 dxdy (x,y)Ω | φn(x,y) | 2 dxdy .
φ1(x,y)=W[φ(x,y)],andφ 2 W (x,y)=W[Gφ(x,y)],(G1),G.
φ2(x,y)=Gφ1(x,y)+W[ φ 2 W (x,y)Gφ1(x,y) ],(radians).
Gφ1(x,y)=φ2(x,y)+φe(x,y).
Gφ1(x,y)=2πk(x,y)+φ 2 W (x,y)+φe(x,y).
 φ2(x,y)=Gφ1(x,y)+W[φ 2 W (x,y)Gφ1(x,y)], φ2(x,y)=Gφ1(x,y)+W[φ 2 W (x,y)2πk(x,y)φ 2 W (x,y)φe(x,y)], φ2(x,y)=Gφ1(x,y)+W[2πk(x,y)φe(x,y)].
φ2(x,y)=Gφ1(x,y)φe(x,y), φ2(x,y)=φ2(x,y)+φe(x,y)φe(x,y), φ2(x,y)=φ2(x,y).
φe(x,y)=[ φ2(x,y)Gφ1(x,y) ](π,π).
I1 (x,y) ReferencePlane =a(x,y)+b(x,y)cos[ ω 1 x+φn(x,y) ], I2 (x,y) ReferencePlane =a(x,y)+b(x,y)cos[ 7 ω 1 x+φn(x,y) ],(G=7).
I1(x,y)=a(x,y)+b(x,y)cos[ ω 1 x+ ω 1 tan(θ)h(x,y)+φn(x,y) ], I2(x,y)=a(x,y)+b(x,y)cos[ 7 ω 1 x+7 ω 1 tan(θ)h(x,y)+φn(x,y) ];(G=7).
φ1(x,y)=angle{ exp[ i ω 1 tan(θ)h(x,y)+iφn(x,y) ] };φ1(π,π), φ 2 W (x,y)=angle{ exp[ i7 ω 1 tan(θ)h(x,y)+iφn(x,y) ] };(G=7).
φ2(x,y)=7φ1(x,y)+W[ φ 2 W (x,y)7φ1(x,y) ];radians,(G=7).
h1(x,y)= φ1(x,y) ω 1 tan(θ) cm;h2(x,y)= φ2(x,y) 7 ω 1 tan(θ) cm.
h1(x,y)=h(x,y)+ φn(x,y) ω 1 tan(θ) ;h2(x,y)=h(x,y)+ φn(x,y) 7 ω 1 tan(θ) ;(G=7).
φ2(x,y)=Gφ1(x,y)+W[ φ 2 W (x,y)Gφ1(x,y) ],(G1,G).
φe(x,y)=φ2(x,y)Gφ1(x,y)=W[ φ 2 W (x,y)Gφ1(x,y) ].
φ k+1 (x,y)= φ k (x,y)+W[ φ k+1 (x,y) φ W k (x,y) ];sup| φ W 0 (x,y) |<λ.
sup| φ k+1 (x,y) φ k (x,y) |<λ,k={0,1,,N1}.

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