Abstract

We focus on recovering the 2D Euclidean structure further for camera calibration from the projections of N parallel similar conics in this paper. This work demonstrates that the conic dual to the absolute points (CDAP) is the general form of the conic dual to the circular points, so it encodes the 2D Euclidean structure. However, the geometric size of the conic should be known if we utilize the CDAP. Under some special conditions (concentric conics), we proposed the rank-1 and rank-2 constraints. Our work relaxes the problem conditions and gives a more general framework than before. Experiments with simulated and real data are carried out to show the validity of the proposed algorithm.

© 2014 Optical Society of America

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  1. X. Meng and Z. Hu, “A new easy camera calibration technique based on circular points,” Pattern Recogn. 36, 1155–1164 (2003).
    [CrossRef]
  2. Y. Wu, H. Zhu, Z. Hu, and F. Wu, “Coplanar circles, quasi-affine invariance and calibration,” Image Vis. Comput. 24, 319–326 (2006).
  3. P. Gurdjos, P. Sturm, and Y. Wu, “Euclidean structure from N ≥ 2 parallel circles: theory and algorithms,” in Proceedings of European Conference on Computer Vision (Springer, 2006), pp. 238–252.
  4. J. Kim, P. Gurdjos, and I. Kweon, “Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27, 637–642 (2005).
    [CrossRef]
  5. P. Gurdjos, J. Kim, and I. Kweon, “Euclidean structure from confocal conics: theory and application to camera calibration,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2006), pp. 1214–1221.
  6. X. Ying and H. Zha, “Camera calibration using principal-axes aligned conics,” in Proceedings of Asian Conference on Computer Vision (Springer, 2007), pp. 138–148.
  7. Z. Zhao, “Conics with a common axis of symmetry: properties and applications to camera calibration,” in Proceedings of International Joint Conference on Artificial Intelligence (AAAI, 2011), pp. 2079–2084.
  8. A. Sugimoto, “A linear algorithm for computing the homography from conics in correspondence,” J. Math. Imaging Vis. 13, 115–130 (2000).
  9. P. Mudigonda, C. V. Jawahar, and P. J. Narayanan, “Geometric structure computing from conics,” in Proceedings of Indian Conference on Computer Vision, Graphics and Image Processing (Allied Publishers Private Limited, 2004), pp. 9–14.
  10. C. Yang, F. Sun, and Z. Hu, “Planar conic based camera calibration,” in Proceedings of International Conference on Pattern Recognition (IEEE Computer Society, 2000), pp. 555–558.
  11. A. W. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least squares fitting of ellipses,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 476–480 (1999).
    [CrossRef]
  12. F. Feschet and L. Tougne, “Optimal time computation of the tangent of a discrete curve: application to the curvature,” in Proceedings of International Conference on Discrete Geometry for Computer Imagery (Springer, 1999), pp. 31–40.
  13. R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge University, 2003).
  14. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334 (2000).
    [CrossRef]
  15. Z. Zhao, Y. Liu, and Z. Zhang, “Camera calibration with 3 non-collinear points under special motions,” IEEE Trans. Image Process. 17, 2393–2402 (2008).
    [CrossRef]
  16. J. Harris, Algebraic Geometry, A First Course (Springer-Verlag, 1992).
  17. R. Horn and C. Johnson, Matrix Analysis (Cambridge University, 1991).
  18. R. Wolf, J. Duchateau, P. Cinquin, and S. Voros, “3D tracking of laparoscopic instruments using statistical and geometric modeling,” in Proceedings of Medical Image Computing and Computer-Assisted Intervention (Springer, 2011), pp. 203–210.

2008

Z. Zhao, Y. Liu, and Z. Zhang, “Camera calibration with 3 non-collinear points under special motions,” IEEE Trans. Image Process. 17, 2393–2402 (2008).
[CrossRef]

2006

Y. Wu, H. Zhu, Z. Hu, and F. Wu, “Coplanar circles, quasi-affine invariance and calibration,” Image Vis. Comput. 24, 319–326 (2006).

2005

J. Kim, P. Gurdjos, and I. Kweon, “Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27, 637–642 (2005).
[CrossRef]

2003

X. Meng and Z. Hu, “A new easy camera calibration technique based on circular points,” Pattern Recogn. 36, 1155–1164 (2003).
[CrossRef]

2000

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334 (2000).
[CrossRef]

A. Sugimoto, “A linear algorithm for computing the homography from conics in correspondence,” J. Math. Imaging Vis. 13, 115–130 (2000).

1999

A. W. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least squares fitting of ellipses,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 476–480 (1999).
[CrossRef]

Cinquin, P.

R. Wolf, J. Duchateau, P. Cinquin, and S. Voros, “3D tracking of laparoscopic instruments using statistical and geometric modeling,” in Proceedings of Medical Image Computing and Computer-Assisted Intervention (Springer, 2011), pp. 203–210.

Duchateau, J.

R. Wolf, J. Duchateau, P. Cinquin, and S. Voros, “3D tracking of laparoscopic instruments using statistical and geometric modeling,” in Proceedings of Medical Image Computing and Computer-Assisted Intervention (Springer, 2011), pp. 203–210.

Feschet, F.

F. Feschet and L. Tougne, “Optimal time computation of the tangent of a discrete curve: application to the curvature,” in Proceedings of International Conference on Discrete Geometry for Computer Imagery (Springer, 1999), pp. 31–40.

Fisher, R. B.

A. W. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least squares fitting of ellipses,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 476–480 (1999).
[CrossRef]

Fitzgibbon, A. W.

A. W. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least squares fitting of ellipses,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 476–480 (1999).
[CrossRef]

Gurdjos, P.

J. Kim, P. Gurdjos, and I. Kweon, “Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27, 637–642 (2005).
[CrossRef]

P. Gurdjos, P. Sturm, and Y. Wu, “Euclidean structure from N ≥ 2 parallel circles: theory and algorithms,” in Proceedings of European Conference on Computer Vision (Springer, 2006), pp. 238–252.

P. Gurdjos, J. Kim, and I. Kweon, “Euclidean structure from confocal conics: theory and application to camera calibration,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2006), pp. 1214–1221.

Harris, J.

J. Harris, Algebraic Geometry, A First Course (Springer-Verlag, 1992).

Hartley, R.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge University, 2003).

Horn, R.

R. Horn and C. Johnson, Matrix Analysis (Cambridge University, 1991).

Hu, Z.

Y. Wu, H. Zhu, Z. Hu, and F. Wu, “Coplanar circles, quasi-affine invariance and calibration,” Image Vis. Comput. 24, 319–326 (2006).

X. Meng and Z. Hu, “A new easy camera calibration technique based on circular points,” Pattern Recogn. 36, 1155–1164 (2003).
[CrossRef]

C. Yang, F. Sun, and Z. Hu, “Planar conic based camera calibration,” in Proceedings of International Conference on Pattern Recognition (IEEE Computer Society, 2000), pp. 555–558.

Jawahar, C. V.

P. Mudigonda, C. V. Jawahar, and P. J. Narayanan, “Geometric structure computing from conics,” in Proceedings of Indian Conference on Computer Vision, Graphics and Image Processing (Allied Publishers Private Limited, 2004), pp. 9–14.

Johnson, C.

R. Horn and C. Johnson, Matrix Analysis (Cambridge University, 1991).

Kim, J.

J. Kim, P. Gurdjos, and I. Kweon, “Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27, 637–642 (2005).
[CrossRef]

P. Gurdjos, J. Kim, and I. Kweon, “Euclidean structure from confocal conics: theory and application to camera calibration,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2006), pp. 1214–1221.

Kweon, I.

J. Kim, P. Gurdjos, and I. Kweon, “Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27, 637–642 (2005).
[CrossRef]

P. Gurdjos, J. Kim, and I. Kweon, “Euclidean structure from confocal conics: theory and application to camera calibration,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2006), pp. 1214–1221.

Liu, Y.

Z. Zhao, Y. Liu, and Z. Zhang, “Camera calibration with 3 non-collinear points under special motions,” IEEE Trans. Image Process. 17, 2393–2402 (2008).
[CrossRef]

Meng, X.

X. Meng and Z. Hu, “A new easy camera calibration technique based on circular points,” Pattern Recogn. 36, 1155–1164 (2003).
[CrossRef]

Mudigonda, P.

P. Mudigonda, C. V. Jawahar, and P. J. Narayanan, “Geometric structure computing from conics,” in Proceedings of Indian Conference on Computer Vision, Graphics and Image Processing (Allied Publishers Private Limited, 2004), pp. 9–14.

Narayanan, P. J.

P. Mudigonda, C. V. Jawahar, and P. J. Narayanan, “Geometric structure computing from conics,” in Proceedings of Indian Conference on Computer Vision, Graphics and Image Processing (Allied Publishers Private Limited, 2004), pp. 9–14.

Pilu, M.

A. W. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least squares fitting of ellipses,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 476–480 (1999).
[CrossRef]

Sturm, P.

P. Gurdjos, P. Sturm, and Y. Wu, “Euclidean structure from N ≥ 2 parallel circles: theory and algorithms,” in Proceedings of European Conference on Computer Vision (Springer, 2006), pp. 238–252.

Sugimoto, A.

A. Sugimoto, “A linear algorithm for computing the homography from conics in correspondence,” J. Math. Imaging Vis. 13, 115–130 (2000).

Sun, F.

C. Yang, F. Sun, and Z. Hu, “Planar conic based camera calibration,” in Proceedings of International Conference on Pattern Recognition (IEEE Computer Society, 2000), pp. 555–558.

Tougne, L.

F. Feschet and L. Tougne, “Optimal time computation of the tangent of a discrete curve: application to the curvature,” in Proceedings of International Conference on Discrete Geometry for Computer Imagery (Springer, 1999), pp. 31–40.

Voros, S.

R. Wolf, J. Duchateau, P. Cinquin, and S. Voros, “3D tracking of laparoscopic instruments using statistical and geometric modeling,” in Proceedings of Medical Image Computing and Computer-Assisted Intervention (Springer, 2011), pp. 203–210.

Wolf, R.

R. Wolf, J. Duchateau, P. Cinquin, and S. Voros, “3D tracking of laparoscopic instruments using statistical and geometric modeling,” in Proceedings of Medical Image Computing and Computer-Assisted Intervention (Springer, 2011), pp. 203–210.

Wu, F.

Y. Wu, H. Zhu, Z. Hu, and F. Wu, “Coplanar circles, quasi-affine invariance and calibration,” Image Vis. Comput. 24, 319–326 (2006).

Wu, Y.

Y. Wu, H. Zhu, Z. Hu, and F. Wu, “Coplanar circles, quasi-affine invariance and calibration,” Image Vis. Comput. 24, 319–326 (2006).

P. Gurdjos, P. Sturm, and Y. Wu, “Euclidean structure from N ≥ 2 parallel circles: theory and algorithms,” in Proceedings of European Conference on Computer Vision (Springer, 2006), pp. 238–252.

Yang, C.

C. Yang, F. Sun, and Z. Hu, “Planar conic based camera calibration,” in Proceedings of International Conference on Pattern Recognition (IEEE Computer Society, 2000), pp. 555–558.

Ying, X.

X. Ying and H. Zha, “Camera calibration using principal-axes aligned conics,” in Proceedings of Asian Conference on Computer Vision (Springer, 2007), pp. 138–148.

Zha, H.

X. Ying and H. Zha, “Camera calibration using principal-axes aligned conics,” in Proceedings of Asian Conference on Computer Vision (Springer, 2007), pp. 138–148.

Zhang, Z.

Z. Zhao, Y. Liu, and Z. Zhang, “Camera calibration with 3 non-collinear points under special motions,” IEEE Trans. Image Process. 17, 2393–2402 (2008).
[CrossRef]

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334 (2000).
[CrossRef]

Zhao, Z.

Z. Zhao, Y. Liu, and Z. Zhang, “Camera calibration with 3 non-collinear points under special motions,” IEEE Trans. Image Process. 17, 2393–2402 (2008).
[CrossRef]

Z. Zhao, “Conics with a common axis of symmetry: properties and applications to camera calibration,” in Proceedings of International Joint Conference on Artificial Intelligence (AAAI, 2011), pp. 2079–2084.

Zhu, H.

Y. Wu, H. Zhu, Z. Hu, and F. Wu, “Coplanar circles, quasi-affine invariance and calibration,” Image Vis. Comput. 24, 319–326 (2006).

Zisserman, A.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge University, 2003).

IEEE Trans. Image Process.

Z. Zhao, Y. Liu, and Z. Zhang, “Camera calibration with 3 non-collinear points under special motions,” IEEE Trans. Image Process. 17, 2393–2402 (2008).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

A. W. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least squares fitting of ellipses,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 476–480 (1999).
[CrossRef]

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334 (2000).
[CrossRef]

J. Kim, P. Gurdjos, and I. Kweon, “Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27, 637–642 (2005).
[CrossRef]

Image Vis. Comput.

Y. Wu, H. Zhu, Z. Hu, and F. Wu, “Coplanar circles, quasi-affine invariance and calibration,” Image Vis. Comput. 24, 319–326 (2006).

J. Math. Imaging Vis.

A. Sugimoto, “A linear algorithm for computing the homography from conics in correspondence,” J. Math. Imaging Vis. 13, 115–130 (2000).

Pattern Recogn.

X. Meng and Z. Hu, “A new easy camera calibration technique based on circular points,” Pattern Recogn. 36, 1155–1164 (2003).
[CrossRef]

Other

F. Feschet and L. Tougne, “Optimal time computation of the tangent of a discrete curve: application to the curvature,” in Proceedings of International Conference on Discrete Geometry for Computer Imagery (Springer, 1999), pp. 31–40.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge University, 2003).

J. Harris, Algebraic Geometry, A First Course (Springer-Verlag, 1992).

R. Horn and C. Johnson, Matrix Analysis (Cambridge University, 1991).

R. Wolf, J. Duchateau, P. Cinquin, and S. Voros, “3D tracking of laparoscopic instruments using statistical and geometric modeling,” in Proceedings of Medical Image Computing and Computer-Assisted Intervention (Springer, 2011), pp. 203–210.

P. Mudigonda, C. V. Jawahar, and P. J. Narayanan, “Geometric structure computing from conics,” in Proceedings of Indian Conference on Computer Vision, Graphics and Image Processing (Allied Publishers Private Limited, 2004), pp. 9–14.

C. Yang, F. Sun, and Z. Hu, “Planar conic based camera calibration,” in Proceedings of International Conference on Pattern Recognition (IEEE Computer Society, 2000), pp. 555–558.

P. Gurdjos, P. Sturm, and Y. Wu, “Euclidean structure from N ≥ 2 parallel circles: theory and algorithms,” in Proceedings of European Conference on Computer Vision (Springer, 2006), pp. 238–252.

P. Gurdjos, J. Kim, and I. Kweon, “Euclidean structure from confocal conics: theory and application to camera calibration,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2006), pp. 1214–1221.

X. Ying and H. Zha, “Camera calibration using principal-axes aligned conics,” in Proceedings of Asian Conference on Computer Vision (Springer, 2007), pp. 138–148.

Z. Zhao, “Conics with a common axis of symmetry: properties and applications to camera calibration,” in Proceedings of International Joint Conference on Artificial Intelligence (AAAI, 2011), pp. 2079–2084.

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Figures (7)

Fig. 1.
Fig. 1.

Two conics C1 and C2 have parallel principal axes. Their degenerate conics consist of line pairs with the form of C1λiC2.

Fig. 2.
Fig. 2.

(a) Rank-1 envelope ccT. (b) Rank-2 envelope C*.

Fig. 3.
Fig. 3.

Performance of the proposed algorithm in the simulated experiments.

Fig. 4.
Fig. 4.

Augmented reality experiments. (a) Nonconcentric ellipses. (b) Concentric ellipses.

Fig. 5.
Fig. 5.

Calibration tool with the PAP pattern in the endoscope operation.

Fig. 6.
Fig. 6.

Sample results of tool tracking.

Fig. 7.
Fig. 7.

Histograms of estimated errors in tool tracking.

Tables (4)

Tables Icon

Table 1. Determine the Relative Position of C1 and C2

Tables Icon

Table 2 Recovering the Euclidean Structure on PAP Conics

Tables Icon

Table 2. Camera Calibration Results of Logitech CCD Camera

Tables Icon

Table 3. Camera Calibration Results of the Endoscope

Equations (34)

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

d[uv1]=K[Rt][XYZ1]withK=[fusu00fvv0001],
H=K[r1r2t],
C=[A0AX00BBY0AX0BY0AX02+BY021],
C˜HTCH1,
C1=[A000B0001]C2=[A0AX00BBY0AX0BY0AX02+BY02f],
λ˜λ=[λ1λ2λ3]=[(f+1AX02BY02Π2f)(f+1AX02BY02+Π2f)1],
f=λ˜3λ˜1λ˜2.
d=AX02+BY02=(λ˜1λ˜3)(λ˜2λ˜3)λ˜1λ˜2.
C˜(λ˜3)HT[00AX000BY0AX0BY0f1(AX02+BY02)]H1.
ξ˜HTξ=HT[AX0BY0f1(AX02+BY02)2].
IA=[1,e21,0]TJA=[1,e21,0]T,
λ˜1:v˜1H[(f1+AX02+BY02+Π)X0f1+AX02+BY02+Π(f1+AX02+BY02+Π)Y0f1+AX02+BY02+Π1],
λ˜2:v˜2H[(f1+AX02+BY02Π)X0f1+AX02+BY02Π(f1+AX02+BY02Π)Y0f1+AX02+BY02Π1].
C*=IAJAT+JAIAT=[11e20].
C˜*l˜=H(IAJAT+JAIAT)HTHTl=H(IA(JATl)+JA(IATl))=0.
u˜1TC˜*u˜2=0,u˜1TC˜*u˜1u˜2TC˜*u˜2=0.
(u˜1±iu˜2)TC˜*(u˜1±iu˜2)=0,
u˜1TC˜*u˜2=0,
u˜1TC˜*u˜1u˜2TC˜*u˜2=0.
C˜*=U[110]UT,
A=[Qt0T1],
C˜*0l˜=0,
u˜1TC˜*0u˜2=0,u˜1TC˜*0u˜11|e21|u˜2TC˜*0u˜2=0.
C1*[A1000B10001],C2*[A1000B1000f1],
C˜i*=μiHCi*HT,i=1,2,.
β1=μ1/μ2(with multiplicity2),β2=fμ1/μ2.
C˜*(β1)=C˜1*β1C˜2*Hdiag(0,0,1)HT=c˜c˜T,
C˜*(β2)=C˜1*β2C˜2*Hdiag(1,1e2,0)HT=C˜*.
Ȟ=L3(HH)D3,
minȞi=1Ndist2(C˜i*^,ȞCi*^).
vech(N)=(N1,1,N1,2,N2,2,N1,3,Nn,n)T.
vec(N)=Dnvech(N),vech(N)=Lnvec(N).
D3=[100000000010100000000010000001000100000001010000000001]T,L3=diag(1,1/2,1,1/2,1/2,1)D3T.
U^={vech(U)ifUis conic(symmetric matrix)vech(UUT)ifUis point(vector).

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