Super-sensitive two-wavelength fringe projection profilometry with 2-sensitivities temporal unwrapping

Optics and Lasers in Engineering 106 (2018) 68–74

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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng

Super-sensitive two-wavelength fringe projection profilometry with 2-sensitivities temporal unwrapping Manuel Servin∗, Moises Padilla, Guillermo Garnica Centro de Investigaciones en Optica A.C., 37150 Leon, Guanajuato, Mexico

a r t i c l e

i n f o

Keywords: Two-wavelength profilometry Phase demodulation Phase metrology Fringe analysis Temporal unwrapping

a b s t r a c t Since the early 1970s, optical two-wavelength phase-metrology (TWPM) has been used in a wide variety of experimental set ups. In TWPM one may compute the phase-sum and the phase-difference of two close phase measurements. Early TWPM optically computed the phase difference and phase sum by double exposure holography. However soon after, TWPM became almost synonymous to calculating the phase-difference only. The more sensitive phase-sum was largely forgotten. The standard application for phase-difference TWPM is to extend the phase measurement depth without phase-unwrapping for discontinuous phase-objects. This phase-difference, while non-wrapped, decreases however the signal-to-noise ratio (SNR) of the estimated phase. On the other hand, the phase-sum increases the phase sensitivity, and the SNR of the estimated phase. In spite of these two great advantages, the use of the phase-sum in TWPM has been almost ignored. In this paper we review and set the stage for digital TWPM for super-sensitive phase-sum estimation. This is coupled with two-sensitivity phase-unwrapping to obtain extended-range super-sensitive fringe-projection profilometry estimations. Here we mathematically prove, and experimentally show that using the phase-sum one obtains a huge increase in SNR with respect to using the phase-difference alone. The pioneer works on double exposure TWPM holography that uses the phase-difference and phase-sum are also properly acknowledged. Finally, two experimental results from fringe-projection profilometry that clearly show the huge SNR gain of the phase-sum, with respect to the phase-difference is now mathematically well established. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction The use of the phase-sum and phase-difference to increase or decrease, the sensitivity in two-wavelength phase-metrology (TWPM) was proposed for the first time by Mustafin and Seleznev in 1970 [1]. Mustafin and Seleanev published the equivalent wavelength formulas for the phase-difference as 𝜆1 𝜆2 ∕(𝜆1 − 𝜆2 ), and the equivalent lambda formula 𝜆1 𝜆2 ∕(𝜆1 + 𝜆2 ) for the phase-sum. At about the same time Weigl [2,3] also published a two-wavelength supersensitive analog holography technique. Weigl submitted his first paper in May 1970, but it was published until February 1971 [2]. To the best of our knowledge, these three papers are the pioneer works that make explicit use of the phase-sums (𝜑1 + 𝜑2 ) and phase-difference (𝜑1 − 𝜑2 ) in analog holographic doubleexposure TWPM [1–3]. In 1971, J. C. Wyant implemented a TWPM using the phase-difference only, for interferometric testing of aspheric optical elements by double-exposure holography [4]. The TWPM using the phase-difference only was applied for the first time using digital interferometry by C. Polhemus in 1973 using an IBM-370 computer [5]. Similarly, Y. Y. Cheng and J. C. Wyant described TWPM phase-difference

∗

techniques using digital phase-shifting algorithms (PSA) in 1984–1985 [6,7]. Afterwards, Onodera and Ishii used Fourier phase-demodulation for profiling a mechanical structure with long equivalent-wavelength 𝜆1 𝜆2 ∕|𝜆1 − 𝜆2 | [8]; unfortunately, their results were over-smoothed due to excessive Fourier spatial filtering. Phase-difference TWPM techniques working with two close-sensitive phases {𝜑1 , 𝜑2 }, 𝜑1 ≈ 𝜑2 (large 𝜆1 𝜆2 ∕|𝜆1 − 𝜆2 |), has been also applied to holographic microscopy [9]; extended range optical metrology [10]; parallel two-step digital holography [11]; multi-wavelength extended-range contouring [12], and two-wavelength surface profiling [13], to name a few. Particularly in profilometry, TWPM using the phase-difference only, has been called, among other names, twofrequency profilometry [14]; profilometry without phase unwrapping [15]; dual-frequency shape measurement [16]; large-depth discontinuous objects profilometry [17]; deflectometry of composite fringe phase retrieval [18]; absolute-height fringe-projection profilometry [19]; dualwavelength two-steps phase-shifting demodulation [20]; two-fringe patterns absolute-phase recovery [21]. Even though these methods were applied to different optical phasemetrology experiments, all of them share the same mathematical back-

Corresponding author. E-mail address: [email protected] (M. Servin).

https://doi.org/10.1016/j.optlaseng.2018.02.012 Received 28 November 2017; Received in revised form 25 January 2018; Accepted 18 February 2018 0143-8166/© 2018 Elsevier Ltd. All rights reserved.

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Optics and Lasers in Engineering 106 (2018) 68–74

ground; they all use the phase difference 𝜑2 − 𝜑1 only. This produces a non-wrapped extended-range phase-measurements because the phasedifference equivalent wavelength is very large 𝜆1 𝜆2 ∕|𝜆1 − 𝜆2 | [1–4]; this feature is so useful that TWPM became almost synonymous to phase-difference only, for more than four decades (e.g., [4–60]). Summarizing, in TWPM where 𝜑2 ≈ 𝜑1 , the phase-sum 𝜑2 + 𝜑1 give super phase-sensitivity with equivalent wavelength of 𝜆1 𝜆2 ∕(𝜆1 + 𝜆2 ). In contrast, their phase-difference 𝜑2 − 𝜑1 , give extended-range non-wrapped phase-estimation with equivalent wavelength of 𝜆1 𝜆2 ∕|𝜆1 − 𝜆2 | [1-62]. The details are discussed in the next section. While analyzing the cited references [1-62], we notice that the phase-difference 𝜑2 − 𝜑1 technique has been cited collectively more than 2000 times [4–50]. In contrast, the use of the phase-sum 𝜑2 + 𝜑1 technique has been cited about 50 times. Among these 50 citations there are two reviews from 1980 and 1991 [51,52], but most of the time the phase-sum was simply listed as a possible application for dualwavelength lasers (e.g., [53–57]).Only five applications that explicity use both the phase-sum and phase-difference are reported to this day [58–62]; but none of them (including an early version of this manuscript at arXiv) cite the pioneer works by Mustafin, Seleznev, and Weigl [1– 3]. Therefore we decided, in this manuscript to pay due recognition for these unfairly ignored pioneer works [1-3]. Here we mathematically prove, and later confirm with experimental fringe projection profilometry, that the main characteristics of TWPM are: (a) The SNR of the phase sum 𝜑𝑆 = 𝜑1 + 𝜑2 is very high compared to that of the phase-difference 𝜑𝐷 = 𝜑2 − 𝜑1 . (b) The phase-noise limitations to unwrap the phase-sum 𝜑1 + 𝜑2 , taking the non-wrapped (𝜑1 − 𝜑2 ) ∈ (−𝜋, 𝜋) as stepping stone. The plan of this paper is as follows. Section 2 reviews the state of the art in TWPM up to mid-2017. In Section 3 we derive the phase-sum 𝜑𝑆 = 𝜑1 + 𝜑2 and phase-difference 𝜑𝐷 = 𝜑2 − 𝜑1 working with demodulated analytic signals (the most natural and convenient approach in our opinion). In Section 4 we analyze the SNR for {𝜑S , 𝜑D }. Section 5 contains our close-form formula for error-free unwrapping of 𝜑1 + 𝜑2 from non-wrapped 𝜑2 − 𝜑1 in presence of additive white-Gaussian noise (AWGN). In Section 6 we use fringe-projection profilometry to exemplify our TWPM theory. Finally, our conclusions are listed in Section 7.

Eq. (1) we may find two synthetic-wavelengths {ΛD ,ΛS } as, 𝜆1 𝜆2 ; |𝜆1 − 𝜆2 |

for

Λ𝑆 =

𝜆1 𝜆2 ; |𝜆1 + 𝜆2 |

for

) 2𝜋 2𝜋 − 𝑤; 𝜆1 𝜆2 ( ) 2𝜋 2𝜋 𝜑1 + 𝜑2 = + 𝑤. 𝜆1 𝜆2 𝜑1 − 𝜑2 =

(2)

The wavelength ΛD is much longer than either {𝜆1 , 𝜆2 }, or ΛD >> {𝜆1 , 𝜆2 }. In contrast, ΛS is shorter than {𝜆1 , 𝜆2 }; or Λ𝑆 = (𝜆1 + 𝜀)∕(2 + 𝜀) ≈ 𝜆1 /2; for 𝜆2 = (1 + 𝜀)𝜆1 and ɛ ≈ 0. Given that 𝜑𝑆 = 𝐺𝜑𝐷 (G >> 1.0), the phase-sum is highly wrapped: 𝜑𝑊 = 𝑊 (𝜑𝑆 ), being 𝑊 (𝑥) = Arg[exp(𝑖 𝑥)]. However 𝜑𝑊 can be un𝑆 𝑆 wrapped from 𝜑D by two-sensitivity temporal phase-unwrapping as [50,63]: [ ] 𝜑𝑆 = 𝐺𝜑𝐷 + 𝑊 𝜑𝑊 − 𝐺𝜑𝐷 ; 𝑆 { ( )} 𝜑𝐷 , 𝜑𝑆 − 𝐺𝜑𝐷 ∈ (−𝜋, 𝜋).

(3)

This simple two-sensitivity unwrapping formula was first published in [63], and predates the page-long unwrapping algorithm in [60]. Eq. (3) unwraps 𝜑𝑊 directly from 𝜑𝐷 ∈ (−𝜋, 𝜋). This is the status in TWPM 𝑆 as mid-2017. 3. Sensitivity gain between the phase-sum and phase-difference Two phase-demodulated analytic signals can be obtained using the fringe-patterns in Eq. (1) and the least-squares M-step PSA as [64]: 𝐴1 (𝑥, 𝑦)𝑒𝑖 𝐴2 (𝑥, 𝑦)𝑒

𝜑1 (𝑥,𝑦)

𝑖 𝜑2 (𝑥,𝑦)

( ) 2𝜋 𝐼1 𝑥, 𝑦, 𝑚 𝑒𝑖 𝑀 𝑚=0 𝑀−1 ) ∑ ( 2𝜋 = 𝐼2 𝑥, 𝑦, 𝑚 𝑒𝑖 𝑀 𝑚=0 =

𝑀−1 ∑

2𝜋 𝑚 𝑀

, (4)

2𝜋 𝑚 𝑀

.

Then, the following products are computed, [ ]∗ 𝐴1 𝑒𝑖 𝜑1 𝐴2 𝑒𝑖 𝜑2 = 𝐴1 𝐴2 𝑒𝑖 [𝜑1 −𝜑2 ] ; [ ] 𝐴1 𝑒𝑖 𝜑1 𝐴2 𝑒𝑖 𝜑2 = 𝐴1 𝐴2 𝑒𝑖 [𝜑1 +𝜑2 ] .

(5)

Obtaining,

2. State of the art in two-wavelength phase metrology From 1971 to 2012, TWPM was almost synonymous to demodulating two close-sensitive phases {𝜑1 , 𝜑2 }, 𝜑1 ≈ 𝜑2 , to obtain a longwavelength, non-wrapped phase-difference 𝜑𝐷 = (𝜑2 − 𝜑1 ) ∈ (−𝜋, 𝜋). In 2013, Di et al. [58] proposed to use the phase-sum 𝜑𝑆 = 𝜑2 + 𝜑1 to obtain higher sensitivity and higher SNR phase measurements, ignoring the existence of the pioneering papers [1–3]. As mid-2017, this paper has been cited only twice in peer-reviewed journals: a self-citation in 2015 [59], and in 2017 by Xiong et al. [60]. This is unfortunate because we consider that combining 𝜑𝑆 = (𝜑2 + 𝜑1 ) and 𝜑𝐷 = (𝜑2 − 𝜑1 ) should be the standard practice in modern digital TWPM. As described in an early version of this manuscript posted at the arXiv repository [61], and afterwards pointed also by Wang et al. [62], the super-sensitive phase-sum can be unwrapped directly from the lowersensitive, non-wrapped phase-difference. It should be noted that Di et al. [58,59] only dealt with continuous wavefronts so 𝜑S was spatially unwrapped, and no need for temporal unwrapping was required. Let us start by considering two fringe patterns with closewavelengths phase-modulation (𝜆1 ≈ 𝜆2 ) as, ] 2𝜋 𝐼1 (𝑥, 𝑦, Δ) = 𝑎(𝑥, 𝑦) + 𝑏(𝑥, 𝑦) cos 𝑤(𝑥, 𝑦) + Δ ; 𝜆1 [ ] 2𝜋 𝐼2 (𝑥, 𝑦, Δ) = 𝑎(𝑥, 𝑦) + 𝑏(𝑥, 𝑦) cos 𝑤(𝑥, 𝑦) + Δ . 𝜆2

(

Λ𝐷 =

𝜆1 𝜆2 2𝜋 ; 𝑤; Λ𝐷 = |𝜆 − 𝜆 | Λ𝐷 1 2| | [ ] [ ] 𝜆1 𝜆2 2𝜋 . = 𝑊 𝜑1 + 𝜑2 = 𝑊 𝑤 ; Λ𝑆 = |𝜆 + 𝜆 | Λ𝑆 1 2| |

𝜑𝐷 = 𝜑1 − 𝜑2 = 𝜑𝑊 𝑆

(6)

The sensitivity gain G, between {𝜑S , 𝜑D } is therefore, 𝐺=

𝜑𝑆 𝜆 + 𝜆2 Λ = 𝐷. = 1 |𝜆 − 𝜆 | 𝜑𝐷 Λ𝑆 2| | 1

(7)

That is, the phase-sum 𝜑S (x, y) is 𝐺 = (Λ𝐷 ∕Λ𝑆 ) times more sensitive than 𝜑D (x, y). This result was implicitly used before [58–60], but not with a clearer description that an explicit formula provides as we do here. The SNR for 𝜑S (x, y) is also closely related to the phase-sensitivity increase 𝐺 = (Λ𝐷 ∕Λ𝑆 ). 4. SNR between the phase-sum and phase-difference In practice, the demodulated phases {𝜑1 , 𝜑2 } are corrupted by additive white noise {n1 , n2 } as [64],

[

𝜑1 (𝑥, 𝑦) → 𝜑1 (𝑥, 𝑦) + 𝑛1 (𝑥, 𝑦), 𝜑2 (𝑥, 𝑦) → 𝜑2 (𝑥, 𝑦) + 𝑛2 (𝑥, 𝑦).

(1)

(8)

We will assume {n1 , n2 } are uncorrelated samples of a zero-mean AWGN with variance 𝜎 2 = 𝐸{𝑛21 } = 𝐸{𝑛22 }; being E{ · } the ensemble average [65]. Then the noisy phase-difference and phase-sum are,

Here a(x, y) is the background; b(x, y) the contrast; w(x, y) the wavefront under measurement, and Δ is a piston phase-shifting. From 69

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(

) 2𝜋 𝑤 + 𝑛2 − 𝑛1 ; Λ𝐷 ( ) 2𝜋 𝜑𝑆 = 𝜑2 + 𝜑1 = 𝑤 + 𝑛2 + 𝑛1 . Λ𝑆

𝜑𝐷 = 𝜑2 − 𝜑1 =

(9)

In a Banach space, (1∕𝐴Ω ) ∬ |𝑓 |2 𝑑Ω represents the average power of f; being 𝐴Ω the area of well-defined fringes Ω. Consequently, the SNR for {𝜑D , 𝜑S } are, ( SNR(𝜑2 − 𝜑1 ) =

SNR(𝜑2 + 𝜑1 ) = Fig. 1. Schematic of a fringe-projection profilometer. The digitizing solid is a sphere segment and 𝜃 is the sensitivity angle of this profilometer.

2𝜋 Λ𝐷

)2

∬(𝑥,𝑦)∈Ω |𝑤|2 𝑑Ω

2 ∬(𝑥,𝑦)∈Ω ||𝑛2 − 𝑛1 || 𝑑Ω ( )2 2𝜋 ∬(𝑥,𝑦)∈Ω |𝑤|2 𝑑Ω Λ 𝑆

2 ∬(𝑥,𝑦)∈Ω ||𝑛2 + 𝑛1 || 𝑑Ω

;

.

(10)

Assuming that {n1 , n2 } are ergodic and stationary, the noise-power of both {𝑛2 − 𝑛1 }, and {𝑛2 + 𝑛1 } are the same [65], so the SNR gain between 𝜑S (x, y) and 𝜑D (x, y) is, ( ) SNR(𝜑1 + 𝜑2 ) Λ𝐷 2 = = 𝐺2 . (11) SNR(𝜑1 − 𝜑2 ) Λ𝑆 Thus, 𝜑S (x, y) has G2 times higher SNR than 𝜑D (x, y). As far as we know, this result has not been published before. Particularly [58–60] do not analyze the SNR power gain between the phase-sum 𝜑𝑆 = (𝜑2 + 𝜑1 ) and the phase-difference 𝜑𝐷 = (𝜑2 − 𝜑1 ). In fringe-projection profilometry one normally has high SNR fringes, so the application of TWPM has no major problems. In contrast, the fringe phase-noise is much higher in Electronic Speckle Pattern Interferometry (ESPI), and a narrow lowpass filtering must be applied before using the TWPM technique herein reported. 5. Error-free temporal unwrapping of the noisy phase-sum Let us analyze the impact of small phase-noises on the phase-sum and phase-difference: 𝜑𝑊 → (𝜑𝑊 + 𝑛𝑆 ) and 𝜑𝐷 → (𝜑𝐷 + 𝑛𝐷 ), where the 𝑆 𝑆 phase-noise amplitude norms {|nS |max , |nD |max } are assumed to be small, {|nS |max << 𝜋, |nD |max << 𝜋}. Applying the noisy phases (𝜑𝐷 + 𝑛𝐷 ) and (𝜑𝑊 + 𝑛𝑆 ) to our temporal unwrapper in Eq. (3), we obtain: 𝑆 [ ( )] 𝜑𝑆 + 𝑛𝑆 = 𝐺(𝜑𝐷 + 𝑛𝐷 ) + 𝑊 (𝜑𝑊 (12) 𝑆 + 𝑛𝑆 ) − 𝐺 𝜑𝐷 + 𝑛𝐷 . The conditions for error-free unwrapping of noisy phases then become: ( ) 𝜑𝐷 + 𝑛𝐷 ∈ (−𝜋, 𝜋); (13) [ ( )] 𝜑𝑆 + 𝑛𝑆 − 𝐺 𝜑𝐷 + 𝑛𝐷 ∈ (−𝜋, 𝜋).

Fig. 2. Digital photograph of a metallic spherical cap used as calibration object.

Fig. 3. A metallic spherical cap being illuminated by linear fringes and their phase demodulation. The phase-shift among successive fringe patterns is 2𝜋/4. The upper row has lower phase-sensitivity fringes than the lower row. The right column shows the demodulated wrapped phases. 70

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Fig. 4. The upper row shows the phase-difference while the lower row the phase-sum. The phase difference has about 0.7 lambda sensitivity (0.7×2𝜋) and nonwrapped. The phase-sum is however wrapped about 9-times (9×2𝜋).

Fig. 5. 3D rendering of the recovered unwrapped phase-sum of the spherical solid. The left-side uses grayscale pseudo-color, while the right-side uses the fringes’ contrast function as texture. Fig. 7. Digital photograph of a spiral fluorescent lamp under analysis.

6. Application to phase-sum fringe-projection profilometry Here we present two examples of super-sensitive TWPM applied to fringe-projection profilometry. These examples fully agree with the three previous papers that use the phase-sum in other phase-metrology areas [58–60]. Digital fringe-projection profilometry has been known for many years, see for example [66]. A set up for 3D profilometry of solids is shown in Fig. 1. We used a 800 × 600 pixels, 3-LCD multimedia projector (EPSON s17); a 1600 × 1200 pixels monochrome CCD digital (iDS UI-2250). The CCD camera lens used was a 6.5-52 mm (KOWA LMVZ655). The mathematical model for the fringes I1 (x, y) and I2 (x, y) taken by the CCD camera are: for 𝑚 = {0, 1, 2, 3}, [ ] 2𝜋 𝐼1 (𝑚) = 𝑎 + 𝑏 cos 𝑢1 tan (𝜃)ℎ(𝑥, 𝑦) + 𝑢1 𝑥 + 𝑚 ; 4 ] (15) [ 2𝜋 𝐼2 (𝑚) = 𝑎 + 𝑏 cos 𝑢2 tan (𝜃)ℎ(𝑥, 𝑦) + 𝑢2 𝑥 + 𝑚 . 4

Fig. 6. Central cuts of the digitized spherical cap. The red trace is the phasedifference and the blue trace is the unwrapped phase-sum. The blue trace has significant lower phase-noise. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The (x, y) dependency on {a,b,I} was omitted for simplicity. h(x, y) represents the solid’s height-profile and 𝜃 is the sensitivity angle. The spatial carriers {u1 , u2 } are in radians/pixel. The sensitivity gain between the phase-sum and phase-difference is, 𝜑 𝑢 + 𝑢2 . 𝐺= 𝑆 = 1 (16) |𝑢 − 𝑢 | 𝜑𝐷 2| | 1

Substituting the sensitivity gain between the phase sum and difference, 𝜑𝑆 − 𝐺𝜑𝐷 = 0 (or 𝜑𝑆 = 𝐺𝜑𝐷 ), the second condition in Eq. (13) depends only on the noise amplitudes and sensitivity gain: (

) 𝑛𝑆 − 𝐺𝑛𝐷 ∈ (−𝜋, 𝜋);

𝐺=

Λ𝐷 >> 1.0 Λ𝑆

(14)

All previous results apply taking (Λ𝐷 ∕Λ𝑆 ) → (𝑢1 + 𝑢2 )∕|𝑢1 − 𝑢2 |.

This is a pure phase-noise restrictive condition for error-free twowavelength (two-sensitivities) unwrapping of noisy measured phases, and has not been reported before. However using a PSA with a large number of phase steps, we can obtain low enough phase-difference noise.

6.1. Profilometer calibration with a spherical cap The metallic spherical segment in Fig. 2 is used as calibrating solid. Fig. 3 shows the two sets of carrier fringe patterns with very similar 71

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Fig. 8. The spiral lamp illuminated by two close-sensitive carrier-frequency fringes. The lower carrier fringes are shown in the first row, while the higher carrier fringes are shown in the second row. The wrapped phases demodulated by 4-step PSA are shown at the right column.

Fig. 9. The upper row shows the non-wrapped phase-difference while the lower row shows the highly wrapped phase-sum. The phase-sum has much higher sensitivity, so it is highly wrapped.

spatial-frequencies (similar phase sensitivities) having 𝑢1 = 2𝜋∕7 and 𝑢2 = 2𝜋∕6 radians/pixel. Fig. 4 shows the phase-subtraction (upper row) and the phaseaddition (lower row) of the two demodulated phases. The phasesensitivity gain G between the phase-sum and phase-difference is given by, 𝐺=

𝑢1 + 𝑢2 (2𝜋∕6 + 2𝜋∕7)radians∕pixel = = 13. |𝑢1 − 𝑢2 | |2𝜋∕6 − 2𝜋∕7|radians∕pixel | |

The upper row spatial carrier is 𝑢1 = 2𝜋∕9 radians/pixel, while the lower row spatial carrier is 𝑢2 = 2𝜋∕8 radians/pixel. Fig. 9 shows the phase-difference 𝜑D (x, y) and phase-sum 𝜑S (x, y) of the demodulated phase-maps. The phase sensitivity gain G between the phase-sum and phase-difference for this experiment is given by 𝐺=

(17)

𝑢1 + 𝑢2 (2𝜋∕8 + 2𝜋∕9)radians∕pixel = = 17. |𝑢 − 𝑢 | |2𝜋∕8 − 2𝜋∕9|radians∕pixel 2| | 1

(18)

Fig. 10 shows the unwrapped super-sensitive phase-sum. Fig. 11 shows two phase-cuts of the digitized spiral fluorescent lamp; A regular zoom-in photograph is shown in the far right to see the joint between these two plastic pieces. This joint depression is hardly seen in the phase-difference because it is immersed in measuring phase noise.

Fig. 5 shows the unwrapped phase-sum 𝜑S (x, y) taking as first estimation the non-wrapped 𝜑D (x, y). In Fig. 6 we show a central phase-cut of 𝜑D (x, y) and 𝜑S (x, y) to gauge the different amounts of degrading phasenoise. In the zoomed-in detail one may see the difference in phase-noise amplitude corresponding to the noisier phase-difference in red, and the blue phase-sum.

7. Summary and conclusions 6.2. Digitization of a fluorescent spiral lamp We have presented a general mathematical theory for optical supersensitivity two-wavelength phase-metrology (TWPM). The TWPM technique consists in taking two close-sensitivity phase-measurements {𝜑1 ,

Fig. 7 shows our next test object: a spiral fluorescent lamp. Fig. 8 shows phase-shifted linear-fringes with two different spatial carriers. 72

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Fig. 10. 3D rendering of the unwrapped super-sensitive phase-sum. The left-side uses grayscale pseudo-color, while the right-side uses the fringes’ contrast function as texture.

Fig. 11. Comparison of the SNR between the phase-difference (in red), and the super-sensitive phase-sum (in blue). We have zoomed-in to clearly see the depression at the joining of the two plastic pieces shown in the photograph-detail. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

𝜑2 }, 𝜑1 ≈ 𝜑2 . Then one obtains their phase-sum (𝜑1 + 𝜑2 ), and phasedifference (𝜑1 − 𝜑2 ). If {𝜑1 , 𝜑2 } are close enough, one obtains a nonwrapped phase-difference 𝜑𝐷 = (𝜑2 − 𝜑1 ) ∈ (−𝜋, 𝜋). We derived formulas to quantify the signal-to-noise ratio (SNR) for the phase-difference and phase-sum in TWPM. We also gave the conditions for error-free phase-unwrapping of the higher-sensitivity noisy phase-sum (𝜑1 + 𝜑2 ) taking the non-wrapped noisy phase-difference 𝜑𝐷 ∈ (−𝜋, 𝜋) as stepping stone. We also recognized the pioneer works from early 1970s that proposed the use of the phase-sums and the phase-differences for TWPM, which have been almost ignored to this day. Finally, two experimental results in fringe-projection profilometry were shown, digitizing continuous and discontinuous (piece-wise continuous) solids, to illustrate the ease of implementation and flexibility of this super-sensitive fringeprojection profilometry technique using (𝜑1 + 𝜑2 ) and (𝜑1 − 𝜑2 ).

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