Abstract

In this work a detailed analysis of the problem of imaging of objects lying in the plane tilted with respect to the optical axis of the rotationally symmetrical optical system is performed by means of geometrical optics theory. It is shown that the fulfillment of the so called Scheimpflug condition (Scheimpflug rule) does not guarantee the sharp image of the object as it is usually declared because of the fact that due to the dependence of aberrations of real optical systems on the object distance the image becomes blurred. The f-number of a given optical system also varies with the object distance. It is shown the influence of above mentioned effects on the accuracy of the laser triangulation sensors measurements. A detailed analysis of laser triangulation sensors, based on geometrical optics theory, is performed and relations for the calculation of measurement errors and construction parameters of laser triangulation sensors are derived.

© 2013 OSA

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  1. Improvements in Enlarging or like Cameras, British Patent No.1139, 1901.
  2. Improved Method and Apparatus for the Systematic Alteration or Distortion of Plane Pictures and Images by Means of Lenses and Mirrors for Photography and for other Purposes, British Patent No.1196, 1904.
  3. L. Larmore, Introduction to Photographic Principles (Dover Publications, 1965).
  4. S. F. Ray, Applied Photographic Optics (Focal Press, 2002).
  5. http://www.linhof.de/index-e.html/
  6. http://www.micro-epsilon.com/
  7. R. G. Dorsch, G. Häusler, and J. M. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt.33(7), 1306–1314 (1994).
    [CrossRef] [PubMed]
  8. R. Leach, Optical Measurement of Surface Topography (Springer, 2011).
  9. K. Harding, Handbook of Optical Dimensional Metrology (Taylor & Francis, 2013).
  10. K. Žbontar, M. Mihelj, B. Podobnik, F. Povše, and M. Munih, “Dynamic symmetrical pattern projection based laser triangulation sensor for precise surface position measurement of various material types,” Appl. Opt.52(12), 2750–2760 (2013).
    [CrossRef] [PubMed]
  11. H.-Y. Feng, Y. Liu, and F. Xi, “Analysis of digitizing errors of a laser scanning system,” Precis. Eng.25(3), 185–191 (2001).
    [CrossRef]
  12. R.-T. Lee and F.-J. Shiou, “Multi-beam laser probe for measuring position and orientation of freeform surface,” Measurement44(1), 1–10 (2011).
    [CrossRef]
  13. J. Liu, L. Tian, and L. Li, “Light power density distribution of image spot of laser triangulation measuring,” Opt. Lasers Eng.29(6), 457–463 (1998).
    [CrossRef]
  14. Lei Shen, Dinggen Li, and Feng Luo, “A study on laser speckle correlation method applied in triangulation displacement measurement,” Optik (submitted) 2013.
    [CrossRef]
  15. H. Wang, “Long-range optical triangulation utilising collimated probe beam,” Opt. Lasers Eng.23(1), 41–52 (1995).
    [CrossRef]
  16. V. Lombardo, T. Marzulli, C. Pappalettere, and P. Sforza, “A time-of-scan laser triangulation technique for distance measurements,” Opt. Lasers Eng.39(2), 247–254 (2003).
    [CrossRef]
  17. G. Wang, B. Zheng, X. Li, Z. Houkes, and P. P. L. Regtien, “Modelling and calibration of the laser beam-scanning triangulation measurement system,” Robot. Auton. Syst.40(4), 267–277 (2002).
    [CrossRef]
  18. B. Muralikrishnan, W. Ren, D. Everett, E. Stanfield, and T. Doiron, “Performance evaluation experiments on a laser spot triangulation probe,” Measurement45(3), 333–343 (2012).
    [CrossRef]
  19. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  20. H. Gross, Handbook of Optical Systems: Fundamentals of Technical Optics (Wiley 2005).
  21. A. Miks and J. Novak, “Estimation of accuracy of optical measuring systems with respect to object distance,” Opt. Express19(15), 14300–14314 (2011).
    [CrossRef] [PubMed]
  22. A. Miks and J. Novak, “Dependence of camera lens induced radial distortion and circle of confusion on object position,” Opt. Laser Technol.44(4), 1043–1049 (2012).
    [CrossRef]
  23. A. Miks, Applied Optics (Czech Technical University, 2009).
  24. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University, 1970).
  25. W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic Press, 1974).
  26. M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
  27. M. Herzberger, Strahlenoptik (Verlag von Julius Springer, Berlin, 1931).
  28. C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc.65B, 429–437 (1952).
  29. R. Baribeau and M. Rioux, “Centroid fluctuations of speckled targets,” Appl. Opt.30(26), 3752–3755 (1991).
    [CrossRef] [PubMed]
  30. F. Träger, Handbook of Laser and Optics (Springer, 2007).
  31. B. E. A. Saleh and M. C. Teich, Fundamental of Photonics (John Wiley & Sons, 2007).
  32. P. E. Yoder, Jr., Opto-Mechanical Systems Design (CRC, 2006).
  33. M. M. Rusinov, Юстировка оптических приборов (Недра, 1969).
  34. J. Picht, Meß - und Prüfmethoden der Optischen Fertigung (Akademie-Verlag, 1953).
  35. F. Hansen, Justierung (VEB Verlag Technik, 1967).

2013

2012

B. Muralikrishnan, W. Ren, D. Everett, E. Stanfield, and T. Doiron, “Performance evaluation experiments on a laser spot triangulation probe,” Measurement45(3), 333–343 (2012).
[CrossRef]

A. Miks and J. Novak, “Dependence of camera lens induced radial distortion and circle of confusion on object position,” Opt. Laser Technol.44(4), 1043–1049 (2012).
[CrossRef]

2011

A. Miks and J. Novak, “Estimation of accuracy of optical measuring systems with respect to object distance,” Opt. Express19(15), 14300–14314 (2011).
[CrossRef] [PubMed]

R.-T. Lee and F.-J. Shiou, “Multi-beam laser probe for measuring position and orientation of freeform surface,” Measurement44(1), 1–10 (2011).
[CrossRef]

2003

V. Lombardo, T. Marzulli, C. Pappalettere, and P. Sforza, “A time-of-scan laser triangulation technique for distance measurements,” Opt. Lasers Eng.39(2), 247–254 (2003).
[CrossRef]

2002

G. Wang, B. Zheng, X. Li, Z. Houkes, and P. P. L. Regtien, “Modelling and calibration of the laser beam-scanning triangulation measurement system,” Robot. Auton. Syst.40(4), 267–277 (2002).
[CrossRef]

2001

H.-Y. Feng, Y. Liu, and F. Xi, “Analysis of digitizing errors of a laser scanning system,” Precis. Eng.25(3), 185–191 (2001).
[CrossRef]

1998

J. Liu, L. Tian, and L. Li, “Light power density distribution of image spot of laser triangulation measuring,” Opt. Lasers Eng.29(6), 457–463 (1998).
[CrossRef]

1995

H. Wang, “Long-range optical triangulation utilising collimated probe beam,” Opt. Lasers Eng.23(1), 41–52 (1995).
[CrossRef]

1994

1991

1952

C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc.65B, 429–437 (1952).

Baribeau, R.

Doiron, T.

B. Muralikrishnan, W. Ren, D. Everett, E. Stanfield, and T. Doiron, “Performance evaluation experiments on a laser spot triangulation probe,” Measurement45(3), 333–343 (2012).
[CrossRef]

Dorsch, R. G.

Everett, D.

B. Muralikrishnan, W. Ren, D. Everett, E. Stanfield, and T. Doiron, “Performance evaluation experiments on a laser spot triangulation probe,” Measurement45(3), 333–343 (2012).
[CrossRef]

Feng, H.-Y.

H.-Y. Feng, Y. Liu, and F. Xi, “Analysis of digitizing errors of a laser scanning system,” Precis. Eng.25(3), 185–191 (2001).
[CrossRef]

Häusler, G.

Herrmann, J. M.

Houkes, Z.

G. Wang, B. Zheng, X. Li, Z. Houkes, and P. P. L. Regtien, “Modelling and calibration of the laser beam-scanning triangulation measurement system,” Robot. Auton. Syst.40(4), 267–277 (2002).
[CrossRef]

Lee, R.-T.

R.-T. Lee and F.-J. Shiou, “Multi-beam laser probe for measuring position and orientation of freeform surface,” Measurement44(1), 1–10 (2011).
[CrossRef]

Li, L.

J. Liu, L. Tian, and L. Li, “Light power density distribution of image spot of laser triangulation measuring,” Opt. Lasers Eng.29(6), 457–463 (1998).
[CrossRef]

Li, X.

G. Wang, B. Zheng, X. Li, Z. Houkes, and P. P. L. Regtien, “Modelling and calibration of the laser beam-scanning triangulation measurement system,” Robot. Auton. Syst.40(4), 267–277 (2002).
[CrossRef]

Liu, J.

J. Liu, L. Tian, and L. Li, “Light power density distribution of image spot of laser triangulation measuring,” Opt. Lasers Eng.29(6), 457–463 (1998).
[CrossRef]

Liu, Y.

H.-Y. Feng, Y. Liu, and F. Xi, “Analysis of digitizing errors of a laser scanning system,” Precis. Eng.25(3), 185–191 (2001).
[CrossRef]

Lombardo, V.

V. Lombardo, T. Marzulli, C. Pappalettere, and P. Sforza, “A time-of-scan laser triangulation technique for distance measurements,” Opt. Lasers Eng.39(2), 247–254 (2003).
[CrossRef]

Marzulli, T.

V. Lombardo, T. Marzulli, C. Pappalettere, and P. Sforza, “A time-of-scan laser triangulation technique for distance measurements,” Opt. Lasers Eng.39(2), 247–254 (2003).
[CrossRef]

Mihelj, M.

Miks, A.

A. Miks and J. Novak, “Dependence of camera lens induced radial distortion and circle of confusion on object position,” Opt. Laser Technol.44(4), 1043–1049 (2012).
[CrossRef]

A. Miks and J. Novak, “Estimation of accuracy of optical measuring systems with respect to object distance,” Opt. Express19(15), 14300–14314 (2011).
[CrossRef] [PubMed]

Munih, M.

Muralikrishnan, B.

B. Muralikrishnan, W. Ren, D. Everett, E. Stanfield, and T. Doiron, “Performance evaluation experiments on a laser spot triangulation probe,” Measurement45(3), 333–343 (2012).
[CrossRef]

Novak, J.

A. Miks and J. Novak, “Dependence of camera lens induced radial distortion and circle of confusion on object position,” Opt. Laser Technol.44(4), 1043–1049 (2012).
[CrossRef]

A. Miks and J. Novak, “Estimation of accuracy of optical measuring systems with respect to object distance,” Opt. Express19(15), 14300–14314 (2011).
[CrossRef] [PubMed]

Pappalettere, C.

V. Lombardo, T. Marzulli, C. Pappalettere, and P. Sforza, “A time-of-scan laser triangulation technique for distance measurements,” Opt. Lasers Eng.39(2), 247–254 (2003).
[CrossRef]

Podobnik, B.

Povše, F.

Regtien, P. P. L.

G. Wang, B. Zheng, X. Li, Z. Houkes, and P. P. L. Regtien, “Modelling and calibration of the laser beam-scanning triangulation measurement system,” Robot. Auton. Syst.40(4), 267–277 (2002).
[CrossRef]

Ren, W.

B. Muralikrishnan, W. Ren, D. Everett, E. Stanfield, and T. Doiron, “Performance evaluation experiments on a laser spot triangulation probe,” Measurement45(3), 333–343 (2012).
[CrossRef]

Rioux, M.

Rusinov, M. M.

M. M. Rusinov, Юстировка оптических приборов (Недра, 1969).

Sforza, P.

V. Lombardo, T. Marzulli, C. Pappalettere, and P. Sforza, “A time-of-scan laser triangulation technique for distance measurements,” Opt. Lasers Eng.39(2), 247–254 (2003).
[CrossRef]

Shiou, F.-J.

R.-T. Lee and F.-J. Shiou, “Multi-beam laser probe for measuring position and orientation of freeform surface,” Measurement44(1), 1–10 (2011).
[CrossRef]

Stanfield, E.

B. Muralikrishnan, W. Ren, D. Everett, E. Stanfield, and T. Doiron, “Performance evaluation experiments on a laser spot triangulation probe,” Measurement45(3), 333–343 (2012).
[CrossRef]

Tian, L.

J. Liu, L. Tian, and L. Li, “Light power density distribution of image spot of laser triangulation measuring,” Opt. Lasers Eng.29(6), 457–463 (1998).
[CrossRef]

Wang, G.

G. Wang, B. Zheng, X. Li, Z. Houkes, and P. P. L. Regtien, “Modelling and calibration of the laser beam-scanning triangulation measurement system,” Robot. Auton. Syst.40(4), 267–277 (2002).
[CrossRef]

Wang, H.

H. Wang, “Long-range optical triangulation utilising collimated probe beam,” Opt. Lasers Eng.23(1), 41–52 (1995).
[CrossRef]

Wynne, C. G.

C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc.65B, 429–437 (1952).

Xi, F.

H.-Y. Feng, Y. Liu, and F. Xi, “Analysis of digitizing errors of a laser scanning system,” Precis. Eng.25(3), 185–191 (2001).
[CrossRef]

Žbontar, K.

Zheng, B.

G. Wang, B. Zheng, X. Li, Z. Houkes, and P. P. L. Regtien, “Modelling and calibration of the laser beam-scanning triangulation measurement system,” Robot. Auton. Syst.40(4), 267–277 (2002).
[CrossRef]

Appl. Opt.

Measurement

B. Muralikrishnan, W. Ren, D. Everett, E. Stanfield, and T. Doiron, “Performance evaluation experiments on a laser spot triangulation probe,” Measurement45(3), 333–343 (2012).
[CrossRef]

R.-T. Lee and F.-J. Shiou, “Multi-beam laser probe for measuring position and orientation of freeform surface,” Measurement44(1), 1–10 (2011).
[CrossRef]

Opt. Express

Opt. Laser Technol.

A. Miks and J. Novak, “Dependence of camera lens induced radial distortion and circle of confusion on object position,” Opt. Laser Technol.44(4), 1043–1049 (2012).
[CrossRef]

Opt. Lasers Eng.

J. Liu, L. Tian, and L. Li, “Light power density distribution of image spot of laser triangulation measuring,” Opt. Lasers Eng.29(6), 457–463 (1998).
[CrossRef]

H. Wang, “Long-range optical triangulation utilising collimated probe beam,” Opt. Lasers Eng.23(1), 41–52 (1995).
[CrossRef]

V. Lombardo, T. Marzulli, C. Pappalettere, and P. Sforza, “A time-of-scan laser triangulation technique for distance measurements,” Opt. Lasers Eng.39(2), 247–254 (2003).
[CrossRef]

Precis. Eng.

H.-Y. Feng, Y. Liu, and F. Xi, “Analysis of digitizing errors of a laser scanning system,” Precis. Eng.25(3), 185–191 (2001).
[CrossRef]

Proc. Phys. Soc.

C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc.65B, 429–437 (1952).

Robot. Auton. Syst.

G. Wang, B. Zheng, X. Li, Z. Houkes, and P. P. L. Regtien, “Modelling and calibration of the laser beam-scanning triangulation measurement system,” Robot. Auton. Syst.40(4), 267–277 (2002).
[CrossRef]

Other

Lei Shen, Dinggen Li, and Feng Luo, “A study on laser speckle correlation method applied in triangulation displacement measurement,” Optik (submitted) 2013.
[CrossRef]

A. Miks, Applied Optics (Czech Technical University, 2009).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University, 1970).

W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic Press, 1974).

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

M. Herzberger, Strahlenoptik (Verlag von Julius Springer, Berlin, 1931).

Improvements in Enlarging or like Cameras, British Patent No.1139, 1901.

Improved Method and Apparatus for the Systematic Alteration or Distortion of Plane Pictures and Images by Means of Lenses and Mirrors for Photography and for other Purposes, British Patent No.1196, 1904.

L. Larmore, Introduction to Photographic Principles (Dover Publications, 1965).

S. F. Ray, Applied Photographic Optics (Focal Press, 2002).

http://www.linhof.de/index-e.html/

http://www.micro-epsilon.com/

F. Träger, Handbook of Laser and Optics (Springer, 2007).

B. E. A. Saleh and M. C. Teich, Fundamental of Photonics (John Wiley & Sons, 2007).

P. E. Yoder, Jr., Opto-Mechanical Systems Design (CRC, 2006).

M. M. Rusinov, Юстировка оптических приборов (Недра, 1969).

J. Picht, Meß - und Prüfmethoden der Optischen Fertigung (Akademie-Verlag, 1953).

F. Hansen, Justierung (VEB Verlag Technik, 1967).

R. Leach, Optical Measurement of Surface Topography (Springer, 2011).

K. Harding, Handbook of Optical Dimensional Metrology (Taylor & Francis, 2013).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

H. Gross, Handbook of Optical Systems: Fundamentals of Technical Optics (Wiley 2005).

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Figures (3)

Fig. 1
Fig. 1

Imaging of object in tilted plane by ideal optical system.

Fig. 2
Fig. 2

Imaging of object in tilted plane by diffraction limited optical system.

Fig. 3
Fig. 3

Imaging of object in tilted plane by real optical system with aberrations.

Tables (1)

Tables Icon

Table 1 Triangulation Sensor Example

Equations (38)

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

m= y y = q f = f q ,
q q = f 2 ,
tanα= y q A q B .
tanβ= y q A q B = y q A q B f 2 ( q A q B ) =tanα q A f = tanα m A ,
tanα= h q A f ,tanβ= h q A + f = h q A f ( q A f ) .
h q A f ( q A f ) =( h q A f ) q A f
h= h .
y ( q A t)=t f tanα
t= y q A f tanα+ y = d q A sinβ dsinβ f tanα = d q A d f 2 q A cosβ .
d=( f 2 q A cosβ )( 1 q A q B ).
δd=( f 2 cosβ ) δ q B q B 2 .
δ q B = q B 2 ( cosβ f 2 )δd.
f 4 + ( q A tanα) 2 f 2 ( q A d) 2 ( 1 q A /t ) 2 =0.
tan ω m = p C D p C 2 + y 2 D 2 /4 ,tan ω m = p C D p C 2 + y 2 D 2 /4 .
tan ω s 2 = D 2 p C 2 + y 2 ,tan ω s 2 = D 2 p C 2 + y 2 .
tan ω 0 2 = D 2 p A ,tan ω 0 2 = D 2 p A .
θ m = ω m /2, θ m = ω m /2, θ s = ω s /2, θ s = ω s /2, θ 0 = ω 0 /2, θ 0 = ω 0 /2.
F= 1 2sin θ .
d A =2.4λF,
δ y =mδy+k( a 1 g 4 S I a 2 g 3 g p S II + a 3 g 2 g P 2 S III + a 4 S IV a 5 g g P 3 S V ), δ x =mδx+k( b 1 g 4 S I b 2 g 3 g p S II + b 3 g 2 g P 2 S III + b 4 S IV ),
k= 1 2 n g p 1 3 , a 1 = y P1 ( y P1 2 + x P1 2 ), a 2 =(3 y P1 2 + x P1 2 )y, a 3 =3 y P1 y 2 , a 4 = n 2 y P1 y 2 p 1 2 , a 5 = y 3 , b 1 = x P1 ( y P1 2 + x P1 2 ), b 2 =2 y P1 x P1 y, b 3 = x P1 y 2 , b 4 = n 2 x P1 y 2 p 1 2 ,
S=BG,
S=( g 4 S I g P g 3 S II g P 2 g 2 S III S IV g P 3 g S V ),B=( b 11 b 12 b 13 b 14 b 15 0 b 22 b 23 b 24 b 25 0 0 b 33 b 34 b 35 0 0 0 0 b 45 0 0 0 b 54 b 55 ),G=( g 4 g 3 g 2 g 1 ),
b 11 = S I , b 12 =4 g P ( S I + S II )n f , b 13 =6 g P 2 ( S I 2 S II + S III )+2 n 2 f 2 S IV , b 14 =4 g P 3 ( S I +3 S II 3 S III + S V )4 n 2 f 2 g P S IV +3n f , b 15 = g P 4 ( S I 4 S II +6 S III 4 S V + S VI )+2 n 2 f 2 g P 2 S IV + g P n f ( g P 2 3), b 22 = g P S II , b 23 =3 g P 2 ( S II + S III )+n f (n f S IV g P ), b 24 =3 g P 3 ( S II 2 S III + S V )2n f (n f g P S IV 1), b 25 = g P 4 ( S II +3 S III 3 S V + S VI )+ n 2 f 2 g P 2 S IV + g P n f ( g P 2 2), b 33 = g P 2 S III , b 34 =2 g P 3 ( S V S III )n f ( g P 2 1), b 35 = g P 4 ( S III + S VI 2 S V )+n f g P ( g P 2 1), b 45 = S IV , b 54 = g P 3 S V , b 55 = g P 4 ( S V + S VI ).
h 1 = s 1 /g, σ 1 =1/g, h P1 = s P1 / g P , σ P1 =1/ g P ,
h 1 = f , σ 1 =0, h P1 = s P1 / g P , σ P1 =1/ g P ,
B= f ( 0 n 0 3n n g P ( g P 2 3) 0 0 n g P 2n n g P ( g P 2 2) 0 0 0 n( g P 2 1) n g P ( g P 2 1) 0 0 0 0 0 0 0 0 0 0 ).
δ x =(1/2g)[ S I 0 A 3 cosφ S II 0 A 2 sin2φtgw+ S III 0 Acosφt g 2 w], δ y =(1/2g)[ S I 0 A 3 sinφ S II 0 A 2 (1+2 sin 2 φ)tgw+3 S III 0 Asinφt g 2 w],
S I 0 = f [ g 3 3g g P ( g P 2 3)], S II 0 = f [ g 2 g P 2g g P ( g P 2 2)], S III 0 = f [g( g P 2 1) g P ( g P 2 1)].
δ x =0,δ y = 1 2g A M 2 tgw S II o ,
d c = A M 6g 9 A M 4 S I 0 2 +48 A M 2 S I 0 S III 0 t g 2 w+24 A M 2 S II 0 2 t g 2 w+90 S III 0 2 t g 4 w ,
A M = 1 2 F 0 (gg ) P ,tgw= y f ( g P g ) .
δ y = y m m P + m 2 (12 m P 2 ) 8 F 0 2 ( m P m) 2 ,
y = f ( m B m A )tanβ= f ( m B / m A 1)tanα.
δ y B = y m B m P + m B 2 (12 m P 2 ) 8 F 0 2 ( m P m B ) 2 .
δ d B = δ y B 1+ tan 2 β tan w tanβ = δ y B 1+ tan 2 β tanw m P ( 1+ δ y B y )tanβ δ y B 1+ tan 2 β tanw m P tanβ .
δ q B = q B 2 ( cosβ f 2 )δ d B .
δ q speckle = C 2π λ sinαsinu ,

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