Generic nonlinear error compensation algorithm for phase measuring profilometry

Xin Yu; Shanshan Lai; Yuankun Liu; Wenjing Chen; Junpeng Xue; Qican Zhang

doi:10.3788/COL202119.101201

Journals >Chinese Optics Letters >Volume 19 >Issue 10 >Page 101201 > Article

Chinese Optics Letters
Vol. 19, Issue 10, 101201 (2021)

Generic nonlinear error compensation algorithm for phase measuring profilometry

Xin Yu¹, Shanshan Lai¹, Yuankun Liu^1,*, Wenjing Chen¹..., Junpeng Xue^1,2 and Qican Zhang¹|Show fewer author(s)

Author Affiliations

¹Opto-Electronic Department, Sichuan University, Chengdu 610065, China

²School of Aeronautics and Astronautics, Sichuan University, Chengdu 610065, China

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DOI: 10.3788/COL202119.101201 Cite this Article Set citation alerts

Xin Yu, Shanshan Lai, Yuankun Liu, Wenjing Chen, Junpeng Xue, Qican Zhang, "Generic nonlinear error compensation algorithm for phase measuring profilometry," Chin. Opt. Lett. 19, 101201 (2021) Copy Citation Text

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Abstract

In this Letter, the periodical errors, which are caused by the nonlinear effect of the commercial projector and camera, are analyzed as a more generic single-coefficient model. The probability density function of the wrapped phase distributions is used as a tool to find the compensation coefficient. When the compensation coefficient is detected, on the premise of ensuring accuracy, a correlation algorithm process is used to replace the traditional iterative process. Therefore, the proposed algorithm improves the efficiency of coefficient detection dramatically. Both computer simulation and experiment show the effectiveness of this method.

Keywords

nonlinear response phase error probability density function three-dimensional sensing

I_{n} (x, y) = a (x, y) + b (x, y) \cos [φ (x, y) + δ_{n}],

(1)

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I_{n}^{c} (x, y) = R (x, y) {[I_{n} (x, y)]}^{γ} + I_{0},

(2)

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I_{n}^{c} = A + \sum_{i = 1}^{\infty} B_{i} \cos [i (φ + δ_{n})],

(3)

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φ = - \arctan (\frac{\sum_{n = 0}^{N - 1} I_{n} \sin δ_{n}}{\sum_{n = 0}^{N - 1} I_{n} \cos δ_{n}}) .

(4)

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φ' = - \arctan {\frac{\sum_{n = 0}^{2} {A + \sum_{i = 1}^{5} B_{i} \cos [i (φ + δ_{n})]} \sin δ_{n}}{\sum_{n = 0}^{2} {A + \sum_{i = 1}^{5} B_{i} \cos [i (φ + δ_{n})]} \cos δ_{n}}} .

(5)

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Δ φ = φ' - φ .

(6)

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Δ φ = - \arctan [\frac{(B_{2} - B_{4}) \sin (3 φ) + B_{5} \sin (6 φ)}{B_{1} + (B_{2} + B_{4}) \cos (3 φ) + B_{5} \cos (6 φ)}] .

(7)

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Δ φ = - \arctan [\frac{(K_{2} - K_{4}) \sin (3 φ) + K_{5} \sin (6 φ)}{1 + (K_{2} + K_{4}) \cos (3 φ) + K_{5} \cos (6 φ)}],

(8)

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Δ φ ≅ - \arctan [\frac{K_{2} \sin (3 φ)}{1 + K_{2} \cos (3 φ)}] .

(9)

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Δ φ ≅ \frac{- 12 K_{2} \sin (3 φ) - 12 K_{2}^{2} \sin (6 φ) - 4 K_{2}^{3} \sin (9 φ)}{12 - 18 K_{2}^{2} + (36 K_{2} - 3 K_{2}^{3}) \cos (3 φ) + 18 K_{2}^{2} \cos (6 φ) + 3 K_{2}^{3} \cos (9 φ)} .

(10)

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Δ φ = - c_{1} \sin (3 φ) - c_{2} \sin (6 φ) .

(11)

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F (m) = P {2 π m / M - π \leq φ_{m} < 2 π (m + 1) / M - π},

(12)

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R_{p} = \frac{\sum F \cdot F_{p}}{\sum F^{2} \cdot \sum F_{p}^{2}},

(13)

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Xin Yu, Shanshan Lai, Yuankun Liu, Wenjing Chen, Junpeng Xue, Qican Zhang, "Generic nonlinear error compensation algorithm for phase measuring profilometry," Chin. Opt. Lett. 19, 101201 (2021)

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