Abstract

Lens distortion parameters vary with the distance between the object point and the image plane. We propose an analytical model of depth-dependent distortion for large depth-of-field digital cameras used for high accuracy photogrammetry. Compared with the magnification-dependent model, the proposed one does not need focusing operation during calibration, thus eliminates focusing errors and guarantees the stability of camera interior parameters. Compared with the widely used constant distortion parameter model, the proposed model reduces the maximum distortion variation from 8.0 μm to 0.9 μm at 20 mm radial distance when the depth changes from 2.46 m to 4.51 m for the 35 mm lens, and from 23.0 μm to 3.6 μm when the depth changes from 2.07 m to 4.17 m for the 50 mm lens. Additionally, when applied to photogrammetry bundle adjustment, the proposed model reduces length measurement standard deviation from 0.055 mm to 0.028 mm in a measurement volume of 7.0 m × 3.5 m × 2.5m compared with the constant parameter model.

© 2017 Optical Society of America

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References

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  1. T. Luhmann, S. Robson, and S. Kyle, Close-range photogrammetry and 3D imaging (Walter de Gruyter, 2014).
  2. T. Luhmann, C. S. Fraser, and H. G. Maas, “Sensor modelling and camera calibration for close-range photogrammetry,” ISPRS J. Photogramm. 115, 37–46 (2016).
    [Crossref]
  3. C. S. Fraser, “Automatic camera calibration in close range photogrammetry,” Photogram. Eng. Remote Sens. 79(4), 381–388 (2013).
    [Crossref]
  4. T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
    [Crossref] [PubMed]
  5. F. Xing, Y. Dong, and Z. You, “Laboratory calibration of star tracker with brightness independent star identification strategy,” Opt. Eng. 45(6), 063604 (2006).
    [Crossref]
  6. Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE. T. Pattern. Anal. 22(11), 1330–1334 (2000).
  7. F. Remondino and C. S. Fraser, “Digital camera calibration methods: considerations and comparisons,” Int. Arch. Photogram. 36(5), 266–272 (2006).
  8. T. Luhmann, “Close range photogrammetry for industrial applications,” ISPRS J. Photogramm. 65(6), 558–569 (2010).
    [Crossref]
  9. A. A. Magill, “Variation in distortion with magnification,” J. Opt. Soc. Am. 45(3), 148–149 (1955).
    [Crossref]
  10. D. C. Brown, “Close-range camera calibration,” Photogram. Eng. 37, 855–866 (1971).
  11. J. G. Fryer and D. C. Brown, “Lens distortion for close-range photogrammetry,” Photogram. Eng. Rem. S. 52(1), 51–58 (1986).
  12. J. G. Fryer and C. S. Fraser, “On the calibration of underwater cameras,” Photogram. Rec. 12(67), 73–85 (1986).
    [Crossref]
  13. J. G. Fryer, T. A. Clarke, and J. Chen, “Lens distortion for simple C-mount lenses,” Int. Arch. Photogramm. Remote Sens. 30, 97–101 (1994).
  14. L. Alvarez, L. Gómez, and J. R. Sendra, “Accurate depth dependent lens distortion models: an application to planar view scenarios,” J. Math. Imaging Vis. 39(1), 75–85 (2011).
    [Crossref]
  15. C. S. Fraser and M. R. Shortis, “Variation of distortion within the photographic field,” Photogram. Eng. Rem. S. 58(6), 851–855 (1992).

2016 (1)

T. Luhmann, C. S. Fraser, and H. G. Maas, “Sensor modelling and camera calibration for close-range photogrammetry,” ISPRS J. Photogramm. 115, 37–46 (2016).
[Crossref]

2013 (2)

C. S. Fraser, “Automatic camera calibration in close range photogrammetry,” Photogram. Eng. Remote Sens. 79(4), 381–388 (2013).
[Crossref]

T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
[Crossref] [PubMed]

2011 (1)

L. Alvarez, L. Gómez, and J. R. Sendra, “Accurate depth dependent lens distortion models: an application to planar view scenarios,” J. Math. Imaging Vis. 39(1), 75–85 (2011).
[Crossref]

2010 (1)

T. Luhmann, “Close range photogrammetry for industrial applications,” ISPRS J. Photogramm. 65(6), 558–569 (2010).
[Crossref]

2006 (2)

F. Remondino and C. S. Fraser, “Digital camera calibration methods: considerations and comparisons,” Int. Arch. Photogram. 36(5), 266–272 (2006).

F. Xing, Y. Dong, and Z. You, “Laboratory calibration of star tracker with brightness independent star identification strategy,” Opt. Eng. 45(6), 063604 (2006).
[Crossref]

2000 (1)

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE. T. Pattern. Anal. 22(11), 1330–1334 (2000).

1994 (1)

J. G. Fryer, T. A. Clarke, and J. Chen, “Lens distortion for simple C-mount lenses,” Int. Arch. Photogramm. Remote Sens. 30, 97–101 (1994).

1992 (1)

C. S. Fraser and M. R. Shortis, “Variation of distortion within the photographic field,” Photogram. Eng. Rem. S. 58(6), 851–855 (1992).

1986 (2)

J. G. Fryer and D. C. Brown, “Lens distortion for close-range photogrammetry,” Photogram. Eng. Rem. S. 52(1), 51–58 (1986).

J. G. Fryer and C. S. Fraser, “On the calibration of underwater cameras,” Photogram. Rec. 12(67), 73–85 (1986).
[Crossref]

1971 (1)

D. C. Brown, “Close-range camera calibration,” Photogram. Eng. 37, 855–866 (1971).

1955 (1)

Alvarez, L.

L. Alvarez, L. Gómez, and J. R. Sendra, “Accurate depth dependent lens distortion models: an application to planar view scenarios,” J. Math. Imaging Vis. 39(1), 75–85 (2011).
[Crossref]

Brown, D. C.

J. G. Fryer and D. C. Brown, “Lens distortion for close-range photogrammetry,” Photogram. Eng. Rem. S. 52(1), 51–58 (1986).

D. C. Brown, “Close-range camera calibration,” Photogram. Eng. 37, 855–866 (1971).

Chen, J.

J. G. Fryer, T. A. Clarke, and J. Chen, “Lens distortion for simple C-mount lenses,” Int. Arch. Photogramm. Remote Sens. 30, 97–101 (1994).

Clarke, T. A.

J. G. Fryer, T. A. Clarke, and J. Chen, “Lens distortion for simple C-mount lenses,” Int. Arch. Photogramm. Remote Sens. 30, 97–101 (1994).

Dong, Y.

F. Xing, Y. Dong, and Z. You, “Laboratory calibration of star tracker with brightness independent star identification strategy,” Opt. Eng. 45(6), 063604 (2006).
[Crossref]

Fraser, C. S.

T. Luhmann, C. S. Fraser, and H. G. Maas, “Sensor modelling and camera calibration for close-range photogrammetry,” ISPRS J. Photogramm. 115, 37–46 (2016).
[Crossref]

C. S. Fraser, “Automatic camera calibration in close range photogrammetry,” Photogram. Eng. Remote Sens. 79(4), 381–388 (2013).
[Crossref]

F. Remondino and C. S. Fraser, “Digital camera calibration methods: considerations and comparisons,” Int. Arch. Photogram. 36(5), 266–272 (2006).

C. S. Fraser and M. R. Shortis, “Variation of distortion within the photographic field,” Photogram. Eng. Rem. S. 58(6), 851–855 (1992).

J. G. Fryer and C. S. Fraser, “On the calibration of underwater cameras,” Photogram. Rec. 12(67), 73–85 (1986).
[Crossref]

Fryer, J. G.

J. G. Fryer, T. A. Clarke, and J. Chen, “Lens distortion for simple C-mount lenses,” Int. Arch. Photogramm. Remote Sens. 30, 97–101 (1994).

J. G. Fryer and D. C. Brown, “Lens distortion for close-range photogrammetry,” Photogram. Eng. Rem. S. 52(1), 51–58 (1986).

J. G. Fryer and C. S. Fraser, “On the calibration of underwater cameras,” Photogram. Rec. 12(67), 73–85 (1986).
[Crossref]

Gómez, L.

L. Alvarez, L. Gómez, and J. R. Sendra, “Accurate depth dependent lens distortion models: an application to planar view scenarios,” J. Math. Imaging Vis. 39(1), 75–85 (2011).
[Crossref]

Luhmann, T.

T. Luhmann, C. S. Fraser, and H. G. Maas, “Sensor modelling and camera calibration for close-range photogrammetry,” ISPRS J. Photogramm. 115, 37–46 (2016).
[Crossref]

T. Luhmann, “Close range photogrammetry for industrial applications,” ISPRS J. Photogramm. 65(6), 558–569 (2010).
[Crossref]

Maas, H. G.

T. Luhmann, C. S. Fraser, and H. G. Maas, “Sensor modelling and camera calibration for close-range photogrammetry,” ISPRS J. Photogramm. 115, 37–46 (2016).
[Crossref]

Magill, A. A.

Remondino, F.

F. Remondino and C. S. Fraser, “Digital camera calibration methods: considerations and comparisons,” Int. Arch. Photogram. 36(5), 266–272 (2006).

Sendra, J. R.

L. Alvarez, L. Gómez, and J. R. Sendra, “Accurate depth dependent lens distortion models: an application to planar view scenarios,” J. Math. Imaging Vis. 39(1), 75–85 (2011).
[Crossref]

Shortis, M. R.

C. S. Fraser and M. R. Shortis, “Variation of distortion within the photographic field,” Photogram. Eng. Rem. S. 58(6), 851–855 (1992).

Sun, T.

T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
[Crossref] [PubMed]

Xing, F.

T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
[Crossref] [PubMed]

F. Xing, Y. Dong, and Z. You, “Laboratory calibration of star tracker with brightness independent star identification strategy,” Opt. Eng. 45(6), 063604 (2006).
[Crossref]

You, Z.

T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
[Crossref] [PubMed]

F. Xing, Y. Dong, and Z. You, “Laboratory calibration of star tracker with brightness independent star identification strategy,” Opt. Eng. 45(6), 063604 (2006).
[Crossref]

Zhang, Z. Y.

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE. T. Pattern. Anal. 22(11), 1330–1334 (2000).

IEEE. T. Pattern. Anal. (1)

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE. T. Pattern. Anal. 22(11), 1330–1334 (2000).

Int. Arch. Photogram. (1)

F. Remondino and C. S. Fraser, “Digital camera calibration methods: considerations and comparisons,” Int. Arch. Photogram. 36(5), 266–272 (2006).

Int. Arch. Photogramm. Remote Sens. (1)

J. G. Fryer, T. A. Clarke, and J. Chen, “Lens distortion for simple C-mount lenses,” Int. Arch. Photogramm. Remote Sens. 30, 97–101 (1994).

ISPRS J. Photogramm. (2)

T. Luhmann, “Close range photogrammetry for industrial applications,” ISPRS J. Photogramm. 65(6), 558–569 (2010).
[Crossref]

T. Luhmann, C. S. Fraser, and H. G. Maas, “Sensor modelling and camera calibration for close-range photogrammetry,” ISPRS J. Photogramm. 115, 37–46 (2016).
[Crossref]

J. Math. Imaging Vis. (1)

L. Alvarez, L. Gómez, and J. R. Sendra, “Accurate depth dependent lens distortion models: an application to planar view scenarios,” J. Math. Imaging Vis. 39(1), 75–85 (2011).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

F. Xing, Y. Dong, and Z. You, “Laboratory calibration of star tracker with brightness independent star identification strategy,” Opt. Eng. 45(6), 063604 (2006).
[Crossref]

Photogram. Eng. (1)

D. C. Brown, “Close-range camera calibration,” Photogram. Eng. 37, 855–866 (1971).

Photogram. Eng. Rem. S. (2)

J. G. Fryer and D. C. Brown, “Lens distortion for close-range photogrammetry,” Photogram. Eng. Rem. S. 52(1), 51–58 (1986).

C. S. Fraser and M. R. Shortis, “Variation of distortion within the photographic field,” Photogram. Eng. Rem. S. 58(6), 851–855 (1992).

Photogram. Eng. Remote Sens. (1)

C. S. Fraser, “Automatic camera calibration in close range photogrammetry,” Photogram. Eng. Remote Sens. 79(4), 381–388 (2013).
[Crossref]

Photogram. Rec. (1)

J. G. Fryer and C. S. Fraser, “On the calibration of underwater cameras,” Photogram. Rec. 12(67), 73–85 (1986).
[Crossref]

Sensors (Basel) (1)

T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
[Crossref] [PubMed]

Other (1)

T. Luhmann, S. Robson, and S. Kyle, Close-range photogrammetry and 3D imaging (Walter de Gruyter, 2014).

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Figures (10)

Fig. 1
Fig. 1

A special measurement scene involving targets with large depth variation. (a) One photo of the measurement scene. The background RRTs and coded RRTs are marked with ‘○’ and ‘△’ respectively while the foreground RRTs are marked with ‘□’; (b) the vertical plan of this scene and the entire bar positions.

Fig. 2
Fig. 2

The coplanar RRF calibration frame and the line patterns. (a) One image of the normally illuminated calibration patterns; (b) One image of the flash light illuminated patterns.

Fig. 3
Fig. 3

The coded RRTs on the frame plane in Fig. 2 and the defined world reference system.

Fig. 4
Fig. 4

Mechanism of the automated orientation adjustment system.

Fig. 5
Fig. 5

Vertical and horizontal images and the extracted line patterns at a specific depth. (a) The vertical line patterns image; (b) The extracted vertical observations; (c) The horizontal line patterns image; and (d) The extracted horizontal observations.

Fig. 6
Fig. 6

The process of the proposed calibration method.

Fig. 7
Fig. 7

Calibration images of the 35 mm lens at six depths. (a) 2.46 m; (b) 2.90 m; (c) 3.15 m; (d) 3.60 m; (e) 4.19 m; (f) 4.51 m.

Fig. 8
Fig. 8

Radial and decentering distortion profiles calibrated for the 50 mm lens at different depths.

Fig. 9
Fig. 9

Radial and decentering distortion profiles measured for the 35 mm lens.

Fig. 10
Fig. 10

Length measurement results.

Tables (6)

Tables Icon

Table 1 Calibrated distortion values of the 50 mm lens.

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Table 2 Calibrated distortion values of the 35 mm lens.

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Table 3 RMSE estimation of one calibration for the 35 mm lens (b1 and b2 are calibrated but not exhibited).

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Table 4 Comparison between calibrated and computed radial distortions for the 50 mm lens.

Tables Icon

Table 5 Comparison between calibrated and computed radial distortions for the 35 mm lens.

Tables Icon

Table 6 Comparison of length measurement results between two models

Equations (22)

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x x p +Δx=c R 11 (X X 0 )+ R 12 (Y Y 0 )+ R 13 (Z Z 0 ) R 31 (X X 0 )+ R 32 (Y Y 0 )+ R 33 (Z Z 0 ) y y p +Δy=c R 21 (X X 0 )+ R 22 (Y Y 0 )+ R 23 (Z Z 0 ) R 31 (X X 0 )+ R 32 (Y Y 0 )+ R 33 (Z Z 0 ) .
Δx= x ¯ ( K 1 r 2 + K 2 r 4 + K 3 r 6 )+ P 1 (2 x ¯ 2 + r 2 )+2 P 2 x ¯ y ¯ + b 1 x ¯ + b 2 y ¯ Δy= y ¯ ( K 1 r 2 + K 2 r 4 + K 3 r 6 )+ P 2 (2 y ¯ 2 + r 2 )+2 P 1 x ¯ y ¯ , x ¯ =x x p y ¯ =y y p r= x ¯ 2 + y ¯ 2
δ r s =δ r m s δ r + ,
δ r s = α s δ r s 1 ( 1 α s )δ r s 2
α s = s 2 s s 2 s 1 s 1 c s c ,
δ r s, s = 1 γ s, s δ r s , and γ s, s = c s c s ,
δ d x_s = P 1s ( r s 2 +2 x s 2 )+2 P 2s x s y s = c c s [ P 1 ( r s 2 +2 x s 2 )+2 P 2 x s y s ] δ d y_s = P 2s ( r s 2 +2 y s 2 )+2 P 1s x s y s = c c s [ P 2 ( r s 2 +2 y s 2 )+2 P 1 x s y s ],
k s, s p ={ k s, s min + α s, s min ( s p )( k s - k s, s min ) if s p <s k s, s max + α s, s max ( s p )( k s - k s, s max ) otherwise ,
δ r s, s 1 = c s c s 1 δ r s 1 .
δ r s, s 1 = c s c s 1 ( k 1 s 1 r s 1 3 + k 2 s 1 r s 1 5 + k 3 s 1 r s 1 7 ).
δ r s, s 1 = c s 2 c s 1 2 k 1 s 1 r s 3 + c s 4 c s 1 4 k 2 s 1 r s 5 + c s 6 c s 1 6 k 3 s 1 r s 7 .
δ r s, s 1 = k 1s, s 1 r s 3 + k 2s, s 1 r s 5 + k 3s, s 1 r s 7 .
k 1 s 1 = c s 1 2 c s 2 k 1s, s 1 k 2 s 1 = c s 1 4 c s 4 k 2s, s 1 k 3 s 1 = c s 1 6 c s 6 k 3s, s 1 ,
k 1 s 2 = c s 2 2 c s 2 k 1s, s 2 k 2 s 2 = c s 2 4 c s 4 k 2s, s 2 k 3 s 2 = c s 2 6 c s 6 k 3s, s 2 .
δ r s = k 1 s r s 3 + k 2 s r s 5 + k 3 s r s 7 = α s δ r s 1 +( 1 α s )δ r s 2 ,
k 1 s = α s k 1 s 1 +( 1 α s ) k 1 s 2 k 2 s = α s k 2 s 1 +( 1 α s ) k 2 s 2 . k 3 s = α s k 3 s 1 +( 1 α s ) k 3 s 2
δ r s, s = c s 2 c s 2 k 1 s r s 3 + c s 4 c s 4 k 2 s r s 5 + c s 6 c s 6 k 3 s r s 7 .
k 1s, s = α s C s 2 C s 1 2 k 1s, s 1 +(1 α s ) C s 2 C s 2 2 k 1s, s 2 k 2s, s = α s C s 4 C s 1 4 k 2s, s 1 +(1 α s ) C s 4 C s 2 4 k 2s, s 2 . k 3s, s = α s C s 6 C s 1 6 k 3s, s 1 +(1 α s ) C s 6 C s 2 6 k 3s, s 2
δ d x_ s = c c s [ P 1 ( r s 2 +2 x s 2 )+2 P 2 x s y s ] δ d y_ s = c c s [ P 2 ( r s 2 +2 y s 2 )+2 P 1 x s y s ].
δ d x_s, s = c s c s δ d x_ s = c c s c s 2 [ P 1 ( r s 2 +2 x s 2 )+2 P 2 x s y s ] δ d y_s, s = c s c s δ d y_ s = c c s c s 2 [ P 2 ( r s 2 +2 y s 2 )+2 P 1 x s y s ].
δ d x_s, s = P 1s, s ( r s 2 +2 x s 2 )+2 P 2s, s x s y s = c c s [ P 1 ( r s 2 +2 x s 2 )+2 P 2 x s y s ] δ d y_s, s = P 2s, s ( r s 2 +2 y s 2 )+2 P 1s, s x s y s = c c s [ P 2 ( r s 2 +2 y s 2 )+2 P 1 x s y s ].
P 1s, s = P 1s , P 2s, s = P 2s .

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