Abstract

Fish-eye cameras are widely used on many occasions due to their ultrawide field of view (about 180°). In this paper, we present a high-precision two-step calibration method to calibrate fish-eye cameras. The two steps are the global polynomial projection model fitting and local line-fitting calibration optimization. In the first step, we obtain the projection model of the fish-eye camera and apply a quartic polynomial to fit the projection model over the entire image. In the second step, the fish-eye image is partitioned into several sections and line fitting is adopted in each section in order to further reduce the residual error of the first calibration step. Experiments show that the new method is able to correct the distortion of the real scene image well. In addition, its average reprojection error is 0.15 pixel better than 0.40 pixel of the general projection model described. The reason that higher calibration precision is obtained is that this method not only considers the global projection model of the fish-eye camera but also considers the local characteristics, such as small tangential distortion and asymmetry.

© 2013 Optical Society of America

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  1. C. Hughes, P. Denny, E. Jones, and M. Glavin, “Accuracy of fish-eye lens models,” Appl. Opt. 49, 3338–3347 (2010).
    [CrossRef]
  2. X. Song, L. Liu, and J. Tang, “High-accuracy angle detection for ultrawide field-of-view acquisition in wireless optical links,” Opt. Eng. 47, 025010 (2008).
    [CrossRef]
  3. D. E. Stevenson and M. M. Fleck, “Robot aerobics: four easy steps to a more flexible calibration,” in Fifth International Conference on Computer Vision (IEEE, 1995), pp. 34–39.
  4. T. N. Mundhenk, M. J. Rivett, X. Q. Liao, and E. L. Hall, “Techniques for fish-eye lens calibration using a minimal number of measurements,” Proc. SPIE 4197, 181–190 (2000).
    [CrossRef]
  5. M. Ahmed and A. Farag, “Nonmetric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process. 14, 1215–1230 (2005).
    [CrossRef]
  6. S. B. Kang, “Semi-automatic methods for recovering radial distortion parameters from a single image,” Technical Reports Series CRL 97/3 (Cambridge Research Labs, 1997).
  7. Q. T. Luong and O. Faugeras, “Self-calibration of a moving camera from point correspondences and fundamental matrices,” Int. J. Comput. Vis. 22, 261–289 (1997).
    [CrossRef]
  8. T. J. Herbert, “Calibration of fish-eye lenses by inversion of area projections,” Appl. Opt. 25, 1875–1876 (1986).
    [CrossRef]
  9. H. Bakstein and T. Pajdla, “Panoramic mosaicing with a 180° field of view lens,” in Proceedings of the IEEE Omni-directional Vision Workshop (IEEE, 2002), pp. 60–67.
  10. S. Shah and J. K. Aggarwal, “Intrinsic parameter calibration procedure for a (high-distortion) fish-eye lens camera with distortion model and accuracy estimation,” Pattern Recogn. 29, 1775–1788 (1996).
    [CrossRef]
  11. D. Scaramuzza, A. Martinelli, and R. Siegwart, “A toolbox for easily calibrating omnidirectional cameras,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2006), pp. 5695–5701.
  12. D. Scaramuzza, A. Martinelli, and R. Siegwart, “A flexible technique for accurate omnidirectional camera calibration and structure from motion,” in Proceedings of the Fourth IEEE International Conference on Computer Vision Systems (IEEE, 2006), p. 45.
  13. J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14, 965–980 (1992).
    [CrossRef]
  14. B. Caprile and V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vis. 4, 127–139 (1990).
    [CrossRef]
  15. S. Baker and S. Nayar, “A theory of single-viewpoint catadioptric image formation,” Int. J. Comput. Vis. 35, 175–196(1999).
    [CrossRef]
  16. F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13, 14–24 (2001).
    [CrossRef]
  17. J. Han, X. Yu, and W. Zhang, “Algorithm of the Delaunay triangulation net interpolated feature points for borehole data,” in 2010 Second International Workshop on Education Technology and Computer Science (ETCS) (IEEE, 2010), Vol. 1, pp. 447–450.
  18. R. Sibsom, “A brief description of natural neighbor interpolation,” in V. Barnett, ed., Interpreting Multivariate Data (Wiley, 1981), pp. 21–36.
  19. D. F. Watson, “Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes,” Comput. J. 24, 167–172 (1981).
    [CrossRef]

2010

2008

X. Song, L. Liu, and J. Tang, “High-accuracy angle detection for ultrawide field-of-view acquisition in wireless optical links,” Opt. Eng. 47, 025010 (2008).
[CrossRef]

2005

M. Ahmed and A. Farag, “Nonmetric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process. 14, 1215–1230 (2005).
[CrossRef]

2001

F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13, 14–24 (2001).
[CrossRef]

2000

T. N. Mundhenk, M. J. Rivett, X. Q. Liao, and E. L. Hall, “Techniques for fish-eye lens calibration using a minimal number of measurements,” Proc. SPIE 4197, 181–190 (2000).
[CrossRef]

1999

S. Baker and S. Nayar, “A theory of single-viewpoint catadioptric image formation,” Int. J. Comput. Vis. 35, 175–196(1999).
[CrossRef]

1997

Q. T. Luong and O. Faugeras, “Self-calibration of a moving camera from point correspondences and fundamental matrices,” Int. J. Comput. Vis. 22, 261–289 (1997).
[CrossRef]

1996

S. Shah and J. K. Aggarwal, “Intrinsic parameter calibration procedure for a (high-distortion) fish-eye lens camera with distortion model and accuracy estimation,” Pattern Recogn. 29, 1775–1788 (1996).
[CrossRef]

1992

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14, 965–980 (1992).
[CrossRef]

1990

B. Caprile and V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vis. 4, 127–139 (1990).
[CrossRef]

1986

1981

D. F. Watson, “Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes,” Comput. J. 24, 167–172 (1981).
[CrossRef]

Aggarwal, J. K.

S. Shah and J. K. Aggarwal, “Intrinsic parameter calibration procedure for a (high-distortion) fish-eye lens camera with distortion model and accuracy estimation,” Pattern Recogn. 29, 1775–1788 (1996).
[CrossRef]

Ahmed, M.

M. Ahmed and A. Farag, “Nonmetric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process. 14, 1215–1230 (2005).
[CrossRef]

Baker, S.

S. Baker and S. Nayar, “A theory of single-viewpoint catadioptric image formation,” Int. J. Comput. Vis. 35, 175–196(1999).
[CrossRef]

Bakstein, H.

H. Bakstein and T. Pajdla, “Panoramic mosaicing with a 180° field of view lens,” in Proceedings of the IEEE Omni-directional Vision Workshop (IEEE, 2002), pp. 60–67.

Caprile, B.

B. Caprile and V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vis. 4, 127–139 (1990).
[CrossRef]

Cohen, P.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14, 965–980 (1992).
[CrossRef]

Denny, P.

Devernay, F.

F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13, 14–24 (2001).
[CrossRef]

Farag, A.

M. Ahmed and A. Farag, “Nonmetric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process. 14, 1215–1230 (2005).
[CrossRef]

Faugeras, O.

F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13, 14–24 (2001).
[CrossRef]

Q. T. Luong and O. Faugeras, “Self-calibration of a moving camera from point correspondences and fundamental matrices,” Int. J. Comput. Vis. 22, 261–289 (1997).
[CrossRef]

Fleck, M. M.

D. E. Stevenson and M. M. Fleck, “Robot aerobics: four easy steps to a more flexible calibration,” in Fifth International Conference on Computer Vision (IEEE, 1995), pp. 34–39.

Glavin, M.

Hall, E. L.

T. N. Mundhenk, M. J. Rivett, X. Q. Liao, and E. L. Hall, “Techniques for fish-eye lens calibration using a minimal number of measurements,” Proc. SPIE 4197, 181–190 (2000).
[CrossRef]

Han, J.

J. Han, X. Yu, and W. Zhang, “Algorithm of the Delaunay triangulation net interpolated feature points for borehole data,” in 2010 Second International Workshop on Education Technology and Computer Science (ETCS) (IEEE, 2010), Vol. 1, pp. 447–450.

Herbert, T. J.

Herniou, M.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14, 965–980 (1992).
[CrossRef]

Hughes, C.

Jones, E.

Kang, S. B.

S. B. Kang, “Semi-automatic methods for recovering radial distortion parameters from a single image,” Technical Reports Series CRL 97/3 (Cambridge Research Labs, 1997).

Liao, X. Q.

T. N. Mundhenk, M. J. Rivett, X. Q. Liao, and E. L. Hall, “Techniques for fish-eye lens calibration using a minimal number of measurements,” Proc. SPIE 4197, 181–190 (2000).
[CrossRef]

Liu, L.

X. Song, L. Liu, and J. Tang, “High-accuracy angle detection for ultrawide field-of-view acquisition in wireless optical links,” Opt. Eng. 47, 025010 (2008).
[CrossRef]

Luong, Q. T.

Q. T. Luong and O. Faugeras, “Self-calibration of a moving camera from point correspondences and fundamental matrices,” Int. J. Comput. Vis. 22, 261–289 (1997).
[CrossRef]

Martinelli, A.

D. Scaramuzza, A. Martinelli, and R. Siegwart, “A toolbox for easily calibrating omnidirectional cameras,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2006), pp. 5695–5701.

D. Scaramuzza, A. Martinelli, and R. Siegwart, “A flexible technique for accurate omnidirectional camera calibration and structure from motion,” in Proceedings of the Fourth IEEE International Conference on Computer Vision Systems (IEEE, 2006), p. 45.

Mundhenk, T. N.

T. N. Mundhenk, M. J. Rivett, X. Q. Liao, and E. L. Hall, “Techniques for fish-eye lens calibration using a minimal number of measurements,” Proc. SPIE 4197, 181–190 (2000).
[CrossRef]

Nayar, S.

S. Baker and S. Nayar, “A theory of single-viewpoint catadioptric image formation,” Int. J. Comput. Vis. 35, 175–196(1999).
[CrossRef]

Pajdla, T.

H. Bakstein and T. Pajdla, “Panoramic mosaicing with a 180° field of view lens,” in Proceedings of the IEEE Omni-directional Vision Workshop (IEEE, 2002), pp. 60–67.

Rivett, M. J.

T. N. Mundhenk, M. J. Rivett, X. Q. Liao, and E. L. Hall, “Techniques for fish-eye lens calibration using a minimal number of measurements,” Proc. SPIE 4197, 181–190 (2000).
[CrossRef]

Scaramuzza, D.

D. Scaramuzza, A. Martinelli, and R. Siegwart, “A toolbox for easily calibrating omnidirectional cameras,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2006), pp. 5695–5701.

D. Scaramuzza, A. Martinelli, and R. Siegwart, “A flexible technique for accurate omnidirectional camera calibration and structure from motion,” in Proceedings of the Fourth IEEE International Conference on Computer Vision Systems (IEEE, 2006), p. 45.

Shah, S.

S. Shah and J. K. Aggarwal, “Intrinsic parameter calibration procedure for a (high-distortion) fish-eye lens camera with distortion model and accuracy estimation,” Pattern Recogn. 29, 1775–1788 (1996).
[CrossRef]

Sibsom, R.

R. Sibsom, “A brief description of natural neighbor interpolation,” in V. Barnett, ed., Interpreting Multivariate Data (Wiley, 1981), pp. 21–36.

Siegwart, R.

D. Scaramuzza, A. Martinelli, and R. Siegwart, “A toolbox for easily calibrating omnidirectional cameras,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2006), pp. 5695–5701.

D. Scaramuzza, A. Martinelli, and R. Siegwart, “A flexible technique for accurate omnidirectional camera calibration and structure from motion,” in Proceedings of the Fourth IEEE International Conference on Computer Vision Systems (IEEE, 2006), p. 45.

Song, X.

X. Song, L. Liu, and J. Tang, “High-accuracy angle detection for ultrawide field-of-view acquisition in wireless optical links,” Opt. Eng. 47, 025010 (2008).
[CrossRef]

Stevenson, D. E.

D. E. Stevenson and M. M. Fleck, “Robot aerobics: four easy steps to a more flexible calibration,” in Fifth International Conference on Computer Vision (IEEE, 1995), pp. 34–39.

Tang, J.

X. Song, L. Liu, and J. Tang, “High-accuracy angle detection for ultrawide field-of-view acquisition in wireless optical links,” Opt. Eng. 47, 025010 (2008).
[CrossRef]

Torre, V.

B. Caprile and V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vis. 4, 127–139 (1990).
[CrossRef]

Watson, D. F.

D. F. Watson, “Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes,” Comput. J. 24, 167–172 (1981).
[CrossRef]

Weng, J.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14, 965–980 (1992).
[CrossRef]

Yu, X.

J. Han, X. Yu, and W. Zhang, “Algorithm of the Delaunay triangulation net interpolated feature points for borehole data,” in 2010 Second International Workshop on Education Technology and Computer Science (ETCS) (IEEE, 2010), Vol. 1, pp. 447–450.

Zhang, W.

J. Han, X. Yu, and W. Zhang, “Algorithm of the Delaunay triangulation net interpolated feature points for borehole data,” in 2010 Second International Workshop on Education Technology and Computer Science (ETCS) (IEEE, 2010), Vol. 1, pp. 447–450.

Appl. Opt.

Comput. J.

D. F. Watson, “Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes,” Comput. J. 24, 167–172 (1981).
[CrossRef]

IEEE Trans. Image Process.

M. Ahmed and A. Farag, “Nonmetric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process. 14, 1215–1230 (2005).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14, 965–980 (1992).
[CrossRef]

Int. J. Comput. Vis.

B. Caprile and V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vis. 4, 127–139 (1990).
[CrossRef]

S. Baker and S. Nayar, “A theory of single-viewpoint catadioptric image formation,” Int. J. Comput. Vis. 35, 175–196(1999).
[CrossRef]

Q. T. Luong and O. Faugeras, “Self-calibration of a moving camera from point correspondences and fundamental matrices,” Int. J. Comput. Vis. 22, 261–289 (1997).
[CrossRef]

Mach. Vis. Appl.

F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13, 14–24 (2001).
[CrossRef]

Opt. Eng.

X. Song, L. Liu, and J. Tang, “High-accuracy angle detection for ultrawide field-of-view acquisition in wireless optical links,” Opt. Eng. 47, 025010 (2008).
[CrossRef]

Pattern Recogn.

S. Shah and J. K. Aggarwal, “Intrinsic parameter calibration procedure for a (high-distortion) fish-eye lens camera with distortion model and accuracy estimation,” Pattern Recogn. 29, 1775–1788 (1996).
[CrossRef]

Proc. SPIE

T. N. Mundhenk, M. J. Rivett, X. Q. Liao, and E. L. Hall, “Techniques for fish-eye lens calibration using a minimal number of measurements,” Proc. SPIE 4197, 181–190 (2000).
[CrossRef]

Other

H. Bakstein and T. Pajdla, “Panoramic mosaicing with a 180° field of view lens,” in Proceedings of the IEEE Omni-directional Vision Workshop (IEEE, 2002), pp. 60–67.

S. B. Kang, “Semi-automatic methods for recovering radial distortion parameters from a single image,” Technical Reports Series CRL 97/3 (Cambridge Research Labs, 1997).

D. Scaramuzza, A. Martinelli, and R. Siegwart, “A toolbox for easily calibrating omnidirectional cameras,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2006), pp. 5695–5701.

D. Scaramuzza, A. Martinelli, and R. Siegwart, “A flexible technique for accurate omnidirectional camera calibration and structure from motion,” in Proceedings of the Fourth IEEE International Conference on Computer Vision Systems (IEEE, 2006), p. 45.

D. E. Stevenson and M. M. Fleck, “Robot aerobics: four easy steps to a more flexible calibration,” in Fifth International Conference on Computer Vision (IEEE, 1995), pp. 34–39.

J. Han, X. Yu, and W. Zhang, “Algorithm of the Delaunay triangulation net interpolated feature points for borehole data,” in 2010 Second International Workshop on Education Technology and Computer Science (ETCS) (IEEE, 2010), Vol. 1, pp. 447–450.

R. Sibsom, “A brief description of natural neighbor interpolation,” in V. Barnett, ed., Interpreting Multivariate Data (Wiley, 1981), pp. 21–36.

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Figures (6)

Fig. 1.
Fig. 1.

Simplified projection model of the fish-eye camera.

Fig. 2.
Fig. 2.

Using the projection polynomial to get the calibrated image.

Fig. 3.
Fig. 3.

Test equipment.

Fig. 4.
Fig. 4.

Second step: (a) extracted chessboard template part, (b) the characteristic point-fitting result, and (c) the NNI interpolation result.

Fig. 5.
Fig. 5.

Calibration results: (a) original image and (b) center, (c) left, (d) top, (e) right, and (f) bottom after calibration.

Fig. 6.
Fig. 6.

Reprojection error: (a) the general projection model and (b) the two-step calibration.

Tables (3)

Tables Icon

Table 1. Parameters of the Testing Devices

Tables Icon

Table 2. Calibration Results of the Global Projection Model

Tables Icon

Table 3. Reprojection Error of the Two Methods

Equations (9)

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

[u,v]T=A·[u,v]T+t,
λ·p=[R|t]·P,λ>0,
f(u,v)=a0+a1ρ+a2ρ2++aNρN,
f(u,v)=a0+a1ρ+a2ρ2+a3ρ3+a4ρ4.
λ·[x,y,z]T=λ·[u,v,w]T=λ·[α·u,α·v,f(u,v)]T=[R|t]·[X,Y,Z,1]T.
λij·[uij,vij,a0+a1ρij+a2ρij2+a3ρij3+a4ρij4]T=[r1i,r2i,ti]·[Xij,Yij,1]T,
E=i=1Kj=1Lmijm^(r1i,r2i,ti,Oc,a0,a1,a2,a3,a4,Mij)2,
E(a,b,d)=i=1k(axi+byid)2.
f(x)=i=1Mwif(pi),

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