Abstract

Deflectometry is widely used to accurately calculate the slopes of any specular reflective surface, ranging from car bodies to nanometer-level mirrors. This paper presents a new deflectometry technique using binary patterns of increasing frequency to retrieve the surface slopes. Binary Pattern Deflectometry allows almost instant, simple, and accurate slope retrieval, which is required for applications using mobile devices. The paper details the theory of this deflectometry method and the challenges of its implementation. Furthermore, the binary pattern method can also be combined with a classic phase-shifting method to eliminate the need of a complex unwrapping algorithm and retrieve the absolute phase, especially in cases like segmented optics, where spatial algorithms have difficulties. Finally, whether it is used as a stand-alone or combined with phase-shifting, the binary patterns can, within seconds, calculate the slopes of any specular reflective surface.

© 2014 Optical Society of America

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2013 (3)

X. Chen, C. Lu, M. Ma, X. Mao, and T. Mei, “Color-coding and phase-shift method for absolute phase measurement,” Opt. Commun. 298, 54–58 (2013).
[CrossRef]

P. Su, M. Khreishi, R. Huang, T. Su, and J. H. Burge, “Precision aspheric optics testing with SCOTS: a deflectometry approach,” Proc. SPIE 8788, 87881E (2013).
[CrossRef]

R. Huang, P. Su, T. Horne, G. Brusa Zappellini, and J. H. Burge, “Measurement of a large deformable aspherical mirror using SCOTS (Software Configurable Optical Test System),” Proc. SPIE 8838, 883807 (2013).
[CrossRef]

2012 (5)

Q. Zhang, X. Su, L. Xiang, and X. Sun, “3-D shape measurement based on complementary gray-code light,” Opt. Lasers Eng. 50, 574–579 (2012).
[CrossRef]

S. Zhang, “Composite phase-shifting algorithm for absolute phase measurement,” Opt. Lasers Eng. 50, 1538–1541 (2012).
[CrossRef]

H. Cui, W. Liao, N. Dai, and X. Cheng, “A flexible phase-shifting method phase marker retrieval,” Measurement 45, 101–108 (2012).
[CrossRef]

P. Su, S. Wang, M. Khreishi, Y. Wang, T. Su, P. Zhou, R. E. Parks, K. Law, M. Rascon, T. Zobrist, H. Martin, and J. H. Burge, “SCOTS: a reverse Hartmann test with high dynamic range for Giant Magellan Telescope primary mirror segments,” Proc. SPIE 8450, 84500W (2012).
[CrossRef]

G. P. Butel, G. A. Smith, and J. H. Burge, “Optimization of dynamic structured illumination for surface slope measurements,” Proc. SPIE 8493, 84930S (2012).
[CrossRef]

2011 (1)

S. Li, S. Liu, and H. Zhang, “3D shape measurement of optical free-form surface based on fringe projection,” Proc. SPIE 8082, 80822Z (2011).
[CrossRef]

2010 (1)

2009 (2)

K. Zhou, M. Zaitsev, and S. Bao, “Reliable two-dimensional phase unwrapping method using region growing and local linear estimation,” Magn. Reson. Med. 62, 1085–1090 (2009).
[CrossRef]

W. Jüptner and T. Bothe, “Sub-nanometer resolution for the inspection of reflective surfaces using white light,” Proc. SPIE 7405, 740502 (2009).
[CrossRef]

2008 (1)

2004 (4)

J. Salvi, J. Pagès, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827–849 (2004).
[CrossRef]

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261 (2004).
[CrossRef]

T. Bothe, W. Li, C. von Kopylow, and W. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

M. C. Knauer, J. Kaminski, and G. Hausler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[CrossRef]

2003 (1)

R. Schöne and O. Schwarz, “Hybrid phase-unwrapping algorithm extended by a minimum-cost-matching strategy,” Proc. SPIE 4933, 305–310 (2003).
[CrossRef]

2002 (1)

1999 (1)

1997 (3)

1995 (1)

D. Bergmann, “New approach for automatic surface reconstruction with coded light,” Proc. SPIE 2572, 2–9 (1995).
[CrossRef]

1994 (1)

T. R. Judge and P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994).
[CrossRef]

1982 (1)

J. L. Posdamer and M. D. Altschuler, “Surface measurement by space encoded projected beam systems,” Comput. Graph. Image Process. 18, 1–17 (1982).
[CrossRef]

Altschuler, M. D.

J. L. Posdamer and M. D. Altschuler, “Surface measurement by space encoded projected beam systems,” Comput. Graph. Image Process. 18, 1–17 (1982).
[CrossRef]

Angel, R. P.

Bao, S.

K. Zhou, M. Zaitsev, and S. Bao, “Reliable two-dimensional phase unwrapping method using region growing and local linear estimation,” Magn. Reson. Med. 62, 1085–1090 (2009).
[CrossRef]

Batlle, J.

J. Salvi, J. Pagès, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827–849 (2004).
[CrossRef]

Bergmann, D.

D. Bergmann, “New approach for automatic surface reconstruction with coded light,” Proc. SPIE 2572, 2–9 (1995).
[CrossRef]

Beyerer, J.

D. Pérard and J. Beyerer, “Three-dimensional measurement of specular free-form surfaces with a structured-lighting reflection technique,” Proc. SPIE 3204, 74–80 (1997).
[CrossRef]

Bioucas-Dias, J.

G. Valadao and J. Bioucas-Dias, “PUMA: phase unwrapping via max flows,” Proceedings of Conference on Telecommunications—ConfTele, Peniche, Portugal (2007), pp. 609–612.

Bothe, T.

W. Jüptner and T. Bothe, “Sub-nanometer resolution for the inspection of reflective surfaces using white light,” Proc. SPIE 7405, 740502 (2009).
[CrossRef]

T. Bothe, W. Li, C. von Kopylow, and W. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Brusa Zappellini, G.

R. Huang, P. Su, T. Horne, G. Brusa Zappellini, and J. H. Burge, “Measurement of a large deformable aspherical mirror using SCOTS (Software Configurable Optical Test System),” Proc. SPIE 8838, 883807 (2013).
[CrossRef]

Bryanston-Cross, P. J.

T. R. Judge and P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994).
[CrossRef]

Burge, J. H.

P. Su, M. Khreishi, R. Huang, T. Su, and J. H. Burge, “Precision aspheric optics testing with SCOTS: a deflectometry approach,” Proc. SPIE 8788, 87881E (2013).
[CrossRef]

R. Huang, P. Su, T. Horne, G. Brusa Zappellini, and J. H. Burge, “Measurement of a large deformable aspherical mirror using SCOTS (Software Configurable Optical Test System),” Proc. SPIE 8838, 883807 (2013).
[CrossRef]

P. Su, S. Wang, M. Khreishi, Y. Wang, T. Su, P. Zhou, R. E. Parks, K. Law, M. Rascon, T. Zobrist, H. Martin, and J. H. Burge, “SCOTS: a reverse Hartmann test with high dynamic range for Giant Magellan Telescope primary mirror segments,” Proc. SPIE 8450, 84500W (2012).
[CrossRef]

G. P. Butel, G. A. Smith, and J. H. Burge, “Optimization of dynamic structured illumination for surface slope measurements,” Proc. SPIE 8493, 84930S (2012).
[CrossRef]

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49, 4404–4412 (2010).
[CrossRef]

Butel, G. P.

G. P. Butel, G. A. Smith, and J. H. Burge, “Optimization of dynamic structured illumination for surface slope measurements,” Proc. SPIE 8493, 84930S (2012).
[CrossRef]

Carocci, M.

Chen, W.

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261 (2004).
[CrossRef]

Chen, X.

X. Chen, C. Lu, M. Ma, X. Mao, and T. Mei, “Color-coding and phase-shift method for absolute phase measurement,” Opt. Commun. 298, 54–58 (2013).
[CrossRef]

Cheng, X.

H. Cui, W. Liao, N. Dai, and X. Cheng, “A flexible phase-shifting method phase marker retrieval,” Measurement 45, 101–108 (2012).
[CrossRef]

Corini, S.

Cui, H.

H. Cui, W. Liao, N. Dai, and X. Cheng, “A flexible phase-shifting method phase marker retrieval,” Measurement 45, 101–108 (2012).
[CrossRef]

Dai, N.

H. Cui, W. Liao, N. Dai, and X. Cheng, “A flexible phase-shifting method phase marker retrieval,” Measurement 45, 101–108 (2012).
[CrossRef]

Docchio, F.

Fujimoto, A.

Hausler, G.

M. C. Knauer, J. Kaminski, and G. Hausler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[CrossRef]

Horne, T.

R. Huang, P. Su, T. Horne, G. Brusa Zappellini, and J. H. Burge, “Measurement of a large deformable aspherical mirror using SCOTS (Software Configurable Optical Test System),” Proc. SPIE 8838, 883807 (2013).
[CrossRef]

Huang, R.

R. Huang, P. Su, T. Horne, G. Brusa Zappellini, and J. H. Burge, “Measurement of a large deformable aspherical mirror using SCOTS (Software Configurable Optical Test System),” Proc. SPIE 8838, 883807 (2013).
[CrossRef]

P. Su, M. Khreishi, R. Huang, T. Su, and J. H. Burge, “Precision aspheric optics testing with SCOTS: a deflectometry approach,” Proc. SPIE 8788, 87881E (2013).
[CrossRef]

Huntley, J. M.

Jing, H.

Judge, T. R.

T. R. Judge and P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994).
[CrossRef]

Jüptner, W.

W. Jüptner and T. Bothe, “Sub-nanometer resolution for the inspection of reflective surfaces using white light,” Proc. SPIE 7405, 740502 (2009).
[CrossRef]

T. Bothe, W. Li, C. von Kopylow, and W. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Kaminski, J.

M. C. Knauer, J. Kaminski, and G. Hausler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[CrossRef]

Khreishi, M.

P. Su, M. Khreishi, R. Huang, T. Su, and J. H. Burge, “Precision aspheric optics testing with SCOTS: a deflectometry approach,” Proc. SPIE 8788, 87881E (2013).
[CrossRef]

P. Su, S. Wang, M. Khreishi, Y. Wang, T. Su, P. Zhou, R. E. Parks, K. Law, M. Rascon, T. Zobrist, H. Martin, and J. H. Burge, “SCOTS: a reverse Hartmann test with high dynamic range for Giant Magellan Telescope primary mirror segments,” Proc. SPIE 8450, 84500W (2012).
[CrossRef]

Knauer, M. C.

M. C. Knauer, J. Kaminski, and G. Hausler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[CrossRef]

Law, K.

P. Su, S. Wang, M. Khreishi, Y. Wang, T. Su, P. Zhou, R. E. Parks, K. Law, M. Rascon, T. Zobrist, H. Martin, and J. H. Burge, “SCOTS: a reverse Hartmann test with high dynamic range for Giant Magellan Telescope primary mirror segments,” Proc. SPIE 8450, 84500W (2012).
[CrossRef]

Lazzari, S.

Li, S.

S. Li, S. Liu, and H. Zhang, “3D shape measurement of optical free-form surface based on fringe projection,” Proc. SPIE 8082, 80822Z (2011).
[CrossRef]

Li, W.

T. Bothe, W. Li, C. von Kopylow, and W. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Liao, W.

H. Cui, W. Liao, N. Dai, and X. Cheng, “A flexible phase-shifting method phase marker retrieval,” Measurement 45, 101–108 (2012).
[CrossRef]

Liu, S.

S. Li, S. Liu, and H. Zhang, “3D shape measurement of optical free-form surface based on fringe projection,” Proc. SPIE 8082, 80822Z (2011).
[CrossRef]

Liu, Y.

Lu, C.

X. Chen, C. Lu, M. Ma, X. Mao, and T. Mei, “Color-coding and phase-shift method for absolute phase measurement,” Opt. Commun. 298, 54–58 (2013).
[CrossRef]

Ma, M.

X. Chen, C. Lu, M. Ma, X. Mao, and T. Mei, “Color-coding and phase-shift method for absolute phase measurement,” Opt. Commun. 298, 54–58 (2013).
[CrossRef]

Mao, X.

X. Chen, C. Lu, M. Ma, X. Mao, and T. Mei, “Color-coding and phase-shift method for absolute phase measurement,” Opt. Commun. 298, 54–58 (2013).
[CrossRef]

Martin, H.

P. Su, S. Wang, M. Khreishi, Y. Wang, T. Su, P. Zhou, R. E. Parks, K. Law, M. Rascon, T. Zobrist, H. Martin, and J. H. Burge, “SCOTS: a reverse Hartmann test with high dynamic range for Giant Magellan Telescope primary mirror segments,” Proc. SPIE 8450, 84500W (2012).
[CrossRef]

Mei, T.

X. Chen, C. Lu, M. Ma, X. Mao, and T. Mei, “Color-coding and phase-shift method for absolute phase measurement,” Opt. Commun. 298, 54–58 (2013).
[CrossRef]

Ohtsubo, J.

Pagès, J.

J. Salvi, J. Pagès, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827–849 (2004).
[CrossRef]

Parks, R. E.

P. Su, S. Wang, M. Khreishi, Y. Wang, T. Su, P. Zhou, R. E. Parks, K. Law, M. Rascon, T. Zobrist, H. Martin, and J. H. Burge, “SCOTS: a reverse Hartmann test with high dynamic range for Giant Magellan Telescope primary mirror segments,” Proc. SPIE 8450, 84500W (2012).
[CrossRef]

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49, 4404–4412 (2010).
[CrossRef]

Pérard, D.

D. Pérard and J. Beyerer, “Three-dimensional measurement of specular free-form surfaces with a structured-lighting reflection technique,” Proc. SPIE 3204, 74–80 (1997).
[CrossRef]

Posdamer, J. L.

J. L. Posdamer and M. D. Altschuler, “Surface measurement by space encoded projected beam systems,” Comput. Graph. Image Process. 18, 1–17 (1982).
[CrossRef]

Rapp, H. H.

H. H. Rapp, Reconstruction of Specular Reflective Surfaces Using Auto-Calibrating Deflectometry (KIT Scientific Publishing, 2012), retrieved from Universität Karlsruhe (Record ID: oai:EVASTAR-Karlsruhe.de:1000030538).

Rascon, M.

P. Su, S. Wang, M. Khreishi, Y. Wang, T. Su, P. Zhou, R. E. Parks, K. Law, M. Rascon, T. Zobrist, H. Martin, and J. H. Burge, “SCOTS: a reverse Hartmann test with high dynamic range for Giant Magellan Telescope primary mirror segments,” Proc. SPIE 8450, 84500W (2012).
[CrossRef]

Rodella, R.

Saldner, H. O.

Salvi, J.

J. Salvi, J. Pagès, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827–849 (2004).
[CrossRef]

Sansoni, G.

Schöne, R.

R. Schöne and O. Schwarz, “Hybrid phase-unwrapping algorithm extended by a minimum-cost-matching strategy,” Proc. SPIE 4933, 305–310 (2003).
[CrossRef]

Schwarz, O.

R. Schöne and O. Schwarz, “Hybrid phase-unwrapping algorithm extended by a minimum-cost-matching strategy,” Proc. SPIE 4933, 305–310 (2003).
[CrossRef]

Smith, G. A.

G. P. Butel, G. A. Smith, and J. H. Burge, “Optimization of dynamic structured illumination for surface slope measurements,” Proc. SPIE 8493, 84930S (2012).
[CrossRef]

Su, P.

P. Su, M. Khreishi, R. Huang, T. Su, and J. H. Burge, “Precision aspheric optics testing with SCOTS: a deflectometry approach,” Proc. SPIE 8788, 87881E (2013).
[CrossRef]

R. Huang, P. Su, T. Horne, G. Brusa Zappellini, and J. H. Burge, “Measurement of a large deformable aspherical mirror using SCOTS (Software Configurable Optical Test System),” Proc. SPIE 8838, 883807 (2013).
[CrossRef]

P. Su, S. Wang, M. Khreishi, Y. Wang, T. Su, P. Zhou, R. E. Parks, K. Law, M. Rascon, T. Zobrist, H. Martin, and J. H. Burge, “SCOTS: a reverse Hartmann test with high dynamic range for Giant Magellan Telescope primary mirror segments,” Proc. SPIE 8450, 84500W (2012).
[CrossRef]

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49, 4404–4412 (2010).
[CrossRef]

Su, T.

P. Su, M. Khreishi, R. Huang, T. Su, and J. H. Burge, “Precision aspheric optics testing with SCOTS: a deflectometry approach,” Proc. SPIE 8788, 87881E (2013).
[CrossRef]

P. Su, S. Wang, M. Khreishi, Y. Wang, T. Su, P. Zhou, R. E. Parks, K. Law, M. Rascon, T. Zobrist, H. Martin, and J. H. Burge, “SCOTS: a reverse Hartmann test with high dynamic range for Giant Magellan Telescope primary mirror segments,” Proc. SPIE 8450, 84500W (2012).
[CrossRef]

Su, X.

Q. Zhang, X. Su, L. Xiang, and X. Sun, “3-D shape measurement based on complementary gray-code light,” Opt. Lasers Eng. 50, 574–579 (2012).
[CrossRef]

Y. Tang, X. Su, Y. Liu, and H. Jing, “3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry,” Opt. Express 16, 15090–15096 (2008).
[CrossRef]

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261 (2004).
[CrossRef]

Sun, X.

Q. Zhang, X. Su, L. Xiang, and X. Sun, “3-D shape measurement based on complementary gray-code light,” Opt. Lasers Eng. 50, 574–579 (2012).
[CrossRef]

Tang, Y.

Valadao, G.

G. Valadao and J. Bioucas-Dias, “PUMA: phase unwrapping via max flows,” Proceedings of Conference on Telecommunications—ConfTele, Peniche, Portugal (2007), pp. 609–612.

von Kopylow, C.

T. Bothe, W. Li, C. von Kopylow, and W. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Wang, L.

Wang, S.

P. Su, S. Wang, M. Khreishi, Y. Wang, T. Su, P. Zhou, R. E. Parks, K. Law, M. Rascon, T. Zobrist, H. Martin, and J. H. Burge, “SCOTS: a reverse Hartmann test with high dynamic range for Giant Magellan Telescope primary mirror segments,” Proc. SPIE 8450, 84500W (2012).
[CrossRef]

Wang, Y.

P. Su, S. Wang, M. Khreishi, Y. Wang, T. Su, P. Zhou, R. E. Parks, K. Law, M. Rascon, T. Zobrist, H. Martin, and J. H. Burge, “SCOTS: a reverse Hartmann test with high dynamic range for Giant Magellan Telescope primary mirror segments,” Proc. SPIE 8450, 84500W (2012).
[CrossRef]

Xiang, L.

Q. Zhang, X. Su, L. Xiang, and X. Sun, “3-D shape measurement based on complementary gray-code light,” Opt. Lasers Eng. 50, 574–579 (2012).
[CrossRef]

Zaitsev, M.

K. Zhou, M. Zaitsev, and S. Bao, “Reliable two-dimensional phase unwrapping method using region growing and local linear estimation,” Magn. Reson. Med. 62, 1085–1090 (2009).
[CrossRef]

Zhang, H.

S. Li, S. Liu, and H. Zhang, “3D shape measurement of optical free-form surface based on fringe projection,” Proc. SPIE 8082, 80822Z (2011).
[CrossRef]

Zhang, Q.

Q. Zhang, X. Su, L. Xiang, and X. Sun, “3-D shape measurement based on complementary gray-code light,” Opt. Lasers Eng. 50, 574–579 (2012).
[CrossRef]

Zhang, S.

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Appl. Opt. (5)

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Figures (13)

Fig. 1.
Fig. 1.

Binary code pattern examples. The patterns Nos. 1–6 above represent the 1D binary code; the patterns Nos. 7–12 represent the 2D binary code. The 1D patterns are independent in x and y. The 2D patterns start with splitting the screen in two (No. 7) and then alternates between x and y frequency in increasing steps. The total number of patterns depends on the number of pixels used.

Fig. 2.
Fig. 2.

Retrieving the pixel screen location using the 1D binary code patterns. Any point on the mirror has a binary code that gives its location in the screen space. The nomenclature ()2 and ()10 designates an integer in base 2 (binary) and base 10 (decimal), respectively.

Fig. 3.
Fig. 3.

Screen coordinates calculation in the 1D case. The points Pj are located on the area of patterns display thanks to their binary code and we calculate xscreen using Rj and the pixel pitch.

Fig. 4.
Fig. 4.

Example of pixel-to-pixel multiplication used to sequentially isolate a screen pixel (a white pixel means 1 and a black pixel 0). The top row represents the pictures taken by the camera, the bottom row the matrix multiplication process. In this example, we consider a mirror pixel (x0,y0). Based on the first picture, the value at (x0,y0) is 1, so the matrix used in the product defined in Eq. (3) is M1. Using the second picture (i=2), α2=0 so the matrix used is (UM2). The intermediate product is, thus, M1(UM2). The same process goes until we reach i=m (m=6 here).

Fig. 5.
Fig. 5.

Screen coordinates calculation in the 2D case. The matrix multiplication isolates a single pixel that gives the coordinates x and y for the mirror pixel (x0,y0). xscreen(xj,yj) and yscreen(xj,yj) are obtained using Eq. (1) that changes the coordinates of the pixel location from the original matrix Sx0,y0 to a real space coordinate system.

Fig. 6.
Fig. 6.

Real data from a spherical mirror. The picture (left) clearly shows the different zones. The y-profile (right) displays a cross section of the picture.

Fig. 7.
Fig. 7.

Two thresholds now create an undetermined region between the 0 and 1 regions, with Δ=((MaxMin)/5).

Fig. 8.
Fig. 8.

Shifted binary patterns. The shifted mirror picture (middle) depicts a shift of half the size of a white stripe compared with mirror picture from Fig. 6 (left, for reference). The profile (right) shows a cross section of the shifted patterns picture.

Fig. 9.
Fig. 9.

Mirror maps representing the screen coordinates for standard (left), shifted (middle) patterns, and final map (right). The white lines show the intermediate regions.

Fig. 10.
Fig. 10.

Contrast obtained in the pictures taken by the camera.

Fig. 11.
Fig. 11.

Matching the square wave to the sine wave gives the correct offsets in decimal. Multiplying the offset values by 2π and adding them to the wrapped phase will give the true phase. In this example, K has 4 different values, (0)10, (1)10, (2)10, and (3)10.

Fig. 12.
Fig. 12.

Experimental setup. The patterns are displayed on the screen (left) and reflected off the mirror (right). Part of the mirror was masked to simulate a segmented mirror (close-up). The camera (left) takes pictures.

Fig. 13.
Fig. 13.

Comparison of the phase maps differences. Map (a) depicts the difference between the full and segmented mirror (spatially unwrapped). Map (b) shows the phase difference between a spatial and temporal unwrapping (full mirror), which appears to be noise. Map (c) depicts the difference between the full and segmented mirror (temporally unwrapped). Map (d) shows the average of two maps temporally unwrapped. All numbers are in radians.

Equations (6)

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

xscreen(xj,yj)=[(Pj)10+Rj]pitch,(xj,yj)Mirror,
ifiodd{Nx=2floor(i/2)Ny=2floor(i/2)+1ifieven{Nx=2floor(i/2)Ny=Nx,
Sx0,y0=i=1m[UMi]1αi[Mi]αi,
XSlopes(x,y)=12zs(xmirror(x,y)xscreen(x,y))+12zc(xmirror(x,y)xcamera(x,y)),
Φtrue(x,y)=Φwrapped(x,y)+2K(x,y)π(x,y)Mirror.
xscreen(x,y)=Φtruex(x,y)2pπN,

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