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.
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Temporal phase unwrapping was introduced by Huntley and Saldner  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 . Temporal unwrapping has also been used for measuring a dynamic sequence of holograms . 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 .
Saldner and Huntley  also applied temporal phase unwrapping to profilometry of discontinuous three-dimensional (3D) objects. Saldner and Huntley  stated that the Nyquist temporal sampling limit, allow them at most 1-wavelength () sensitivity increase per temporal step. Therefore if one wishes to pass from to 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  multi-sensitivity profilometry of 3D discontinuous static objects has been an active research field [4–12]. 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 [3–12] 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 . 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,14] to obtain the 2 demodulated wrapped phase-maps as,Equation (2) shows the two demodulated phases of the 2 fringe-patterns in Eq. (1). The first demodulation is not wrapped because it is less than ; the second phase is highly wrapped because it is scaled-up by G which in practice falls within depending on the quality of and .
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 and by as,Eq. (3) are,14]
4. Temporal phase unwrapping of static surfaces with 2-measuring sensitivities
No matter which digital phase demodulation method one uses  the last step is always phase unwrapping. We are assuming that we end up with 2 wrapped phase measurements as,
Let us now display our temporal 2-steps unwrapping formula as,Eq. (7a)). The integer 2D-field which unwraps is . Substituting Eq. (7b) into Eq. (6) and rewriting Eq. (6) for the reader’s convenience one obtains,Eq. (7a) one obtains,Equation (9) is valid whenever the phase unwrapped errordo not exceed , i.e. , that is,Eq. (10) is satisfied. Figure 1 shows graphically the difference between our 2-steps temporal phase unwrapper and the one reported in . Note that if both and were noiseless and distortion-less , the factor G could a large number.
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  to unwrap this sequence of 9 temporal phase-maps. In contrast using the 2-steps algorithm herein described, only the 2 extreme phase-maps are needed.
5. First computer simulation example with G = 10
As Fig. 2 shows an intrinsically discontinuous surface assumed to be 1-wavelenght () “height”, noisy and distorted.
The second surface is wrapped and it is assumed to be noiseless and undistorted. This idealized condition is made to show that our first coarse estimation errors do not propagate towards. Figure 3 shows the unwrapping process where the rounded-plus-sign represents our unwrapping algorithm (Eq. (6)). As Fig. 3 shows, the amplified phase is noisy and distorted while and its unwrapped versionhave neither noise nor distortion.
Next Fig. 4 shows in gray levels, the phases corresponding to and . Observe the centered cut-lines in red for and in blue for. In panel (b) we have superimposed andto have an intuitive estimation of the phase error .
Figure 5 shows that whenever holds we end up with the desired high-sensitive unwrapped phase without the phase errors generated by in our initial coarse approximation.
Finally Fig. 6 shows what happens to the continuous (unwrapped) phase when the condition do not hold; spurious phase jumps start to appear in(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 of until these spurious phase jumps disappear.
6. Second example: fringe-projection profilometry of a discontinuous surface
In the absence of a 3D object, one obtains two pure carrier-fringe patterns projected over the reference plane,Eq. (11) the carrier-frequency in is multiplied by G = 7 however the phase-noise remains the same for both fringe patterns.
Figure 8 shows a computer simulated surfacehaving 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 and are:Eq. (6) as,Eq. (12) one obtains,Equation (16) means that the height is 7-times less-noisy than (Fig. 8). Additionally, Eq. (16) is equivalent to Eq. (4), but now the height-noise reduction is expressed in terms of the noise-amplitude not in terms of its power.
We finally emphasize that this profilometry simulated experiment would have required at least N = 7 phase-maps using standard temporal unwrapping . 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.
We have presented a 2-steps temporal phase-unwrapping algorithm for measuring static surfaces. The first phase comes from a fringe-pattern phase-modulated by less than. The wrapped phase is G-times more sensitive. This algorithm was demonstrated mathematically in section 4 and we repeated it here for the reader’s convenience,Eq. (18)) is not fulfilled spurious phase jumps start to appear in the unwrapped phase (Fig. 6). In this case we must repeat the experiment to decrease the amplifying sensitivity factor, and/or decrease the phase-errors in. Just for completeness let us show the standard algorithm for temporal phase unwrapping [3,14] applied to phase-maps,Eq. (19) and Eq. (20)) temporal phase unwrapper .
- a) One needs 2-sensitivity fringe-patterns phase-modulated fringe-patterns. One modulated by and another one by () of the static surface under test. The amplifying G parameter in practice typically falls within(see Fig. 1).
- b) The demodulated phasemust be less than sensitivity.
- c) The wrapped phase is G-times more sensitive than.
- d) The unwrapped phaseis roughly approximated by
- f) If the phase error (Eq. (10)) lies outside 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 estimation of.
- g) If (for example) we increase the phase sensitivity from to , both the standard temporal phase unwrapper  and our 2-steps one, would give the same unwrapping results. But the standard temporal unwrapper  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-noise. For example a 7-step (with sensitivity increase) using the standard unwrapper  would increase 49-times the signal-to-noise power ratio (see Eq. (4)) with reference to the estimate. This is the same amount of signal-to-noise increase obtained using our 2-step temporal unwrapper using only the and phase-maps sensitivities (see Fig. 8).
In brief, all anti-noise benefits of N-steps standard temporal phase unwrapping  (Eqs. (19) y (20)) are kept unchanged, except that we are using only the 2 extreme phase-maps (see Fig. 1 and Eq. (6)); this holds whenever the condition in Eq. (10) is maintained.
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
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