Abstract

We present a 3D reconstruction of retinal blood vessel trees using two views of fundus images. The problem is addressed by using well known computer vision techniques which consider: 1) The recovery of camera-eyeball model parameters by an auto-calibration method. The camera parameters are found via the solution of simplified Kruppa equations, based on correspondences found by a LMedS optimisation correlation between pairs of eight different views. 2) The extraction of blood vessels and skeletons from two fundus images. 3) The matching of corresponding points of the two skeleton trees. The trees are previously labelled during the analysis of 2D binary images. Finally, 4) the lineal triangulation of matched correspondence points and the surface modelling via generalised cylinders using diameter measurements extracted from the 2D binary images. The method is nearly automatic and it is tested with 2 sets of 10 fundus retinal images, each one taken from different subjects. Results of 3D vein and artery trees reconstructions are shown.

© 2012 OSA

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References

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  1. M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, N. Chapman, A. A. Bharath, and K. H. Parker, “Retinal vascular tree morphology: A semi-automatic quantification,” IEEE Trans. Biomed. Eng. 49, 912–917 (2002).
    [CrossRef]
  2. M. E. Martinez-Perez, A. D. Hughes, S. A. Thom, A. A. Bharath, and K. H. Parker, “Segmentation of blood vessels from red-free and fluorescein retinal images,” Med. Image Anal. 11, 47–61 (2007).
    [CrossRef] [PubMed]
  3. K. Deguchi, J. Noami, and H. Hontani, “3d fundus pattern reconstruction and display from multiple fundus images,” in Proceedings 15th International Conference on Pattern Recognition (IEEE, 2000), pp. 94–97.
    [CrossRef]
  4. K. Deguchi, D. Kawamata, K. Mizutani, H. Hontani, and K. Wakabayashi, “3d fundus shape reconstruction and display from stereo fundus images,” IEICE Trans. Inf. Syst. E83-D, 1408–1414 (2000).
  5. T. E. Choe, G. Medioni, I. Cohen, A. C. Walsh, and S. R. Sadda, “2-D registration and 3-D shape inference of the retinal fundus from fluorescein images,” Med. Image Anal. 12, 174–190 (2008).
    [CrossRef]
  6. D. Liu, N. Wood, X. Xu, N. Witt, A. Hughes, and S. Thom, “3D reconstruction of the retinal arterial tree using subject-specific fundus images,” in Advances in Computational Vision and Medical Image Processing, J. M. R. S. Tavares and R. M. N. Jorge ed. (Springer, 2009), pp. 187–201.
    [CrossRef]
  7. M. E. Martinez-Perez and A. Espinosa-Romero, “3D Reconstruction of Retinal Blood Vessels From Two Views,” in Proceedings of the 4th Indian Conference on Computer Vision, Graphics and Image Processing, B. Chanda, S. Chandran, and L. Davis, ed. (Indian Statistical Insitute, 2004), pp. 258–263.
  8. J. Arnold, J. Gates, and K. Taylor, “Possible errors in the measurement of retinal lesions,” Invest. Ophthalmol. Vis. Sci. 34, 2576–2580 (1993).
    [PubMed]
  9. M. I. A. Lourakis and R. Deriche, “Camera self-calibration using the singular value decomposition of the fundamental matrix: from point correspondences to 3d measurements,” Research Report 3748, INRIA Sophia-Antipolis, (1999).
  10. M. Sonka, Image Processing, Analysis, and Machine Vision (Thomson, 2008).
  11. K. Kanatani, Geometry Computation for Machine Vision (Oxford Science Publications, 1993).
  12. R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge Uiversity Press, 2000).
  13. P. J. Rousseeuw and A. M. Leroy, Robust Regression and Outilier Detection (John Wiley & Sons, 1987).
    [CrossRef]
  14. Z. Zhang, R. Deriche, O. Faugeras, and Q.-T. Luong, “A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry,” Research Report 2273, INRIA Sophia-Antipolis, (1994).
  15. R. I. Hartley, “Estimation of relative camera positions for uncalibrated cameras,” in Proceedings of the 2nd European Conference on Computer Vision, G. Sandini, ed. (Springer-Verlag, 1992), pp. 579–587.
  16. J. Ponce, “Straight homogeneous generalized cylinders: Differential geometry and uniqueness results,” Int. J. Comput. Vis. 4, 79–100 (1990).
    [CrossRef]

2008 (1)

T. E. Choe, G. Medioni, I. Cohen, A. C. Walsh, and S. R. Sadda, “2-D registration and 3-D shape inference of the retinal fundus from fluorescein images,” Med. Image Anal. 12, 174–190 (2008).
[CrossRef]

2007 (1)

M. E. Martinez-Perez, A. D. Hughes, S. A. Thom, A. A. Bharath, and K. H. Parker, “Segmentation of blood vessels from red-free and fluorescein retinal images,” Med. Image Anal. 11, 47–61 (2007).
[CrossRef] [PubMed]

2002 (1)

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, N. Chapman, A. A. Bharath, and K. H. Parker, “Retinal vascular tree morphology: A semi-automatic quantification,” IEEE Trans. Biomed. Eng. 49, 912–917 (2002).
[CrossRef]

2000 (1)

K. Deguchi, D. Kawamata, K. Mizutani, H. Hontani, and K. Wakabayashi, “3d fundus shape reconstruction and display from stereo fundus images,” IEICE Trans. Inf. Syst. E83-D, 1408–1414 (2000).

1993 (1)

J. Arnold, J. Gates, and K. Taylor, “Possible errors in the measurement of retinal lesions,” Invest. Ophthalmol. Vis. Sci. 34, 2576–2580 (1993).
[PubMed]

1990 (1)

J. Ponce, “Straight homogeneous generalized cylinders: Differential geometry and uniqueness results,” Int. J. Comput. Vis. 4, 79–100 (1990).
[CrossRef]

Arnold, J.

J. Arnold, J. Gates, and K. Taylor, “Possible errors in the measurement of retinal lesions,” Invest. Ophthalmol. Vis. Sci. 34, 2576–2580 (1993).
[PubMed]

Bharath, A. A.

M. E. Martinez-Perez, A. D. Hughes, S. A. Thom, A. A. Bharath, and K. H. Parker, “Segmentation of blood vessels from red-free and fluorescein retinal images,” Med. Image Anal. 11, 47–61 (2007).
[CrossRef] [PubMed]

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, N. Chapman, A. A. Bharath, and K. H. Parker, “Retinal vascular tree morphology: A semi-automatic quantification,” IEEE Trans. Biomed. Eng. 49, 912–917 (2002).
[CrossRef]

Chapman, N.

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, N. Chapman, A. A. Bharath, and K. H. Parker, “Retinal vascular tree morphology: A semi-automatic quantification,” IEEE Trans. Biomed. Eng. 49, 912–917 (2002).
[CrossRef]

Choe, T. E.

T. E. Choe, G. Medioni, I. Cohen, A. C. Walsh, and S. R. Sadda, “2-D registration and 3-D shape inference of the retinal fundus from fluorescein images,” Med. Image Anal. 12, 174–190 (2008).
[CrossRef]

Cohen, I.

T. E. Choe, G. Medioni, I. Cohen, A. C. Walsh, and S. R. Sadda, “2-D registration and 3-D shape inference of the retinal fundus from fluorescein images,” Med. Image Anal. 12, 174–190 (2008).
[CrossRef]

Deguchi, K.

K. Deguchi, D. Kawamata, K. Mizutani, H. Hontani, and K. Wakabayashi, “3d fundus shape reconstruction and display from stereo fundus images,” IEICE Trans. Inf. Syst. E83-D, 1408–1414 (2000).

K. Deguchi, J. Noami, and H. Hontani, “3d fundus pattern reconstruction and display from multiple fundus images,” in Proceedings 15th International Conference on Pattern Recognition (IEEE, 2000), pp. 94–97.
[CrossRef]

Deriche, R.

Z. Zhang, R. Deriche, O. Faugeras, and Q.-T. Luong, “A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry,” Research Report 2273, INRIA Sophia-Antipolis, (1994).

M. I. A. Lourakis and R. Deriche, “Camera self-calibration using the singular value decomposition of the fundamental matrix: from point correspondences to 3d measurements,” Research Report 3748, INRIA Sophia-Antipolis, (1999).

Espinosa-Romero, A.

M. E. Martinez-Perez and A. Espinosa-Romero, “3D Reconstruction of Retinal Blood Vessels From Two Views,” in Proceedings of the 4th Indian Conference on Computer Vision, Graphics and Image Processing, B. Chanda, S. Chandran, and L. Davis, ed. (Indian Statistical Insitute, 2004), pp. 258–263.

Faugeras, O.

Z. Zhang, R. Deriche, O. Faugeras, and Q.-T. Luong, “A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry,” Research Report 2273, INRIA Sophia-Antipolis, (1994).

Gates, J.

J. Arnold, J. Gates, and K. Taylor, “Possible errors in the measurement of retinal lesions,” Invest. Ophthalmol. Vis. Sci. 34, 2576–2580 (1993).
[PubMed]

Hartley, R.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge Uiversity Press, 2000).

Hartley, R. I.

R. I. Hartley, “Estimation of relative camera positions for uncalibrated cameras,” in Proceedings of the 2nd European Conference on Computer Vision, G. Sandini, ed. (Springer-Verlag, 1992), pp. 579–587.

Hontani, H.

K. Deguchi, D. Kawamata, K. Mizutani, H. Hontani, and K. Wakabayashi, “3d fundus shape reconstruction and display from stereo fundus images,” IEICE Trans. Inf. Syst. E83-D, 1408–1414 (2000).

K. Deguchi, J. Noami, and H. Hontani, “3d fundus pattern reconstruction and display from multiple fundus images,” in Proceedings 15th International Conference on Pattern Recognition (IEEE, 2000), pp. 94–97.
[CrossRef]

Hughes, A.

D. Liu, N. Wood, X. Xu, N. Witt, A. Hughes, and S. Thom, “3D reconstruction of the retinal arterial tree using subject-specific fundus images,” in Advances in Computational Vision and Medical Image Processing, J. M. R. S. Tavares and R. M. N. Jorge ed. (Springer, 2009), pp. 187–201.
[CrossRef]

Hughes, A. D.

M. E. Martinez-Perez, A. D. Hughes, S. A. Thom, A. A. Bharath, and K. H. Parker, “Segmentation of blood vessels from red-free and fluorescein retinal images,” Med. Image Anal. 11, 47–61 (2007).
[CrossRef] [PubMed]

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, N. Chapman, A. A. Bharath, and K. H. Parker, “Retinal vascular tree morphology: A semi-automatic quantification,” IEEE Trans. Biomed. Eng. 49, 912–917 (2002).
[CrossRef]

Kanatani, K.

K. Kanatani, Geometry Computation for Machine Vision (Oxford Science Publications, 1993).

Kawamata, D.

K. Deguchi, D. Kawamata, K. Mizutani, H. Hontani, and K. Wakabayashi, “3d fundus shape reconstruction and display from stereo fundus images,” IEICE Trans. Inf. Syst. E83-D, 1408–1414 (2000).

Leroy, A. M.

P. J. Rousseeuw and A. M. Leroy, Robust Regression and Outilier Detection (John Wiley & Sons, 1987).
[CrossRef]

Liu, D.

D. Liu, N. Wood, X. Xu, N. Witt, A. Hughes, and S. Thom, “3D reconstruction of the retinal arterial tree using subject-specific fundus images,” in Advances in Computational Vision and Medical Image Processing, J. M. R. S. Tavares and R. M. N. Jorge ed. (Springer, 2009), pp. 187–201.
[CrossRef]

Lourakis, M. I. A.

M. I. A. Lourakis and R. Deriche, “Camera self-calibration using the singular value decomposition of the fundamental matrix: from point correspondences to 3d measurements,” Research Report 3748, INRIA Sophia-Antipolis, (1999).

Luong, Q.-T.

Z. Zhang, R. Deriche, O. Faugeras, and Q.-T. Luong, “A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry,” Research Report 2273, INRIA Sophia-Antipolis, (1994).

Martinez-Perez, M. E.

M. E. Martinez-Perez, A. D. Hughes, S. A. Thom, A. A. Bharath, and K. H. Parker, “Segmentation of blood vessels from red-free and fluorescein retinal images,” Med. Image Anal. 11, 47–61 (2007).
[CrossRef] [PubMed]

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, N. Chapman, A. A. Bharath, and K. H. Parker, “Retinal vascular tree morphology: A semi-automatic quantification,” IEEE Trans. Biomed. Eng. 49, 912–917 (2002).
[CrossRef]

M. E. Martinez-Perez and A. Espinosa-Romero, “3D Reconstruction of Retinal Blood Vessels From Two Views,” in Proceedings of the 4th Indian Conference on Computer Vision, Graphics and Image Processing, B. Chanda, S. Chandran, and L. Davis, ed. (Indian Statistical Insitute, 2004), pp. 258–263.

Medioni, G.

T. E. Choe, G. Medioni, I. Cohen, A. C. Walsh, and S. R. Sadda, “2-D registration and 3-D shape inference of the retinal fundus from fluorescein images,” Med. Image Anal. 12, 174–190 (2008).
[CrossRef]

Mizutani, K.

K. Deguchi, D. Kawamata, K. Mizutani, H. Hontani, and K. Wakabayashi, “3d fundus shape reconstruction and display from stereo fundus images,” IEICE Trans. Inf. Syst. E83-D, 1408–1414 (2000).

Noami, J.

K. Deguchi, J. Noami, and H. Hontani, “3d fundus pattern reconstruction and display from multiple fundus images,” in Proceedings 15th International Conference on Pattern Recognition (IEEE, 2000), pp. 94–97.
[CrossRef]

Parker, K. H.

M. E. Martinez-Perez, A. D. Hughes, S. A. Thom, A. A. Bharath, and K. H. Parker, “Segmentation of blood vessels from red-free and fluorescein retinal images,” Med. Image Anal. 11, 47–61 (2007).
[CrossRef] [PubMed]

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, N. Chapman, A. A. Bharath, and K. H. Parker, “Retinal vascular tree morphology: A semi-automatic quantification,” IEEE Trans. Biomed. Eng. 49, 912–917 (2002).
[CrossRef]

Ponce, J.

J. Ponce, “Straight homogeneous generalized cylinders: Differential geometry and uniqueness results,” Int. J. Comput. Vis. 4, 79–100 (1990).
[CrossRef]

Rousseeuw, P. J.

P. J. Rousseeuw and A. M. Leroy, Robust Regression and Outilier Detection (John Wiley & Sons, 1987).
[CrossRef]

Sadda, S. R.

T. E. Choe, G. Medioni, I. Cohen, A. C. Walsh, and S. R. Sadda, “2-D registration and 3-D shape inference of the retinal fundus from fluorescein images,” Med. Image Anal. 12, 174–190 (2008).
[CrossRef]

Sonka, M.

M. Sonka, Image Processing, Analysis, and Machine Vision (Thomson, 2008).

Stanton, A. V.

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, N. Chapman, A. A. Bharath, and K. H. Parker, “Retinal vascular tree morphology: A semi-automatic quantification,” IEEE Trans. Biomed. Eng. 49, 912–917 (2002).
[CrossRef]

Taylor, K.

J. Arnold, J. Gates, and K. Taylor, “Possible errors in the measurement of retinal lesions,” Invest. Ophthalmol. Vis. Sci. 34, 2576–2580 (1993).
[PubMed]

Thom, S.

D. Liu, N. Wood, X. Xu, N. Witt, A. Hughes, and S. Thom, “3D reconstruction of the retinal arterial tree using subject-specific fundus images,” in Advances in Computational Vision and Medical Image Processing, J. M. R. S. Tavares and R. M. N. Jorge ed. (Springer, 2009), pp. 187–201.
[CrossRef]

Thom, S. A.

M. E. Martinez-Perez, A. D. Hughes, S. A. Thom, A. A. Bharath, and K. H. Parker, “Segmentation of blood vessels from red-free and fluorescein retinal images,” Med. Image Anal. 11, 47–61 (2007).
[CrossRef] [PubMed]

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, N. Chapman, A. A. Bharath, and K. H. Parker, “Retinal vascular tree morphology: A semi-automatic quantification,” IEEE Trans. Biomed. Eng. 49, 912–917 (2002).
[CrossRef]

Wakabayashi, K.

K. Deguchi, D. Kawamata, K. Mizutani, H. Hontani, and K. Wakabayashi, “3d fundus shape reconstruction and display from stereo fundus images,” IEICE Trans. Inf. Syst. E83-D, 1408–1414 (2000).

Walsh, A. C.

T. E. Choe, G. Medioni, I. Cohen, A. C. Walsh, and S. R. Sadda, “2-D registration and 3-D shape inference of the retinal fundus from fluorescein images,” Med. Image Anal. 12, 174–190 (2008).
[CrossRef]

Witt, N.

D. Liu, N. Wood, X. Xu, N. Witt, A. Hughes, and S. Thom, “3D reconstruction of the retinal arterial tree using subject-specific fundus images,” in Advances in Computational Vision and Medical Image Processing, J. M. R. S. Tavares and R. M. N. Jorge ed. (Springer, 2009), pp. 187–201.
[CrossRef]

Wood, N.

D. Liu, N. Wood, X. Xu, N. Witt, A. Hughes, and S. Thom, “3D reconstruction of the retinal arterial tree using subject-specific fundus images,” in Advances in Computational Vision and Medical Image Processing, J. M. R. S. Tavares and R. M. N. Jorge ed. (Springer, 2009), pp. 187–201.
[CrossRef]

Xu, X.

D. Liu, N. Wood, X. Xu, N. Witt, A. Hughes, and S. Thom, “3D reconstruction of the retinal arterial tree using subject-specific fundus images,” in Advances in Computational Vision and Medical Image Processing, J. M. R. S. Tavares and R. M. N. Jorge ed. (Springer, 2009), pp. 187–201.
[CrossRef]

Zhang, Z.

Z. Zhang, R. Deriche, O. Faugeras, and Q.-T. Luong, “A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry,” Research Report 2273, INRIA Sophia-Antipolis, (1994).

Zisserman, A.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge Uiversity Press, 2000).

IEEE Trans. Biomed. Eng. (1)

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, N. Chapman, A. A. Bharath, and K. H. Parker, “Retinal vascular tree morphology: A semi-automatic quantification,” IEEE Trans. Biomed. Eng. 49, 912–917 (2002).
[CrossRef]

IEICE Trans. Inf. Syst. (1)

K. Deguchi, D. Kawamata, K. Mizutani, H. Hontani, and K. Wakabayashi, “3d fundus shape reconstruction and display from stereo fundus images,” IEICE Trans. Inf. Syst. E83-D, 1408–1414 (2000).

Int. J. Comput. Vis. (1)

J. Ponce, “Straight homogeneous generalized cylinders: Differential geometry and uniqueness results,” Int. J. Comput. Vis. 4, 79–100 (1990).
[CrossRef]

Invest. Ophthalmol. Vis. Sci. (1)

J. Arnold, J. Gates, and K. Taylor, “Possible errors in the measurement of retinal lesions,” Invest. Ophthalmol. Vis. Sci. 34, 2576–2580 (1993).
[PubMed]

Med. Image Anal. (2)

M. E. Martinez-Perez, A. D. Hughes, S. A. Thom, A. A. Bharath, and K. H. Parker, “Segmentation of blood vessels from red-free and fluorescein retinal images,” Med. Image Anal. 11, 47–61 (2007).
[CrossRef] [PubMed]

T. E. Choe, G. Medioni, I. Cohen, A. C. Walsh, and S. R. Sadda, “2-D registration and 3-D shape inference of the retinal fundus from fluorescein images,” Med. Image Anal. 12, 174–190 (2008).
[CrossRef]

Other (10)

D. Liu, N. Wood, X. Xu, N. Witt, A. Hughes, and S. Thom, “3D reconstruction of the retinal arterial tree using subject-specific fundus images,” in Advances in Computational Vision and Medical Image Processing, J. M. R. S. Tavares and R. M. N. Jorge ed. (Springer, 2009), pp. 187–201.
[CrossRef]

M. E. Martinez-Perez and A. Espinosa-Romero, “3D Reconstruction of Retinal Blood Vessels From Two Views,” in Proceedings of the 4th Indian Conference on Computer Vision, Graphics and Image Processing, B. Chanda, S. Chandran, and L. Davis, ed. (Indian Statistical Insitute, 2004), pp. 258–263.

K. Deguchi, J. Noami, and H. Hontani, “3d fundus pattern reconstruction and display from multiple fundus images,” in Proceedings 15th International Conference on Pattern Recognition (IEEE, 2000), pp. 94–97.
[CrossRef]

M. I. A. Lourakis and R. Deriche, “Camera self-calibration using the singular value decomposition of the fundamental matrix: from point correspondences to 3d measurements,” Research Report 3748, INRIA Sophia-Antipolis, (1999).

M. Sonka, Image Processing, Analysis, and Machine Vision (Thomson, 2008).

K. Kanatani, Geometry Computation for Machine Vision (Oxford Science Publications, 1993).

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge Uiversity Press, 2000).

P. J. Rousseeuw and A. M. Leroy, Robust Regression and Outilier Detection (John Wiley & Sons, 1987).
[CrossRef]

Z. Zhang, R. Deriche, O. Faugeras, and Q.-T. Luong, “A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry,” Research Report 2273, INRIA Sophia-Antipolis, (1994).

R. I. Hartley, “Estimation of relative camera positions for uncalibrated cameras,” in Proceedings of the 2nd European Conference on Computer Vision, G. Sandini, ed. (Springer-Verlag, 1992), pp. 579–587.

Supplementary Material (6)

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Figures (10)

Fig. 1
Fig. 1

Fundus images, (a) and (c) are two different views from the same eyeball and (b) and (d) are their respective segmented binary images.

Fig. 2
Fig. 2

Flowchart of the whole 3D reconstruction process.

Fig. 3
Fig. 3

Examples of the vanishing point estimation process. (a) shows an image of the calibration frame captured by the fundus camera, (b) shows the edges of the calibration frame obtained from the image in (a), and (c) shows the rectangle fitted to the calibration frame in red lines, and the two pairs of lines whose vanishing points are used to estimate the focal length which are shown in blue.

Fig. 4
Fig. 4

Procedure to capture the fundus images.

Fig. 5
Fig. 5

Process to extract a tree from binary image. (a) and (b) show skeleton and significant points marked over the original image, (b) and (e) show the automated tracking of the tree selected by the operator and (c) and (f) show the extraction of the tree segments from the binary image. Upper raw: subject set 1, lower raw: subject set 2.

Fig. 6
Fig. 6

Matching process: (a) feature points on image 1 (n=55), (b) feature points on image 2 (n=59), (c) putative matches after normalised correlation (n=36) and (d) inlier points after LMedS minimisation (n=29). Images taken from set 1.

Fig. 7
Fig. 7

(a) T2 extracted tree from image 1 from set 1, (b) T2 extracted tree from image 2 from set 1 and (c) matched skeleton points between the two images. (d), (e) and (f) correspond to the same example using a pair of images from set 2.

Fig. 8
Fig. 8

(a) and (b) show the epipolar lines drawn on the image pair from subject 1, used to recover the 3D vascular tree, while (c) and (d) show the same from the image pair from subject 2.

Fig. 9
Fig. 9

(a) and (b) different 3D views of skeleton trees reconstruction from set 1. (c) and (d) same example from set 2.

Fig. 10
Fig. 10

Blood vessel surfaces extracted from the pair of images. (a), (b) and (c) correspond to set 1; (d), (e) and (f) to set 2. Diameter measure of each vessel segment is obtained from the corresponded 2D binary images. (c) and (f) show both trees from each set respectively, where the curvature of each eyeball can be seen.

Equations (15)

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

K = [ α u α u cot θ u 0 0 α v / sin θ v 0 0 0 1 ]
a a + b b + f 2 = 0
m = [ m 1 , m 2 , ( f / f ) m 3 ]
f ^ = f m 1 m 1 + m 2 m 2 m 3 m 3
F ω * F T = [ e ] × ω * [ e ] ×
u 2 T ω * u 2 r 2 v 1 T ω * v 1 = u 1 T ω * u 2 r s v 1 T ω * v 2 = u 1 T ω * u 1 s v 2 T ω * v 2
ω * = K K T
Cor = W 1 * W 2 W 2 2
min θ ^ med i r i 2
σ ^ = 1.4826 ( 1 + 5 / ( n p ) ) med i r i 2 ( θ ^ )
k i = { 1 if | r i / σ ^ | 0 otherwise
ω * = argmin ω ˜ * i = 1 N π 12 2 ( S F i , ω ˜ * ) σ π 12 2 ( S F i , ω ˜ * ) + π 13 2 ( S F i , ω ˜ * ) σ π 13 2 ( S F i , ω ˜ * ) + π 23 2 ( S F i , ω ˜ * ) σ π 23 2 ( S F i , ω ˜ * )
x i = P X i x i = P X i for all i
K m 1 = [ 909.68 0 391.28 0 909.68 295.61 0 0 1 ] ; K m 2 = [ 639.79 0 460.75 0 639.39 247.02 0 0 1 ]
K 1 = [ 998.81 1 372.74 0 905.54 317.42 0 0 1 ] ; K 2 = [ 462.89 0.0090 344.16 0 515.60 329.64 0 0 1 ]

Metrics