Abstract

In this study, a novel two-dimensional spatial coding pattern called two-dimensional Gold matrix method is proposed for general two-dimensional positioning. Considering the difficulty in representing a two-dimensional position in a single binary matrix, constructing a matrix while each submatrix refers to its location is a challenging mathematical problem. The general two-dimensional signal can be labeled by the two-dimensional Gold matrix, which results from a preferred pair of two m-sequences. For a pseudorandom m-sequence, the span-n property of the two-dimensional Gold matrix states that every n×n submatrix is unique and the decoding is fast and convenient. Numerical simulation and a proof-of-principle experiment are performed, and experimental results verified that the two-dimensional Gold matrix method is effective for high resolution and large range two-dimensional measurements.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (3)

2016 (2)

S. Van der Jeught and J. J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).
[Crossref]

V. Horan and B. Stevens, “Locating patterns in the de Bruijn torus,” Discrete Math. 339(4), 1274–1282 (2016).
[Crossref]

2015 (3)

Y. K. Liu, Q. C. Zhang, and X. Y. Su, “3D shape from phase errors by using binary fringe with multi-step phase-shift technique,” Opt. Lasers Eng. 74, 22–27 (2015).
[Crossref]

T. Dziwinski, “A Novel Approach of an Absolute Encoder Coding Pattern,” IEEE Sens. J. 15(1), 397–401 (2015).
[Crossref]

H. Wang, J. Wang, B. Chen, P. Xiao, X. Chen, N. Cai, and B. W. K. Ling, “Absolute optical imaging position encoder,” Measurement 67, 42–50 (2015).
[Crossref]

2014 (2)

M. Schaffer, M. Grosse, B. Harendt, and R. Kowarschik, “Statistical patterns: an approach for high-speed and high-accuracy shape measurements,” Opt. Eng. 53(11), 112205 (2014).
[Crossref]

K. Liu, C. Zhou, S. Wei, S. Wang, X. Fan, and J. Ma, “Optimized stereo matching in binocular three-dimensional measurement system using structured light,” Appl. Opt. 53(26), 6083–6090 (2014).
[Crossref] [PubMed]

2013 (1)

Q. C. Zhang and Z. Y. Wu, “A carrier removal method in Fourier transform profilometry with Zernike polynomials,” Opt. Lasers Eng. 51(3), 253–260 (2013).
[Crossref]

2011 (1)

J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photonics 3(2), 128–160 (2011).
[Crossref]

2010 (2)

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43(8), 2666–2680 (2010).
[Crossref]

2009 (2)

J. Zhang, C. Zhou, and X. Wang, “Three-dimensional profilometry using a Dammann grating,” Appl. Opt. 48(19), 3709–3715 (2009).
[Crossref] [PubMed]

B. Jackson, B. Stevens, and G. Hurlbert, “Research problems on Gray codes and universal cycles,” Discrete Math. 309(17), 5341–5348 (2009).
[Crossref]

Araki, H.

Cai, N.

H. Wang, J. Wang, B. Chen, P. Xiao, X. Chen, N. Cai, and B. W. K. Ling, “Absolute optical imaging position encoder,” Measurement 67, 42–50 (2015).
[Crossref]

Chang, C.

Chen, B.

H. Wang, J. Wang, B. Chen, P. Xiao, X. Chen, N. Cai, and B. W. K. Ling, “Absolute optical imaging position encoder,” Measurement 67, 42–50 (2015).
[Crossref]

Chen, X.

H. Wang, J. Wang, B. Chen, P. Xiao, X. Chen, N. Cai, and B. W. K. Ling, “Absolute optical imaging position encoder,” Measurement 67, 42–50 (2015).
[Crossref]

Dirckx, J. J. J.

S. Van der Jeught and J. J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).
[Crossref]

Dziwinski, T.

T. Dziwinski, “A Novel Approach of an Absolute Encoder Coding Pattern,” IEEE Sens. J. 15(1), 397–401 (2015).
[Crossref]

Fan, X.

Fernandez, S.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43(8), 2666–2680 (2010).
[Crossref]

Fujiwara, M.

Geng, J.

J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photonics 3(2), 128–160 (2011).
[Crossref]

Grosse, M.

M. Schaffer, M. Grosse, B. Harendt, and R. Kowarschik, “Statistical patterns: an approach for high-speed and high-accuracy shape measurements,” Opt. Eng. 53(11), 112205 (2014).
[Crossref]

Harendt, B.

M. Schaffer, M. Grosse, B. Harendt, and R. Kowarschik, “Statistical patterns: an approach for high-speed and high-accuracy shape measurements,” Opt. Eng. 53(11), 112205 (2014).
[Crossref]

Horan, V.

V. Horan and B. Stevens, “Locating patterns in the de Bruijn torus,” Discrete Math. 339(4), 1274–1282 (2016).
[Crossref]

Hurlbert, G.

B. Jackson, B. Stevens, and G. Hurlbert, “Research problems on Gray codes and universal cycles,” Discrete Math. 309(17), 5341–5348 (2009).
[Crossref]

Ikawa, S.

Ito, T.

Jackson, B.

B. Jackson, B. Stevens, and G. Hurlbert, “Research problems on Gray codes and universal cycles,” Discrete Math. 309(17), 5341–5348 (2009).
[Crossref]

Kakue, T.

Kong, L.

Kowarschik, R.

M. Schaffer, M. Grosse, B. Harendt, and R. Kowarschik, “Statistical patterns: an approach for high-speed and high-accuracy shape measurements,” Opt. Eng. 53(11), 112205 (2014).
[Crossref]

Li, Y.

Ling, B. W. K.

H. Wang, J. Wang, B. Chen, P. Xiao, X. Chen, N. Cai, and B. W. K. Ling, “Absolute optical imaging position encoder,” Measurement 67, 42–50 (2015).
[Crossref]

Liu, K.

Liu, R.

Liu, Y. K.

Y. K. Liu, Q. C. Zhang, and X. Y. Su, “3D shape from phase errors by using binary fringe with multi-step phase-shift technique,” Opt. Lasers Eng. 74, 22–27 (2015).
[Crossref]

Llado, X.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43(8), 2666–2680 (2010).
[Crossref]

Ma, J.

Maeda, Y.

Nakayama, H.

Niwase, H.

Oikawa, M.

Pribanic, T.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43(8), 2666–2680 (2010).
[Crossref]

Qi, Y.

Salvi, J.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43(8), 2666–2680 (2010).
[Crossref]

Schaffer, M.

M. Schaffer, M. Grosse, B. Harendt, and R. Kowarschik, “Statistical patterns: an approach for high-speed and high-accuracy shape measurements,” Opt. Eng. 53(11), 112205 (2014).
[Crossref]

Shimobaba, T.

Si, Y.

Stevens, B.

V. Horan and B. Stevens, “Locating patterns in the de Bruijn torus,” Discrete Math. 339(4), 1274–1282 (2016).
[Crossref]

B. Jackson, B. Stevens, and G. Hurlbert, “Research problems on Gray codes and universal cycles,” Discrete Math. 309(17), 5341–5348 (2009).
[Crossref]

Su, X. Y.

Y. K. Liu, Q. C. Zhang, and X. Y. Su, “3D shape from phase errors by using binary fringe with multi-step phase-shift technique,” Opt. Lasers Eng. 74, 22–27 (2015).
[Crossref]

Takada, N.

Tu, C.

Van der Jeught, S.

S. Van der Jeught and J. J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).
[Crossref]

Wang, H.

L. Kong, R. Liu, Y. Si, Z. Wang, C. Tu, Y. Li, and H. Wang, “Time-resolved multiple imaging by detecting photons with changeable wavelengths,” Chin. Opt. Lett. 15(8), 081101 (2017).
[Crossref]

H. Wang, J. Wang, B. Chen, P. Xiao, X. Chen, N. Cai, and B. W. K. Ling, “Absolute optical imaging position encoder,” Measurement 67, 42–50 (2015).
[Crossref]

Wang, J.

H. Wang, J. Wang, B. Chen, P. Xiao, X. Chen, N. Cai, and B. W. K. Ling, “Absolute optical imaging position encoder,” Measurement 67, 42–50 (2015).
[Crossref]

Wang, S.

Wang, X.

Wang, Z.

Wei, S.

Wu, Z. Y.

Q. C. Zhang and Z. Y. Wu, “A carrier removal method in Fourier transform profilometry with Zernike polynomials,” Opt. Lasers Eng. 51(3), 253–260 (2013).
[Crossref]

Xia, J.

Xiao, P.

H. Wang, J. Wang, B. Chen, P. Xiao, X. Chen, N. Cai, and B. W. K. Ling, “Absolute optical imaging position encoder,” Measurement 67, 42–50 (2015).
[Crossref]

Zhang, J.

Zhang, Q. C.

Y. K. Liu, Q. C. Zhang, and X. Y. Su, “3D shape from phase errors by using binary fringe with multi-step phase-shift technique,” Opt. Lasers Eng. 74, 22–27 (2015).
[Crossref]

Q. C. Zhang and Z. Y. Wu, “A carrier removal method in Fourier transform profilometry with Zernike polynomials,” Opt. Lasers Eng. 51(3), 253–260 (2013).
[Crossref]

Zhang, S.

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

Zhou, C.

Adv. Opt. Photonics (1)

J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photonics 3(2), 128–160 (2011).
[Crossref]

Appl. Opt. (2)

Chin. Opt. Lett. (3)

Discrete Math. (2)

V. Horan and B. Stevens, “Locating patterns in the de Bruijn torus,” Discrete Math. 339(4), 1274–1282 (2016).
[Crossref]

B. Jackson, B. Stevens, and G. Hurlbert, “Research problems on Gray codes and universal cycles,” Discrete Math. 309(17), 5341–5348 (2009).
[Crossref]

IEEE Sens. J. (1)

T. Dziwinski, “A Novel Approach of an Absolute Encoder Coding Pattern,” IEEE Sens. J. 15(1), 397–401 (2015).
[Crossref]

Measurement (1)

H. Wang, J. Wang, B. Chen, P. Xiao, X. Chen, N. Cai, and B. W. K. Ling, “Absolute optical imaging position encoder,” Measurement 67, 42–50 (2015).
[Crossref]

Opt. Eng. (1)

M. Schaffer, M. Grosse, B. Harendt, and R. Kowarschik, “Statistical patterns: an approach for high-speed and high-accuracy shape measurements,” Opt. Eng. 53(11), 112205 (2014).
[Crossref]

Opt. Lasers Eng. (4)

S. Van der Jeught and J. J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).
[Crossref]

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

Y. K. Liu, Q. C. Zhang, and X. Y. Su, “3D shape from phase errors by using binary fringe with multi-step phase-shift technique,” Opt. Lasers Eng. 74, 22–27 (2015).
[Crossref]

Q. C. Zhang and Z. Y. Wu, “A carrier removal method in Fourier transform profilometry with Zernike polynomials,” Opt. Lasers Eng. 51(3), 253–260 (2013).
[Crossref]

Pattern Recognit. (1)

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43(8), 2666–2680 (2010).
[Crossref]

Other (1)

S. W. Golomb, “Shift register sequences - A retrospective account,” in Sequences and Their Applications - Seta 2006, G. Gong, T. Helleseth, H. Y. Song, and K. Yang, eds. (Springer-Verlag Berlin, Berlin, 2006), pp. 1–4.

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Figures (12)

Fig. 1
Fig. 1 The encoding principle of the proposed design. (a)The matrix A. (b)The matrix B. (c)The matrix C.
Fig. 2
Fig. 2 Analyzing the effect of errors for some bits.
Fig. 3
Fig. 3 System setup.
Fig. 4
Fig. 4 Encoding process. The special arrangement of the matrix (a) A, (b) B, and (c) C.
Fig. 5
Fig. 5 Glimpse of the designed mask.
Fig. 6
Fig. 6 Imaging processing system.
Fig. 7
Fig. 7 Image processing. (a) Original image. (b) The image after image processing. (c) 10×14 submatrix extracted from the image.
Fig. 8
Fig. 8 The image subdivision for precise two-dimensional positioning.
Fig. 9
Fig. 9 Decoding processing. (a) Submatrix of the matrix A. (b) Submatrix of the matrix B. (c) Submatrix of the matrix C, corresponding to Fig. 9 (a) and Fig. 9 (b). (d) 10×10 submatrix extracted from Fig. 7 (c), corresponding to Fig. 9 (c).
Fig. 10
Fig. 10 Track of the two-dimensional encoder and the linear encoder.
Fig. 11
Fig. 11 Absolute errors on X-axis and Y-axis.
Fig. 12
Fig. 12 An example for a meter measurement range

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

f(x)= i=0 n c i x i , c 0 = c n =1, c i =0,1(i=1,2,...,n1,n2)
a 0 a 1 a n1 , a i =0,1(i=0,1,...,n1)
a 0 a 1 a n1 ={X(i)|i=0,1,...,n1}
( a 0 ) old =X(0)
( a 0 a 1 a n2 ) new = ( a 1 a 2 a n1 ) old
( a n1 ) new = c 1 ( a n1 ) old + c 2 ( a n2 ) old ++ c n ( a o ) old
X(0) X(1) X(2) X( 2 n 2) X(0) X(1) X(2) X( 2 n 2) X(0) X(1) X(2) X( 2 n 2) X(0) X(1) X(2) X( 2 n 2)
Y(0) Y(1) Y(2) Y( 2 n 2) Y( 2 n 2) Y(0) Y(1) Y( 2 n 3) Y( 2 n 3) Y( 2 n 2) Y(0) Y( 2 n 4) Y(1) Y(2) Y(3) Y(0)
X(0)+Y(0) X(1)+Y(1) X(2)+Y(2) X( 2 n 2)+Y( 2 n 2) X(0)+Y( 2 n 2) X(1)+Y(0) X(2)+Y(1) X( 2 n 2)+Y( 2 n 3) X(0)+Y( 2 n 3) X(1)+Y( 2 n 2) X(2)+Y(0) X( 2 n 2)+Y( 2 n 4) X(0)+Y(1) X(1)+Y(2) X(2)+Y(3) X( 2 n 2)+Y(0)
X(c)+Y(r) X(c+1)+Y(r+1) X(c+2)+Y(r+2) X(c+n1)+Y(r+n1) X(c)+Y(r1) X(c+1)+Y(r) X(c+2)+Y(r+1) X(c+n1)+Y(r+n2) X(c)+Y(r2) X(c+1)+Y(r1) X(c+2)+Y(r) X(c+n1)+Y(r+n3) X(c)+Y(rn+1) X(c+1)+Y(rn+2) X(c+2)+Y(rn+3) X(c+n1)+Y(r)
{ x={ cr+1, 0<cr+1< 2 n 1 cr+ 2 n , others y=c
{ L x =xD+Δ L x =xD+ N x d L y =yD+Δ L y =yD+ N y d
{ L x =(x+ N x /N )D L y =(y+ N y /N )D

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