Abstract

With improvements in fundus imaging technology and the increasing use of digital images in screening and diagnosis, the issue of automated analysis of retinal images is gaining more serious attention. We consider the problem of retinal vessel segmentation, a key issue in automated analysis of digital fundus images. We propose a texture-based vessel segmentation algorithm based on the notion of textons. Using a weak statistical learning approach, we construct textons for retinal vasculature by designing filters that are specifically tuned to the structural and photometric properties of retinal vessels. We evaluate the performance of the proposed approach using a standard database of retinal images. On the DRIVE data set, the proposed method produced an average performance of 0.9568 specificity at 0.7346 sensitivity. This compares well with the best-published results on the data set 0.9773 specificity at 0.7194 sensitivity [Proc. SPIE 5370, 648 (2004) ].

© 2007 Optical Society of America

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  1. J. J. Wang, B. Taylor, T. Y. Wong, B. Chua, E. Rochtchina, R. Klein, and P. Mitchell, "Retinal vessel diameters and obesity: a population-based study in older persons," Obes. Res. 14, 206-214 (2006).
    [CrossRef]
  2. P. Mitchell, H. Leung, J. J. Wang, E. Rochtchina, A. J. Lee, T. Y. Wong, and R. Klein, "Retinal vessel diameter and open-angle glaucoma: the Blue Mountains eye study," Ophthalmology 112, 245-250 (2005).
    [CrossRef] [PubMed]
  3. H. Leung, J. J. Wang, E. Rochtchina, T. Y. Wong, R. Klein, and P. Mitchell, "Impact of current and past blood pressure on retinal arteriolar diameter in older population," J. Hypertens. 22, 1543-1549 (2003).
    [CrossRef]
  4. J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
    [CrossRef] [PubMed]
  5. R. Klein, B. E. K. Klein, M. D. Knudtson, T. Y. Wong, and M. Y. Tsai, "Are inflammatory factors related to retinal vessel caliber? the Beaver Dam eye study," Arch. Ophthalmol. 124, 87-94 (2006).
    [CrossRef] [PubMed]
  6. P. F. Sharpl, J. Olson, F. Strachan, J. Hipwell, A. Ludbrook, M. O'Donnell, S. Wallace, K. Goatman, A. Grant, N. Waugh, K. McHardy, and J. V. Forrester, "The value of digital imaging in diabetic retinopathy," Health Technol. Assess 7, 1-132 (2003), http://www.hta.nhsweb.nhs.uk/fullmono/mon730.pdf.
  7. A. Hunter, J. Lowell, D. Steel, A. Basu, and R. Ryder, Non-linear Filtering for Vascular Segmentation and Detection of Venous Beading, Technical Report (Department of Computer Science, University of Durham, 2002).
  8. J. Staal, M. Abramoff, M. Niemeijer, M. Viergever, and B. van Ginneken, "Ridge-based vessel segmentation in color images of the retina," IEEE Trans. Med. Imaging 23, 501-509 (2004).
    [CrossRef] [PubMed]
  9. T. Leung and J. Malik, "Representing and recognizing the visual appearance of materials using three dimensional textons," Int. J. Comput. Vis. 43, 29-44 (2001).
    [CrossRef]
  10. O. G. Cula and K. J. Dana, "3D texture recognition using bidirectional feature histograms," Int. J. Comput. Vis. 59, 33-60 (2004).
    [CrossRef]
  11. C. Schmid, "Weakly supervised learning of visual models and its application to content-based retrieval," Int. J. Comput. Vis. 56, 7-16 (2004).
    [CrossRef]
  12. M. Varma and A. Zisserman, "A statistical approach to texture classification from single images," Int. J. Comput. Vis. 62, 61-81 (2005).
  13. D. Paton, B. N. Hyman, and J. Justice, Introduction to Ophthalmoscopy (The Upjohn Co., 1976).
  14. I. C. Michealson, Textbook of the Fundus of the Eye (Livingstone, 1980).
  15. M. Niemeijer, J. J. Staal, B. van Ginneken, M. Loog, and M. D. Abramoff, "Comparative study of retinal vessel segmentation methods on a new publicly available database," in SPIE Medical Imaging, J.Michael Fitzpatrick and M.Sonka, eds., Proc. SPIE 5370, 648-656 (2004).
  16. C. Sinthanayothin, J. F. Boyce, T. H. Williamson, and H. L. Cook, "Automated Detection of diabetic retionopathy on digital fundus images," Diabetic Med. 19, 105-112 (2002).
    [CrossRef]
  17. N. Ritter, R. Owens, J. Cooper, R. Eikelboom, and P. van Saarloos, "Registration of stereo and temporal images of the retina," IEEE Trans. Med. Imaging 18, 404-418 (1999).
    [CrossRef]
  18. A. Can, C. Stewart, B. Roysam, and H. Tanenbaum, "A feature-based, robust, hierarchical algorithm for registering pairs of images of the curved human retina," IEEE Trans. Pattern Anal. Mach. Intell. 24, 347-364 (2002).
    [CrossRef]
  19. C. Kirbas and F. Quek, "A review of vessel extraction techniques and algorithms," ACM Comput. Surv. 36, 81-121 (2004).
    [CrossRef]
  20. L. Pedersen, B. Ersbøll, K. Madsen, M. Grunkin, U. Skands, M. Larsen, and N. Christoffersen, "Quantitative measurement of changes in retinal vessel diameter in ocular fundus images," Pattern Recogn. Lett. 21, 1215-1223 (2000).
    [CrossRef]
  21. T. Y. Wong, M. D. Knudtson, R. Klein, B. E. Klein, S. M. Meuer, and L. D. Hubbard, "Computer-assisted measurement of retinal vessel diameter in the Beaver Dam eye study," Ophthalmology 111, 1183-1190 (2004).
    [CrossRef] [PubMed]
  22. W. Hart, M. Goldbaum, B. Cote, P. Kube, and M. Nelson, "Automated measurement of retinal vascular tortuosity," in Proceedings of the AMIA Annual Symposium (AMIA, 1997), pp. 459-463.
  23. F. Bin and T. Y. Yan, "Elastic registration for retinal images based on reconstructed vascular trees," IEEE Trans. Biomed. Eng. 53, 1183-1187 (2006).
    [CrossRef]
  24. L. Gang, O. Chutatape, and S. M. Krishnan, "Detection and measurement of retinal vessels in fundus images using amplitude modified second-order Gaussian filter," IEEE Trans. Biomed. Eng. 49, 168-172 (2002).
    [CrossRef] [PubMed]
  25. A. Hoover, V. Kouznetsova, and M. Goldbaum, "Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response," IEEE Trans. Med. Imaging 19, 203-210 (2000).
    [CrossRef] [PubMed]
  26. X. Jiang and D. Mojon, "Adaptive local thresholding by verification-based multithreshold probing with application to vessel detection in retinal image," IEEE Trans. Pattern Anal. Mach. Intell. 25, 131-137 (2003).
    [CrossRef]
  27. J. Lowell, A. Hunter, D. Steel, A. Basu, R. Ryder, and E. Fletcher, "Optic nerve head segmentation," IEEE Trans. Med. Imaging 23, 256-264 (2004).
    [CrossRef] [PubMed]
  28. S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, and M. Goldbaum, "Detection of blood vessels in retinal images using two-dimensional matched filters," IEEE Trans. Med. Imaging 8, 263-269 (1989).
    [CrossRef] [PubMed]
  29. T. Walter, J. Klein, and F. Zana, "Automatic segmentation and registration of retinal fluorescin angiographies--applications to diabetic retinopathy," IEEE Trans. Inf. Technol. Biomed. 79, 463-491 (1999).
  30. F. Zana and J.-C. Klein, "Segmentation of vessel-like patterns using mathematical morphology and curvature evaluation," IEEE Trans. Image Process. 10, 1010-1019 (2001).
    [CrossRef]
  31. A. Can, H. Shen, J. N. Turner, H. L. Tanenbaum, and B. Roysam, "Rapid automated tracing and feature extraction from live high-resolution retinal fundus images using direct exploratory algorithms," IEEE Trans. Inf. Technol. Biomed. 3, 125-138 (1999).
    [CrossRef]
  32. Y. Tolias and S. Panas, "A fuzzy vessel tracking algorithm for retinal images based on fuzzy clustering," IEEE Trans. Med. Imaging 17, 263-273 (1998).
    [CrossRef] [PubMed]
  33. J. Capowski, J. Kylstra, and S. Freedman, "A numeric index based on spatial-frequency for the tortuosity of retinal vessels and its application to plus disease in retinopathy of prematurity," Retina 15, 490-500 (1995).
    [PubMed]
  34. O. Brinchmann-Hansen and H. Heier, "Theoretical relationships between light streak characteristics and optical properties of retinal vessels," Acta Ophthalmol. Suppl. 179, 33-37 (1986).
  35. X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thom, "A method of vessel tracking for vessel diameter measurement on retinal images," in Proceedings of IEEE International Conference on Image Processing (IEEE, 2001), pp. 881-884.
  36. J. Kittler, M. Hatef, R. P. W. Duin, and J. Matas, "On combining classifiers," IEEE Trans. Pattern Anal. Mach. Intell. 20, 226-239 (1998).
    [CrossRef]

2006

J. J. Wang, B. Taylor, T. Y. Wong, B. Chua, E. Rochtchina, R. Klein, and P. Mitchell, "Retinal vessel diameters and obesity: a population-based study in older persons," Obes. Res. 14, 206-214 (2006).
[CrossRef]

R. Klein, B. E. K. Klein, M. D. Knudtson, T. Y. Wong, and M. Y. Tsai, "Are inflammatory factors related to retinal vessel caliber? the Beaver Dam eye study," Arch. Ophthalmol. 124, 87-94 (2006).
[CrossRef] [PubMed]

F. Bin and T. Y. Yan, "Elastic registration for retinal images based on reconstructed vascular trees," IEEE Trans. Biomed. Eng. 53, 1183-1187 (2006).
[CrossRef]

2005

P. Mitchell, H. Leung, J. J. Wang, E. Rochtchina, A. J. Lee, T. Y. Wong, and R. Klein, "Retinal vessel diameter and open-angle glaucoma: the Blue Mountains eye study," Ophthalmology 112, 245-250 (2005).
[CrossRef] [PubMed]

M. Varma and A. Zisserman, "A statistical approach to texture classification from single images," Int. J. Comput. Vis. 62, 61-81 (2005).

2004

O. G. Cula and K. J. Dana, "3D texture recognition using bidirectional feature histograms," Int. J. Comput. Vis. 59, 33-60 (2004).
[CrossRef]

C. Schmid, "Weakly supervised learning of visual models and its application to content-based retrieval," Int. J. Comput. Vis. 56, 7-16 (2004).
[CrossRef]

C. Kirbas and F. Quek, "A review of vessel extraction techniques and algorithms," ACM Comput. Surv. 36, 81-121 (2004).
[CrossRef]

T. Y. Wong, M. D. Knudtson, R. Klein, B. E. Klein, S. M. Meuer, and L. D. Hubbard, "Computer-assisted measurement of retinal vessel diameter in the Beaver Dam eye study," Ophthalmology 111, 1183-1190 (2004).
[CrossRef] [PubMed]

J. Staal, M. Abramoff, M. Niemeijer, M. Viergever, and B. van Ginneken, "Ridge-based vessel segmentation in color images of the retina," IEEE Trans. Med. Imaging 23, 501-509 (2004).
[CrossRef] [PubMed]

J. Lowell, A. Hunter, D. Steel, A. Basu, R. Ryder, and E. Fletcher, "Optic nerve head segmentation," IEEE Trans. Med. Imaging 23, 256-264 (2004).
[CrossRef] [PubMed]

2003

X. Jiang and D. Mojon, "Adaptive local thresholding by verification-based multithreshold probing with application to vessel detection in retinal image," IEEE Trans. Pattern Anal. Mach. Intell. 25, 131-137 (2003).
[CrossRef]

P. F. Sharpl, J. Olson, F. Strachan, J. Hipwell, A. Ludbrook, M. O'Donnell, S. Wallace, K. Goatman, A. Grant, N. Waugh, K. McHardy, and J. V. Forrester, "The value of digital imaging in diabetic retinopathy," Health Technol. Assess 7, 1-132 (2003), http://www.hta.nhsweb.nhs.uk/fullmono/mon730.pdf.

H. Leung, J. J. Wang, E. Rochtchina, T. Y. Wong, R. Klein, and P. Mitchell, "Impact of current and past blood pressure on retinal arteriolar diameter in older population," J. Hypertens. 22, 1543-1549 (2003).
[CrossRef]

J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
[CrossRef] [PubMed]

2002

A. Can, C. Stewart, B. Roysam, and H. Tanenbaum, "A feature-based, robust, hierarchical algorithm for registering pairs of images of the curved human retina," IEEE Trans. Pattern Anal. Mach. Intell. 24, 347-364 (2002).
[CrossRef]

C. Sinthanayothin, J. F. Boyce, T. H. Williamson, and H. L. Cook, "Automated Detection of diabetic retionopathy on digital fundus images," Diabetic Med. 19, 105-112 (2002).
[CrossRef]

L. Gang, O. Chutatape, and S. M. Krishnan, "Detection and measurement of retinal vessels in fundus images using amplitude modified second-order Gaussian filter," IEEE Trans. Biomed. Eng. 49, 168-172 (2002).
[CrossRef] [PubMed]

2001

F. Zana and J.-C. Klein, "Segmentation of vessel-like patterns using mathematical morphology and curvature evaluation," IEEE Trans. Image Process. 10, 1010-1019 (2001).
[CrossRef]

T. Leung and J. Malik, "Representing and recognizing the visual appearance of materials using three dimensional textons," Int. J. Comput. Vis. 43, 29-44 (2001).
[CrossRef]

2000

L. Pedersen, B. Ersbøll, K. Madsen, M. Grunkin, U. Skands, M. Larsen, and N. Christoffersen, "Quantitative measurement of changes in retinal vessel diameter in ocular fundus images," Pattern Recogn. Lett. 21, 1215-1223 (2000).
[CrossRef]

A. Hoover, V. Kouznetsova, and M. Goldbaum, "Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response," IEEE Trans. Med. Imaging 19, 203-210 (2000).
[CrossRef] [PubMed]

1999

A. Can, H. Shen, J. N. Turner, H. L. Tanenbaum, and B. Roysam, "Rapid automated tracing and feature extraction from live high-resolution retinal fundus images using direct exploratory algorithms," IEEE Trans. Inf. Technol. Biomed. 3, 125-138 (1999).
[CrossRef]

T. Walter, J. Klein, and F. Zana, "Automatic segmentation and registration of retinal fluorescin angiographies--applications to diabetic retinopathy," IEEE Trans. Inf. Technol. Biomed. 79, 463-491 (1999).

N. Ritter, R. Owens, J. Cooper, R. Eikelboom, and P. van Saarloos, "Registration of stereo and temporal images of the retina," IEEE Trans. Med. Imaging 18, 404-418 (1999).
[CrossRef]

1998

J. Kittler, M. Hatef, R. P. W. Duin, and J. Matas, "On combining classifiers," IEEE Trans. Pattern Anal. Mach. Intell. 20, 226-239 (1998).
[CrossRef]

Y. Tolias and S. Panas, "A fuzzy vessel tracking algorithm for retinal images based on fuzzy clustering," IEEE Trans. Med. Imaging 17, 263-273 (1998).
[CrossRef] [PubMed]

1995

J. Capowski, J. Kylstra, and S. Freedman, "A numeric index based on spatial-frequency for the tortuosity of retinal vessels and its application to plus disease in retinopathy of prematurity," Retina 15, 490-500 (1995).
[PubMed]

1989

S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, and M. Goldbaum, "Detection of blood vessels in retinal images using two-dimensional matched filters," IEEE Trans. Med. Imaging 8, 263-269 (1989).
[CrossRef] [PubMed]

1986

O. Brinchmann-Hansen and H. Heier, "Theoretical relationships between light streak characteristics and optical properties of retinal vessels," Acta Ophthalmol. Suppl. 179, 33-37 (1986).

Abramoff, M.

J. Staal, M. Abramoff, M. Niemeijer, M. Viergever, and B. van Ginneken, "Ridge-based vessel segmentation in color images of the retina," IEEE Trans. Med. Imaging 23, 501-509 (2004).
[CrossRef] [PubMed]

Abramoff, M. D.

M. Niemeijer, J. J. Staal, B. van Ginneken, M. Loog, and M. D. Abramoff, "Comparative study of retinal vessel segmentation methods on a new publicly available database," in SPIE Medical Imaging, J.Michael Fitzpatrick and M.Sonka, eds., Proc. SPIE 5370, 648-656 (2004).

Basu, A.

J. Lowell, A. Hunter, D. Steel, A. Basu, R. Ryder, and E. Fletcher, "Optic nerve head segmentation," IEEE Trans. Med. Imaging 23, 256-264 (2004).
[CrossRef] [PubMed]

A. Hunter, J. Lowell, D. Steel, A. Basu, and R. Ryder, Non-linear Filtering for Vascular Segmentation and Detection of Venous Beading, Technical Report (Department of Computer Science, University of Durham, 2002).

Betsholtz, C.

J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
[CrossRef] [PubMed]

Bharath, A.

X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thom, "A method of vessel tracking for vessel diameter measurement on retinal images," in Proceedings of IEEE International Conference on Image Processing (IEEE, 2001), pp. 881-884.

Bin, F.

F. Bin and T. Y. Yan, "Elastic registration for retinal images based on reconstructed vascular trees," IEEE Trans. Biomed. Eng. 53, 1183-1187 (2006).
[CrossRef]

Boyce, J. F.

C. Sinthanayothin, J. F. Boyce, T. H. Williamson, and H. L. Cook, "Automated Detection of diabetic retionopathy on digital fundus images," Diabetic Med. 19, 105-112 (2002).
[CrossRef]

Brinchmann-Hansen, O.

O. Brinchmann-Hansen and H. Heier, "Theoretical relationships between light streak characteristics and optical properties of retinal vessels," Acta Ophthalmol. Suppl. 179, 33-37 (1986).

Can, A.

A. Can, C. Stewart, B. Roysam, and H. Tanenbaum, "A feature-based, robust, hierarchical algorithm for registering pairs of images of the curved human retina," IEEE Trans. Pattern Anal. Mach. Intell. 24, 347-364 (2002).
[CrossRef]

A. Can, H. Shen, J. N. Turner, H. L. Tanenbaum, and B. Roysam, "Rapid automated tracing and feature extraction from live high-resolution retinal fundus images using direct exploratory algorithms," IEEE Trans. Inf. Technol. Biomed. 3, 125-138 (1999).
[CrossRef]

Capowski, J.

J. Capowski, J. Kylstra, and S. Freedman, "A numeric index based on spatial-frequency for the tortuosity of retinal vessels and its application to plus disease in retinopathy of prematurity," Retina 15, 490-500 (1995).
[PubMed]

Chapman, N.

X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thom, "A method of vessel tracking for vessel diameter measurement on retinal images," in Proceedings of IEEE International Conference on Image Processing (IEEE, 2001), pp. 881-884.

Chatterjee, S.

S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, and M. Goldbaum, "Detection of blood vessels in retinal images using two-dimensional matched filters," IEEE Trans. Med. Imaging 8, 263-269 (1989).
[CrossRef] [PubMed]

Chaudhuri, S.

S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, and M. Goldbaum, "Detection of blood vessels in retinal images using two-dimensional matched filters," IEEE Trans. Med. Imaging 8, 263-269 (1989).
[CrossRef] [PubMed]

Christoffersen, N.

L. Pedersen, B. Ersbøll, K. Madsen, M. Grunkin, U. Skands, M. Larsen, and N. Christoffersen, "Quantitative measurement of changes in retinal vessel diameter in ocular fundus images," Pattern Recogn. Lett. 21, 1215-1223 (2000).
[CrossRef]

Chua, B.

J. J. Wang, B. Taylor, T. Y. Wong, B. Chua, E. Rochtchina, R. Klein, and P. Mitchell, "Retinal vessel diameters and obesity: a population-based study in older persons," Obes. Res. 14, 206-214 (2006).
[CrossRef]

Chutatape, O.

L. Gang, O. Chutatape, and S. M. Krishnan, "Detection and measurement of retinal vessels in fundus images using amplitude modified second-order Gaussian filter," IEEE Trans. Biomed. Eng. 49, 168-172 (2002).
[CrossRef] [PubMed]

Cook, H. L.

C. Sinthanayothin, J. F. Boyce, T. H. Williamson, and H. L. Cook, "Automated Detection of diabetic retionopathy on digital fundus images," Diabetic Med. 19, 105-112 (2002).
[CrossRef]

Cooper, J.

N. Ritter, R. Owens, J. Cooper, R. Eikelboom, and P. van Saarloos, "Registration of stereo and temporal images of the retina," IEEE Trans. Med. Imaging 18, 404-418 (1999).
[CrossRef]

Cote, B.

W. Hart, M. Goldbaum, B. Cote, P. Kube, and M. Nelson, "Automated measurement of retinal vascular tortuosity," in Proceedings of the AMIA Annual Symposium (AMIA, 1997), pp. 459-463.

Cula, O. G.

O. G. Cula and K. J. Dana, "3D texture recognition using bidirectional feature histograms," Int. J. Comput. Vis. 59, 33-60 (2004).
[CrossRef]

Damert, A.

J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
[CrossRef] [PubMed]

Dana, K. J.

O. G. Cula and K. J. Dana, "3D texture recognition using bidirectional feature histograms," Int. J. Comput. Vis. 59, 33-60 (2004).
[CrossRef]

Duin, R. P. W.

J. Kittler, M. Hatef, R. P. W. Duin, and J. Matas, "On combining classifiers," IEEE Trans. Pattern Anal. Mach. Intell. 20, 226-239 (1998).
[CrossRef]

Eikelboom, R.

N. Ritter, R. Owens, J. Cooper, R. Eikelboom, and P. van Saarloos, "Registration of stereo and temporal images of the retina," IEEE Trans. Med. Imaging 18, 404-418 (1999).
[CrossRef]

Ersbøll, B.

L. Pedersen, B. Ersbøll, K. Madsen, M. Grunkin, U. Skands, M. Larsen, and N. Christoffersen, "Quantitative measurement of changes in retinal vessel diameter in ocular fundus images," Pattern Recogn. Lett. 21, 1215-1223 (2000).
[CrossRef]

Ferrara, N.

J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
[CrossRef] [PubMed]

Fletcher, E.

J. Lowell, A. Hunter, D. Steel, A. Basu, R. Ryder, and E. Fletcher, "Optic nerve head segmentation," IEEE Trans. Med. Imaging 23, 256-264 (2004).
[CrossRef] [PubMed]

Forrester, J. V.

P. F. Sharpl, J. Olson, F. Strachan, J. Hipwell, A. Ludbrook, M. O'Donnell, S. Wallace, K. Goatman, A. Grant, N. Waugh, K. McHardy, and J. V. Forrester, "The value of digital imaging in diabetic retinopathy," Health Technol. Assess 7, 1-132 (2003), http://www.hta.nhsweb.nhs.uk/fullmono/mon730.pdf.

Freedman, S.

J. Capowski, J. Kylstra, and S. Freedman, "A numeric index based on spatial-frequency for the tortuosity of retinal vessels and its application to plus disease in retinopathy of prematurity," Retina 15, 490-500 (1995).
[PubMed]

Gang, L.

L. Gang, O. Chutatape, and S. M. Krishnan, "Detection and measurement of retinal vessels in fundus images using amplitude modified second-order Gaussian filter," IEEE Trans. Biomed. Eng. 49, 168-172 (2002).
[CrossRef] [PubMed]

Gao, X. W.

X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thom, "A method of vessel tracking for vessel diameter measurement on retinal images," in Proceedings of IEEE International Conference on Image Processing (IEEE, 2001), pp. 881-884.

Gerhardt, H.

J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
[CrossRef] [PubMed]

Goatman, K.

P. F. Sharpl, J. Olson, F. Strachan, J. Hipwell, A. Ludbrook, M. O'Donnell, S. Wallace, K. Goatman, A. Grant, N. Waugh, K. McHardy, and J. V. Forrester, "The value of digital imaging in diabetic retinopathy," Health Technol. Assess 7, 1-132 (2003), http://www.hta.nhsweb.nhs.uk/fullmono/mon730.pdf.

Goldbaum, M.

A. Hoover, V. Kouznetsova, and M. Goldbaum, "Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response," IEEE Trans. Med. Imaging 19, 203-210 (2000).
[CrossRef] [PubMed]

S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, and M. Goldbaum, "Detection of blood vessels in retinal images using two-dimensional matched filters," IEEE Trans. Med. Imaging 8, 263-269 (1989).
[CrossRef] [PubMed]

W. Hart, M. Goldbaum, B. Cote, P. Kube, and M. Nelson, "Automated measurement of retinal vascular tortuosity," in Proceedings of the AMIA Annual Symposium (AMIA, 1997), pp. 459-463.

Grant, A.

P. F. Sharpl, J. Olson, F. Strachan, J. Hipwell, A. Ludbrook, M. O'Donnell, S. Wallace, K. Goatman, A. Grant, N. Waugh, K. McHardy, and J. V. Forrester, "The value of digital imaging in diabetic retinopathy," Health Technol. Assess 7, 1-132 (2003), http://www.hta.nhsweb.nhs.uk/fullmono/mon730.pdf.

Grunkin, M.

L. Pedersen, B. Ersbøll, K. Madsen, M. Grunkin, U. Skands, M. Larsen, and N. Christoffersen, "Quantitative measurement of changes in retinal vessel diameter in ocular fundus images," Pattern Recogn. Lett. 21, 1215-1223 (2000).
[CrossRef]

Haigh, J. J.

J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
[CrossRef] [PubMed]

Haigh, K.

J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
[CrossRef] [PubMed]

Hart, W.

W. Hart, M. Goldbaum, B. Cote, P. Kube, and M. Nelson, "Automated measurement of retinal vascular tortuosity," in Proceedings of the AMIA Annual Symposium (AMIA, 1997), pp. 459-463.

Hatef, M.

J. Kittler, M. Hatef, R. P. W. Duin, and J. Matas, "On combining classifiers," IEEE Trans. Pattern Anal. Mach. Intell. 20, 226-239 (1998).
[CrossRef]

Heier, H.

O. Brinchmann-Hansen and H. Heier, "Theoretical relationships between light streak characteristics and optical properties of retinal vessels," Acta Ophthalmol. Suppl. 179, 33-37 (1986).

Hipwell, J.

P. F. Sharpl, J. Olson, F. Strachan, J. Hipwell, A. Ludbrook, M. O'Donnell, S. Wallace, K. Goatman, A. Grant, N. Waugh, K. McHardy, and J. V. Forrester, "The value of digital imaging in diabetic retinopathy," Health Technol. Assess 7, 1-132 (2003), http://www.hta.nhsweb.nhs.uk/fullmono/mon730.pdf.

Hoover, A.

A. Hoover, V. Kouznetsova, and M. Goldbaum, "Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response," IEEE Trans. Med. Imaging 19, 203-210 (2000).
[CrossRef] [PubMed]

Hubbard, L. D.

T. Y. Wong, M. D. Knudtson, R. Klein, B. E. Klein, S. M. Meuer, and L. D. Hubbard, "Computer-assisted measurement of retinal vessel diameter in the Beaver Dam eye study," Ophthalmology 111, 1183-1190 (2004).
[CrossRef] [PubMed]

Hughes, A.

X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thom, "A method of vessel tracking for vessel diameter measurement on retinal images," in Proceedings of IEEE International Conference on Image Processing (IEEE, 2001), pp. 881-884.

Hunter, A.

J. Lowell, A. Hunter, D. Steel, A. Basu, R. Ryder, and E. Fletcher, "Optic nerve head segmentation," IEEE Trans. Med. Imaging 23, 256-264 (2004).
[CrossRef] [PubMed]

A. Hunter, J. Lowell, D. Steel, A. Basu, and R. Ryder, Non-linear Filtering for Vascular Segmentation and Detection of Venous Beading, Technical Report (Department of Computer Science, University of Durham, 2002).

Hyman, B. N.

D. Paton, B. N. Hyman, and J. Justice, Introduction to Ophthalmoscopy (The Upjohn Co., 1976).

Jiang, X.

X. Jiang and D. Mojon, "Adaptive local thresholding by verification-based multithreshold probing with application to vessel detection in retinal image," IEEE Trans. Pattern Anal. Mach. Intell. 25, 131-137 (2003).
[CrossRef]

Justice, J.

D. Paton, B. N. Hyman, and J. Justice, Introduction to Ophthalmoscopy (The Upjohn Co., 1976).

Katz, N.

S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, and M. Goldbaum, "Detection of blood vessels in retinal images using two-dimensional matched filters," IEEE Trans. Med. Imaging 8, 263-269 (1989).
[CrossRef] [PubMed]

Kirbas, C.

C. Kirbas and F. Quek, "A review of vessel extraction techniques and algorithms," ACM Comput. Surv. 36, 81-121 (2004).
[CrossRef]

Kittler, J.

J. Kittler, M. Hatef, R. P. W. Duin, and J. Matas, "On combining classifiers," IEEE Trans. Pattern Anal. Mach. Intell. 20, 226-239 (1998).
[CrossRef]

Klein, B. E.

T. Y. Wong, M. D. Knudtson, R. Klein, B. E. Klein, S. M. Meuer, and L. D. Hubbard, "Computer-assisted measurement of retinal vessel diameter in the Beaver Dam eye study," Ophthalmology 111, 1183-1190 (2004).
[CrossRef] [PubMed]

Klein, B. E. K.

R. Klein, B. E. K. Klein, M. D. Knudtson, T. Y. Wong, and M. Y. Tsai, "Are inflammatory factors related to retinal vessel caliber? the Beaver Dam eye study," Arch. Ophthalmol. 124, 87-94 (2006).
[CrossRef] [PubMed]

Klein, J.

T. Walter, J. Klein, and F. Zana, "Automatic segmentation and registration of retinal fluorescin angiographies--applications to diabetic retinopathy," IEEE Trans. Inf. Technol. Biomed. 79, 463-491 (1999).

Klein, J.-C.

F. Zana and J.-C. Klein, "Segmentation of vessel-like patterns using mathematical morphology and curvature evaluation," IEEE Trans. Image Process. 10, 1010-1019 (2001).
[CrossRef]

Klein, R.

R. Klein, B. E. K. Klein, M. D. Knudtson, T. Y. Wong, and M. Y. Tsai, "Are inflammatory factors related to retinal vessel caliber? the Beaver Dam eye study," Arch. Ophthalmol. 124, 87-94 (2006).
[CrossRef] [PubMed]

J. J. Wang, B. Taylor, T. Y. Wong, B. Chua, E. Rochtchina, R. Klein, and P. Mitchell, "Retinal vessel diameters and obesity: a population-based study in older persons," Obes. Res. 14, 206-214 (2006).
[CrossRef]

P. Mitchell, H. Leung, J. J. Wang, E. Rochtchina, A. J. Lee, T. Y. Wong, and R. Klein, "Retinal vessel diameter and open-angle glaucoma: the Blue Mountains eye study," Ophthalmology 112, 245-250 (2005).
[CrossRef] [PubMed]

T. Y. Wong, M. D. Knudtson, R. Klein, B. E. Klein, S. M. Meuer, and L. D. Hubbard, "Computer-assisted measurement of retinal vessel diameter in the Beaver Dam eye study," Ophthalmology 111, 1183-1190 (2004).
[CrossRef] [PubMed]

J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
[CrossRef] [PubMed]

H. Leung, J. J. Wang, E. Rochtchina, T. Y. Wong, R. Klein, and P. Mitchell, "Impact of current and past blood pressure on retinal arteriolar diameter in older population," J. Hypertens. 22, 1543-1549 (2003).
[CrossRef]

Knudtson, M. D.

R. Klein, B. E. K. Klein, M. D. Knudtson, T. Y. Wong, and M. Y. Tsai, "Are inflammatory factors related to retinal vessel caliber? the Beaver Dam eye study," Arch. Ophthalmol. 124, 87-94 (2006).
[CrossRef] [PubMed]

T. Y. Wong, M. D. Knudtson, R. Klein, B. E. Klein, S. M. Meuer, and L. D. Hubbard, "Computer-assisted measurement of retinal vessel diameter in the Beaver Dam eye study," Ophthalmology 111, 1183-1190 (2004).
[CrossRef] [PubMed]

Kouznetsova, V.

A. Hoover, V. Kouznetsova, and M. Goldbaum, "Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response," IEEE Trans. Med. Imaging 19, 203-210 (2000).
[CrossRef] [PubMed]

Krishnan, S. M.

L. Gang, O. Chutatape, and S. M. Krishnan, "Detection and measurement of retinal vessels in fundus images using amplitude modified second-order Gaussian filter," IEEE Trans. Biomed. Eng. 49, 168-172 (2002).
[CrossRef] [PubMed]

Kube, P.

W. Hart, M. Goldbaum, B. Cote, P. Kube, and M. Nelson, "Automated measurement of retinal vascular tortuosity," in Proceedings of the AMIA Annual Symposium (AMIA, 1997), pp. 459-463.

Kylstra, J.

J. Capowski, J. Kylstra, and S. Freedman, "A numeric index based on spatial-frequency for the tortuosity of retinal vessels and its application to plus disease in retinopathy of prematurity," Retina 15, 490-500 (1995).
[PubMed]

Larsen, M.

L. Pedersen, B. Ersbøll, K. Madsen, M. Grunkin, U. Skands, M. Larsen, and N. Christoffersen, "Quantitative measurement of changes in retinal vessel diameter in ocular fundus images," Pattern Recogn. Lett. 21, 1215-1223 (2000).
[CrossRef]

Lee, A. J.

P. Mitchell, H. Leung, J. J. Wang, E. Rochtchina, A. J. Lee, T. Y. Wong, and R. Klein, "Retinal vessel diameter and open-angle glaucoma: the Blue Mountains eye study," Ophthalmology 112, 245-250 (2005).
[CrossRef] [PubMed]

Leung, H.

P. Mitchell, H. Leung, J. J. Wang, E. Rochtchina, A. J. Lee, T. Y. Wong, and R. Klein, "Retinal vessel diameter and open-angle glaucoma: the Blue Mountains eye study," Ophthalmology 112, 245-250 (2005).
[CrossRef] [PubMed]

H. Leung, J. J. Wang, E. Rochtchina, T. Y. Wong, R. Klein, and P. Mitchell, "Impact of current and past blood pressure on retinal arteriolar diameter in older population," J. Hypertens. 22, 1543-1549 (2003).
[CrossRef]

Leung, T.

T. Leung and J. Malik, "Representing and recognizing the visual appearance of materials using three dimensional textons," Int. J. Comput. Vis. 43, 29-44 (2001).
[CrossRef]

Loog, M.

M. Niemeijer, J. J. Staal, B. van Ginneken, M. Loog, and M. D. Abramoff, "Comparative study of retinal vessel segmentation methods on a new publicly available database," in SPIE Medical Imaging, J.Michael Fitzpatrick and M.Sonka, eds., Proc. SPIE 5370, 648-656 (2004).

Lowell, J.

J. Lowell, A. Hunter, D. Steel, A. Basu, R. Ryder, and E. Fletcher, "Optic nerve head segmentation," IEEE Trans. Med. Imaging 23, 256-264 (2004).
[CrossRef] [PubMed]

A. Hunter, J. Lowell, D. Steel, A. Basu, and R. Ryder, Non-linear Filtering for Vascular Segmentation and Detection of Venous Beading, Technical Report (Department of Computer Science, University of Durham, 2002).

Ludbrook, A.

P. F. Sharpl, J. Olson, F. Strachan, J. Hipwell, A. Ludbrook, M. O'Donnell, S. Wallace, K. Goatman, A. Grant, N. Waugh, K. McHardy, and J. V. Forrester, "The value of digital imaging in diabetic retinopathy," Health Technol. Assess 7, 1-132 (2003), http://www.hta.nhsweb.nhs.uk/fullmono/mon730.pdf.

Madsen, K.

L. Pedersen, B. Ersbøll, K. Madsen, M. Grunkin, U. Skands, M. Larsen, and N. Christoffersen, "Quantitative measurement of changes in retinal vessel diameter in ocular fundus images," Pattern Recogn. Lett. 21, 1215-1223 (2000).
[CrossRef]

Malik, J.

T. Leung and J. Malik, "Representing and recognizing the visual appearance of materials using three dimensional textons," Int. J. Comput. Vis. 43, 29-44 (2001).
[CrossRef]

Matas, J.

J. Kittler, M. Hatef, R. P. W. Duin, and J. Matas, "On combining classifiers," IEEE Trans. Pattern Anal. Mach. Intell. 20, 226-239 (1998).
[CrossRef]

McHardy, K.

P. F. Sharpl, J. Olson, F. Strachan, J. Hipwell, A. Ludbrook, M. O'Donnell, S. Wallace, K. Goatman, A. Grant, N. Waugh, K. McHardy, and J. V. Forrester, "The value of digital imaging in diabetic retinopathy," Health Technol. Assess 7, 1-132 (2003), http://www.hta.nhsweb.nhs.uk/fullmono/mon730.pdf.

Meuer, S. M.

T. Y. Wong, M. D. Knudtson, R. Klein, B. E. Klein, S. M. Meuer, and L. D. Hubbard, "Computer-assisted measurement of retinal vessel diameter in the Beaver Dam eye study," Ophthalmology 111, 1183-1190 (2004).
[CrossRef] [PubMed]

Michealson, I. C.

I. C. Michealson, Textbook of the Fundus of the Eye (Livingstone, 1980).

Miquerol, L.

J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
[CrossRef] [PubMed]

Mitchell, P.

J. J. Wang, B. Taylor, T. Y. Wong, B. Chua, E. Rochtchina, R. Klein, and P. Mitchell, "Retinal vessel diameters and obesity: a population-based study in older persons," Obes. Res. 14, 206-214 (2006).
[CrossRef]

P. Mitchell, H. Leung, J. J. Wang, E. Rochtchina, A. J. Lee, T. Y. Wong, and R. Klein, "Retinal vessel diameter and open-angle glaucoma: the Blue Mountains eye study," Ophthalmology 112, 245-250 (2005).
[CrossRef] [PubMed]

H. Leung, J. J. Wang, E. Rochtchina, T. Y. Wong, R. Klein, and P. Mitchell, "Impact of current and past blood pressure on retinal arteriolar diameter in older population," J. Hypertens. 22, 1543-1549 (2003).
[CrossRef]

Mojon, D.

X. Jiang and D. Mojon, "Adaptive local thresholding by verification-based multithreshold probing with application to vessel detection in retinal image," IEEE Trans. Pattern Anal. Mach. Intell. 25, 131-137 (2003).
[CrossRef]

Morelli, P. I.

J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
[CrossRef] [PubMed]

Mulhner, U.

J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
[CrossRef] [PubMed]

Nagy, A.

J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
[CrossRef] [PubMed]

Nelson, M.

S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, and M. Goldbaum, "Detection of blood vessels in retinal images using two-dimensional matched filters," IEEE Trans. Med. Imaging 8, 263-269 (1989).
[CrossRef] [PubMed]

W. Hart, M. Goldbaum, B. Cote, P. Kube, and M. Nelson, "Automated measurement of retinal vascular tortuosity," in Proceedings of the AMIA Annual Symposium (AMIA, 1997), pp. 459-463.

Niemeijer, M.

J. Staal, M. Abramoff, M. Niemeijer, M. Viergever, and B. van Ginneken, "Ridge-based vessel segmentation in color images of the retina," IEEE Trans. Med. Imaging 23, 501-509 (2004).
[CrossRef] [PubMed]

M. Niemeijer, J. J. Staal, B. van Ginneken, M. Loog, and M. D. Abramoff, "Comparative study of retinal vessel segmentation methods on a new publicly available database," in SPIE Medical Imaging, J.Michael Fitzpatrick and M.Sonka, eds., Proc. SPIE 5370, 648-656 (2004).

O'Donnell, M.

P. F. Sharpl, J. Olson, F. Strachan, J. Hipwell, A. Ludbrook, M. O'Donnell, S. Wallace, K. Goatman, A. Grant, N. Waugh, K. McHardy, and J. V. Forrester, "The value of digital imaging in diabetic retinopathy," Health Technol. Assess 7, 1-132 (2003), http://www.hta.nhsweb.nhs.uk/fullmono/mon730.pdf.

Olson, J.

P. F. Sharpl, J. Olson, F. Strachan, J. Hipwell, A. Ludbrook, M. O'Donnell, S. Wallace, K. Goatman, A. Grant, N. Waugh, K. McHardy, and J. V. Forrester, "The value of digital imaging in diabetic retinopathy," Health Technol. Assess 7, 1-132 (2003), http://www.hta.nhsweb.nhs.uk/fullmono/mon730.pdf.

Owens, R.

N. Ritter, R. Owens, J. Cooper, R. Eikelboom, and P. van Saarloos, "Registration of stereo and temporal images of the retina," IEEE Trans. Med. Imaging 18, 404-418 (1999).
[CrossRef]

Panas, S.

Y. Tolias and S. Panas, "A fuzzy vessel tracking algorithm for retinal images based on fuzzy clustering," IEEE Trans. Med. Imaging 17, 263-273 (1998).
[CrossRef] [PubMed]

Paton, D.

D. Paton, B. N. Hyman, and J. Justice, Introduction to Ophthalmoscopy (The Upjohn Co., 1976).

Pedersen, L.

L. Pedersen, B. Ersbøll, K. Madsen, M. Grunkin, U. Skands, M. Larsen, and N. Christoffersen, "Quantitative measurement of changes in retinal vessel diameter in ocular fundus images," Pattern Recogn. Lett. 21, 1215-1223 (2000).
[CrossRef]

Quek, F.

C. Kirbas and F. Quek, "A review of vessel extraction techniques and algorithms," ACM Comput. Surv. 36, 81-121 (2004).
[CrossRef]

Ritter, N.

N. Ritter, R. Owens, J. Cooper, R. Eikelboom, and P. van Saarloos, "Registration of stereo and temporal images of the retina," IEEE Trans. Med. Imaging 18, 404-418 (1999).
[CrossRef]

Rochtchina, E.

J. J. Wang, B. Taylor, T. Y. Wong, B. Chua, E. Rochtchina, R. Klein, and P. Mitchell, "Retinal vessel diameters and obesity: a population-based study in older persons," Obes. Res. 14, 206-214 (2006).
[CrossRef]

P. Mitchell, H. Leung, J. J. Wang, E. Rochtchina, A. J. Lee, T. Y. Wong, and R. Klein, "Retinal vessel diameter and open-angle glaucoma: the Blue Mountains eye study," Ophthalmology 112, 245-250 (2005).
[CrossRef] [PubMed]

H. Leung, J. J. Wang, E. Rochtchina, T. Y. Wong, R. Klein, and P. Mitchell, "Impact of current and past blood pressure on retinal arteriolar diameter in older population," J. Hypertens. 22, 1543-1549 (2003).
[CrossRef]

Roysam, B.

A. Can, C. Stewart, B. Roysam, and H. Tanenbaum, "A feature-based, robust, hierarchical algorithm for registering pairs of images of the curved human retina," IEEE Trans. Pattern Anal. Mach. Intell. 24, 347-364 (2002).
[CrossRef]

A. Can, H. Shen, J. N. Turner, H. L. Tanenbaum, and B. Roysam, "Rapid automated tracing and feature extraction from live high-resolution retinal fundus images using direct exploratory algorithms," IEEE Trans. Inf. Technol. Biomed. 3, 125-138 (1999).
[CrossRef]

Ryder, R.

J. Lowell, A. Hunter, D. Steel, A. Basu, R. Ryder, and E. Fletcher, "Optic nerve head segmentation," IEEE Trans. Med. Imaging 23, 256-264 (2004).
[CrossRef] [PubMed]

A. Hunter, J. Lowell, D. Steel, A. Basu, and R. Ryder, Non-linear Filtering for Vascular Segmentation and Detection of Venous Beading, Technical Report (Department of Computer Science, University of Durham, 2002).

Schmid, C.

C. Schmid, "Weakly supervised learning of visual models and its application to content-based retrieval," Int. J. Comput. Vis. 56, 7-16 (2004).
[CrossRef]

Sharpl, P. F.

P. F. Sharpl, J. Olson, F. Strachan, J. Hipwell, A. Ludbrook, M. O'Donnell, S. Wallace, K. Goatman, A. Grant, N. Waugh, K. McHardy, and J. V. Forrester, "The value of digital imaging in diabetic retinopathy," Health Technol. Assess 7, 1-132 (2003), http://www.hta.nhsweb.nhs.uk/fullmono/mon730.pdf.

Shen, H.

A. Can, H. Shen, J. N. Turner, H. L. Tanenbaum, and B. Roysam, "Rapid automated tracing and feature extraction from live high-resolution retinal fundus images using direct exploratory algorithms," IEEE Trans. Inf. Technol. Biomed. 3, 125-138 (1999).
[CrossRef]

Sinthanayothin, C.

C. Sinthanayothin, J. F. Boyce, T. H. Williamson, and H. L. Cook, "Automated Detection of diabetic retionopathy on digital fundus images," Diabetic Med. 19, 105-112 (2002).
[CrossRef]

Skands, U.

L. Pedersen, B. Ersbøll, K. Madsen, M. Grunkin, U. Skands, M. Larsen, and N. Christoffersen, "Quantitative measurement of changes in retinal vessel diameter in ocular fundus images," Pattern Recogn. Lett. 21, 1215-1223 (2000).
[CrossRef]

Staal, J.

J. Staal, M. Abramoff, M. Niemeijer, M. Viergever, and B. van Ginneken, "Ridge-based vessel segmentation in color images of the retina," IEEE Trans. Med. Imaging 23, 501-509 (2004).
[CrossRef] [PubMed]

Staal, J. J.

M. Niemeijer, J. J. Staal, B. van Ginneken, M. Loog, and M. D. Abramoff, "Comparative study of retinal vessel segmentation methods on a new publicly available database," in SPIE Medical Imaging, J.Michael Fitzpatrick and M.Sonka, eds., Proc. SPIE 5370, 648-656 (2004).

Stanton, A.

X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thom, "A method of vessel tracking for vessel diameter measurement on retinal images," in Proceedings of IEEE International Conference on Image Processing (IEEE, 2001), pp. 881-884.

Steel, D.

J. Lowell, A. Hunter, D. Steel, A. Basu, R. Ryder, and E. Fletcher, "Optic nerve head segmentation," IEEE Trans. Med. Imaging 23, 256-264 (2004).
[CrossRef] [PubMed]

A. Hunter, J. Lowell, D. Steel, A. Basu, and R. Ryder, Non-linear Filtering for Vascular Segmentation and Detection of Venous Beading, Technical Report (Department of Computer Science, University of Durham, 2002).

Stewart, C.

A. Can, C. Stewart, B. Roysam, and H. Tanenbaum, "A feature-based, robust, hierarchical algorithm for registering pairs of images of the curved human retina," IEEE Trans. Pattern Anal. Mach. Intell. 24, 347-364 (2002).
[CrossRef]

Strachan, F.

P. F. Sharpl, J. Olson, F. Strachan, J. Hipwell, A. Ludbrook, M. O'Donnell, S. Wallace, K. Goatman, A. Grant, N. Waugh, K. McHardy, and J. V. Forrester, "The value of digital imaging in diabetic retinopathy," Health Technol. Assess 7, 1-132 (2003), http://www.hta.nhsweb.nhs.uk/fullmono/mon730.pdf.

Tanenbaum, H.

A. Can, C. Stewart, B. Roysam, and H. Tanenbaum, "A feature-based, robust, hierarchical algorithm for registering pairs of images of the curved human retina," IEEE Trans. Pattern Anal. Mach. Intell. 24, 347-364 (2002).
[CrossRef]

Tanenbaum, H. L.

A. Can, H. Shen, J. N. Turner, H. L. Tanenbaum, and B. Roysam, "Rapid automated tracing and feature extraction from live high-resolution retinal fundus images using direct exploratory algorithms," IEEE Trans. Inf. Technol. Biomed. 3, 125-138 (1999).
[CrossRef]

Taylor, B.

J. J. Wang, B. Taylor, T. Y. Wong, B. Chua, E. Rochtchina, R. Klein, and P. Mitchell, "Retinal vessel diameters and obesity: a population-based study in older persons," Obes. Res. 14, 206-214 (2006).
[CrossRef]

Thom, S.

X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thom, "A method of vessel tracking for vessel diameter measurement on retinal images," in Proceedings of IEEE International Conference on Image Processing (IEEE, 2001), pp. 881-884.

Tolias, Y.

Y. Tolias and S. Panas, "A fuzzy vessel tracking algorithm for retinal images based on fuzzy clustering," IEEE Trans. Med. Imaging 17, 263-273 (1998).
[CrossRef] [PubMed]

Tsai, M. Y.

R. Klein, B. E. K. Klein, M. D. Knudtson, T. Y. Wong, and M. Y. Tsai, "Are inflammatory factors related to retinal vessel caliber? the Beaver Dam eye study," Arch. Ophthalmol. 124, 87-94 (2006).
[CrossRef] [PubMed]

Tsien, J.

J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
[CrossRef] [PubMed]

Turner, J. N.

A. Can, H. Shen, J. N. Turner, H. L. Tanenbaum, and B. Roysam, "Rapid automated tracing and feature extraction from live high-resolution retinal fundus images using direct exploratory algorithms," IEEE Trans. Inf. Technol. Biomed. 3, 125-138 (1999).
[CrossRef]

van Ginneken, B.

J. Staal, M. Abramoff, M. Niemeijer, M. Viergever, and B. van Ginneken, "Ridge-based vessel segmentation in color images of the retina," IEEE Trans. Med. Imaging 23, 501-509 (2004).
[CrossRef] [PubMed]

M. Niemeijer, J. J. Staal, B. van Ginneken, M. Loog, and M. D. Abramoff, "Comparative study of retinal vessel segmentation methods on a new publicly available database," in SPIE Medical Imaging, J.Michael Fitzpatrick and M.Sonka, eds., Proc. SPIE 5370, 648-656 (2004).

van Saarloos, P.

N. Ritter, R. Owens, J. Cooper, R. Eikelboom, and P. van Saarloos, "Registration of stereo and temporal images of the retina," IEEE Trans. Med. Imaging 18, 404-418 (1999).
[CrossRef]

Varma, M.

M. Varma and A. Zisserman, "A statistical approach to texture classification from single images," Int. J. Comput. Vis. 62, 61-81 (2005).

Viergever, M.

J. Staal, M. Abramoff, M. Niemeijer, M. Viergever, and B. van Ginneken, "Ridge-based vessel segmentation in color images of the retina," IEEE Trans. Med. Imaging 23, 501-509 (2004).
[CrossRef] [PubMed]

Wagner, E. F.

J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
[CrossRef] [PubMed]

Wallace, S.

P. F. Sharpl, J. Olson, F. Strachan, J. Hipwell, A. Ludbrook, M. O'Donnell, S. Wallace, K. Goatman, A. Grant, N. Waugh, K. McHardy, and J. V. Forrester, "The value of digital imaging in diabetic retinopathy," Health Technol. Assess 7, 1-132 (2003), http://www.hta.nhsweb.nhs.uk/fullmono/mon730.pdf.

Walter, T.

T. Walter, J. Klein, and F. Zana, "Automatic segmentation and registration of retinal fluorescin angiographies--applications to diabetic retinopathy," IEEE Trans. Inf. Technol. Biomed. 79, 463-491 (1999).

Wang, J. J.

J. J. Wang, B. Taylor, T. Y. Wong, B. Chua, E. Rochtchina, R. Klein, and P. Mitchell, "Retinal vessel diameters and obesity: a population-based study in older persons," Obes. Res. 14, 206-214 (2006).
[CrossRef]

P. Mitchell, H. Leung, J. J. Wang, E. Rochtchina, A. J. Lee, T. Y. Wong, and R. Klein, "Retinal vessel diameter and open-angle glaucoma: the Blue Mountains eye study," Ophthalmology 112, 245-250 (2005).
[CrossRef] [PubMed]

H. Leung, J. J. Wang, E. Rochtchina, T. Y. Wong, R. Klein, and P. Mitchell, "Impact of current and past blood pressure on retinal arteriolar diameter in older population," J. Hypertens. 22, 1543-1549 (2003).
[CrossRef]

Waugh, N.

P. F. Sharpl, J. Olson, F. Strachan, J. Hipwell, A. Ludbrook, M. O'Donnell, S. Wallace, K. Goatman, A. Grant, N. Waugh, K. McHardy, and J. V. Forrester, "The value of digital imaging in diabetic retinopathy," Health Technol. Assess 7, 1-132 (2003), http://www.hta.nhsweb.nhs.uk/fullmono/mon730.pdf.

Williamson, T. H.

C. Sinthanayothin, J. F. Boyce, T. H. Williamson, and H. L. Cook, "Automated Detection of diabetic retionopathy on digital fundus images," Diabetic Med. 19, 105-112 (2002).
[CrossRef]

Wong, T. Y.

J. J. Wang, B. Taylor, T. Y. Wong, B. Chua, E. Rochtchina, R. Klein, and P. Mitchell, "Retinal vessel diameters and obesity: a population-based study in older persons," Obes. Res. 14, 206-214 (2006).
[CrossRef]

R. Klein, B. E. K. Klein, M. D. Knudtson, T. Y. Wong, and M. Y. Tsai, "Are inflammatory factors related to retinal vessel caliber? the Beaver Dam eye study," Arch. Ophthalmol. 124, 87-94 (2006).
[CrossRef] [PubMed]

P. Mitchell, H. Leung, J. J. Wang, E. Rochtchina, A. J. Lee, T. Y. Wong, and R. Klein, "Retinal vessel diameter and open-angle glaucoma: the Blue Mountains eye study," Ophthalmology 112, 245-250 (2005).
[CrossRef] [PubMed]

T. Y. Wong, M. D. Knudtson, R. Klein, B. E. Klein, S. M. Meuer, and L. D. Hubbard, "Computer-assisted measurement of retinal vessel diameter in the Beaver Dam eye study," Ophthalmology 111, 1183-1190 (2004).
[CrossRef] [PubMed]

H. Leung, J. J. Wang, E. Rochtchina, T. Y. Wong, R. Klein, and P. Mitchell, "Impact of current and past blood pressure on retinal arteriolar diameter in older population," J. Hypertens. 22, 1543-1549 (2003).
[CrossRef]

Yan, T. Y.

F. Bin and T. Y. Yan, "Elastic registration for retinal images based on reconstructed vascular trees," IEEE Trans. Biomed. Eng. 53, 1183-1187 (2006).
[CrossRef]

Zana, F.

F. Zana and J.-C. Klein, "Segmentation of vessel-like patterns using mathematical morphology and curvature evaluation," IEEE Trans. Image Process. 10, 1010-1019 (2001).
[CrossRef]

T. Walter, J. Klein, and F. Zana, "Automatic segmentation and registration of retinal fluorescin angiographies--applications to diabetic retinopathy," IEEE Trans. Inf. Technol. Biomed. 79, 463-491 (1999).

Zisserman, A.

M. Varma and A. Zisserman, "A statistical approach to texture classification from single images," Int. J. Comput. Vis. 62, 61-81 (2005).

ACM Comput. Surv.

C. Kirbas and F. Quek, "A review of vessel extraction techniques and algorithms," ACM Comput. Surv. 36, 81-121 (2004).
[CrossRef]

Acta Ophthalmol. Suppl.

O. Brinchmann-Hansen and H. Heier, "Theoretical relationships between light streak characteristics and optical properties of retinal vessels," Acta Ophthalmol. Suppl. 179, 33-37 (1986).

Arch. Ophthalmol.

R. Klein, B. E. K. Klein, M. D. Knudtson, T. Y. Wong, and M. Y. Tsai, "Are inflammatory factors related to retinal vessel caliber? the Beaver Dam eye study," Arch. Ophthalmol. 124, 87-94 (2006).
[CrossRef] [PubMed]

Dev. Biol.

J. J. Haigh, P. I. Morelli, H. Gerhardt, K. Haigh, J. Tsien, A. Damert, L. Miquerol, U. Mulhner, R. Klein, N. Ferrara, E. F. Wagner, C. Betsholtz, and A. Nagy, "Cortical and retinal defects caused by dosage-dependent reductions in VEGF-A paracrine signaling," Dev. Biol. 262, 225-241 (2003).
[CrossRef] [PubMed]

Diabetic Med.

C. Sinthanayothin, J. F. Boyce, T. H. Williamson, and H. L. Cook, "Automated Detection of diabetic retionopathy on digital fundus images," Diabetic Med. 19, 105-112 (2002).
[CrossRef]

Health Technol. Assess

P. F. Sharpl, J. Olson, F. Strachan, J. Hipwell, A. Ludbrook, M. O'Donnell, S. Wallace, K. Goatman, A. Grant, N. Waugh, K. McHardy, and J. V. Forrester, "The value of digital imaging in diabetic retinopathy," Health Technol. Assess 7, 1-132 (2003), http://www.hta.nhsweb.nhs.uk/fullmono/mon730.pdf.

IEEE Trans. Biomed. Eng.

F. Bin and T. Y. Yan, "Elastic registration for retinal images based on reconstructed vascular trees," IEEE Trans. Biomed. Eng. 53, 1183-1187 (2006).
[CrossRef]

L. Gang, O. Chutatape, and S. M. Krishnan, "Detection and measurement of retinal vessels in fundus images using amplitude modified second-order Gaussian filter," IEEE Trans. Biomed. Eng. 49, 168-172 (2002).
[CrossRef] [PubMed]

IEEE Trans. Image Process.

F. Zana and J.-C. Klein, "Segmentation of vessel-like patterns using mathematical morphology and curvature evaluation," IEEE Trans. Image Process. 10, 1010-1019 (2001).
[CrossRef]

IEEE Trans. Inf. Technol. Biomed.

A. Can, H. Shen, J. N. Turner, H. L. Tanenbaum, and B. Roysam, "Rapid automated tracing and feature extraction from live high-resolution retinal fundus images using direct exploratory algorithms," IEEE Trans. Inf. Technol. Biomed. 3, 125-138 (1999).
[CrossRef]

T. Walter, J. Klein, and F. Zana, "Automatic segmentation and registration of retinal fluorescin angiographies--applications to diabetic retinopathy," IEEE Trans. Inf. Technol. Biomed. 79, 463-491 (1999).

IEEE Trans. Med. Imaging

Y. Tolias and S. Panas, "A fuzzy vessel tracking algorithm for retinal images based on fuzzy clustering," IEEE Trans. Med. Imaging 17, 263-273 (1998).
[CrossRef] [PubMed]

J. Lowell, A. Hunter, D. Steel, A. Basu, R. Ryder, and E. Fletcher, "Optic nerve head segmentation," IEEE Trans. Med. Imaging 23, 256-264 (2004).
[CrossRef] [PubMed]

S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, and M. Goldbaum, "Detection of blood vessels in retinal images using two-dimensional matched filters," IEEE Trans. Med. Imaging 8, 263-269 (1989).
[CrossRef] [PubMed]

A. Hoover, V. Kouznetsova, and M. Goldbaum, "Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response," IEEE Trans. Med. Imaging 19, 203-210 (2000).
[CrossRef] [PubMed]

N. Ritter, R. Owens, J. Cooper, R. Eikelboom, and P. van Saarloos, "Registration of stereo and temporal images of the retina," IEEE Trans. Med. Imaging 18, 404-418 (1999).
[CrossRef]

J. Staal, M. Abramoff, M. Niemeijer, M. Viergever, and B. van Ginneken, "Ridge-based vessel segmentation in color images of the retina," IEEE Trans. Med. Imaging 23, 501-509 (2004).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell.

A. Can, C. Stewart, B. Roysam, and H. Tanenbaum, "A feature-based, robust, hierarchical algorithm for registering pairs of images of the curved human retina," IEEE Trans. Pattern Anal. Mach. Intell. 24, 347-364 (2002).
[CrossRef]

X. Jiang and D. Mojon, "Adaptive local thresholding by verification-based multithreshold probing with application to vessel detection in retinal image," IEEE Trans. Pattern Anal. Mach. Intell. 25, 131-137 (2003).
[CrossRef]

J. Kittler, M. Hatef, R. P. W. Duin, and J. Matas, "On combining classifiers," IEEE Trans. Pattern Anal. Mach. Intell. 20, 226-239 (1998).
[CrossRef]

Int. J. Comput. Vis.

T. Leung and J. Malik, "Representing and recognizing the visual appearance of materials using three dimensional textons," Int. J. Comput. Vis. 43, 29-44 (2001).
[CrossRef]

O. G. Cula and K. J. Dana, "3D texture recognition using bidirectional feature histograms," Int. J. Comput. Vis. 59, 33-60 (2004).
[CrossRef]

C. Schmid, "Weakly supervised learning of visual models and its application to content-based retrieval," Int. J. Comput. Vis. 56, 7-16 (2004).
[CrossRef]

M. Varma and A. Zisserman, "A statistical approach to texture classification from single images," Int. J. Comput. Vis. 62, 61-81 (2005).

J. Hypertens.

H. Leung, J. J. Wang, E. Rochtchina, T. Y. Wong, R. Klein, and P. Mitchell, "Impact of current and past blood pressure on retinal arteriolar diameter in older population," J. Hypertens. 22, 1543-1549 (2003).
[CrossRef]

Obes. Res.

J. J. Wang, B. Taylor, T. Y. Wong, B. Chua, E. Rochtchina, R. Klein, and P. Mitchell, "Retinal vessel diameters and obesity: a population-based study in older persons," Obes. Res. 14, 206-214 (2006).
[CrossRef]

Ophthalmology

P. Mitchell, H. Leung, J. J. Wang, E. Rochtchina, A. J. Lee, T. Y. Wong, and R. Klein, "Retinal vessel diameter and open-angle glaucoma: the Blue Mountains eye study," Ophthalmology 112, 245-250 (2005).
[CrossRef] [PubMed]

T. Y. Wong, M. D. Knudtson, R. Klein, B. E. Klein, S. M. Meuer, and L. D. Hubbard, "Computer-assisted measurement of retinal vessel diameter in the Beaver Dam eye study," Ophthalmology 111, 1183-1190 (2004).
[CrossRef] [PubMed]

Pattern Recogn. Lett.

L. Pedersen, B. Ersbøll, K. Madsen, M. Grunkin, U. Skands, M. Larsen, and N. Christoffersen, "Quantitative measurement of changes in retinal vessel diameter in ocular fundus images," Pattern Recogn. Lett. 21, 1215-1223 (2000).
[CrossRef]

Retina

J. Capowski, J. Kylstra, and S. Freedman, "A numeric index based on spatial-frequency for the tortuosity of retinal vessels and its application to plus disease in retinopathy of prematurity," Retina 15, 490-500 (1995).
[PubMed]

Other

X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thom, "A method of vessel tracking for vessel diameter measurement on retinal images," in Proceedings of IEEE International Conference on Image Processing (IEEE, 2001), pp. 881-884.

W. Hart, M. Goldbaum, B. Cote, P. Kube, and M. Nelson, "Automated measurement of retinal vascular tortuosity," in Proceedings of the AMIA Annual Symposium (AMIA, 1997), pp. 459-463.

A. Hunter, J. Lowell, D. Steel, A. Basu, and R. Ryder, Non-linear Filtering for Vascular Segmentation and Detection of Venous Beading, Technical Report (Department of Computer Science, University of Durham, 2002).

D. Paton, B. N. Hyman, and J. Justice, Introduction to Ophthalmoscopy (The Upjohn Co., 1976).

I. C. Michealson, Textbook of the Fundus of the Eye (Livingstone, 1980).

M. Niemeijer, J. J. Staal, B. van Ginneken, M. Loog, and M. D. Abramoff, "Comparative study of retinal vessel segmentation methods on a new publicly available database," in SPIE Medical Imaging, J.Michael Fitzpatrick and M.Sonka, eds., Proc. SPIE 5370, 648-656 (2004).

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Figures (9)

Fig. 1
Fig. 1

Texton assignment procedure.

Fig. 2
Fig. 2

Intensity profile across a vessel width: (a) without light-reflex modeling, (b) with light reflex modeling.

Fig. 3
Fig. 3

Result of filter grouping procedure using a correlation-based analysis.

Fig. 4
Fig. 4

Filter response for RVF-20, the proposed filter bank specially constructed for retinal vessels.

Fig. 5
Fig. 5

Texton map for a sample fundus image using the three previously proposed filter banks, and the proposed specially selected filters for vessel segmentation: (a) original image, (b) LM-48: 48-filter bank proposed by Leung and Malik,[9] (c) MR8: 38-filter bank used by Varma and Zisserman in MR-8 (Ref. [12]), (d) S-13: 13 rotationally invariant filters used by Schmid,[11] (e) proposed RVF-20: 20 specially selected filters for retinal vessels.

Fig. 6
Fig. 6

Textons extracted from vessel regions in a sample fundus image.

Fig. 7
Fig. 7

Genuine-impostor curve for the proposed texton-based vessel segmentation.

Fig. 8
Fig. 8

ROC curves for vessel segmentation using textons based on different filter banks, including the proposed RVF-20 filter bank. (Note: results for all the texton-based methods are based on our implementation of each approach.)

Fig. 9
Fig. 9

Sample vessel segmentation result using the proposed texton-based approach.

Tables (1)

Tables Icon

Table 1 Comparative Performance with Other Vessel Segmentation Methods a

Equations (20)

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R i ( x , y ) = Ω I ( x p , y q ) F i ( p , q ) d p d q , ( x , y ) Λ ,
F ( x , y , τ , σ ) = F 0 ( τ , σ ) + cos ( π τ σ x 2 + y 2 ) exp ( x 2 + y 2 2 σ 2 ) ,
F σ ( x , y ) = 1 2 π σ exp ( x 2 + y 2 2 σ 2 ) .
DoG σ ( x , y ) = F σ ( x , y ) F σ ( x + δ , y ) .
F σ ( x , y ) x = x σ 2 F σ ( x , y ) ;
F σ ( x , y ) y = y σ 2 F σ ( x , y ) .
med = median ( i , j , k ) Φ { T I D ( i , j , k ) } ,
L ( x , y ) = { 1 , if med T 1 0 , if med T 0 } .
T 1 ( i , j , k ) = { 1 , if T ( i , j , k ) T 1 0 , otherwise } ,
n 1 = ( i , j , k ) Φ T 1 ( i , j , k ) ,
n 0 = ω n 1 = ( i , j , k ) Φ T 0 ( i , j , k ) ,
L ( x , y ) = { 1 , if n 1 n 0 0 , otherwise } .
p k ( i , j ) = 1 φ ( i , j ) φ T 1 ( i , j , k ) .
P ( x , y ) 1 = [ ( i , j ) φ k = 1 m p k ( i , j ) ] 1 ω .
P ( x , y ) 0 = [ ( i , j ) φ k = 1 m ( 1 p k ( i , j ) ) ] 1 ω .
P ( x , y ) 1 = 1 m φ ( i , j ) φ k = 1 m p k ( i , j ) ,
P ( x , y ) 0 = 1 m φ ( i , j ) φ k = 1 m ( 1 p k ( i , j ) ) .
P ( x , y ) 1 = P ( x , y ) 1 P ( x , y ) 1 + P ( x , y ) 0 ,
P ( x , y ) 0 = P ( x , y ) 0 P ( x , y ) 1 + P ( x , y ) 0 ,
L ( x , y ) = { 1 , if P ( x , y ) 1 P ( x , y ) 0 0 , otherwise } .

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