Abstract

Retinal fundus images are widely used in the diagnosis and treatment of various eye diseases, such as diabetic retinopathy and glaucoma. A computer-aided retinal fundus image analysis could provide an immediate detection and characterization of retinal features prior to specialist inspection. This paper proposes an approach to automatically localize the main features in fundus images, such as blood vessels, optic disc, and fovea by exploiting the spatial and geometric relations that govern their distribution within the fundus image. The blood vessels are segmented by scale-space analysis. The average thickness of these blood vessels is then computed using the vessels centerlines and orientations from a Hessian matrix. The optic disc is localized using the circular Hough transform, the parabolic Hough transform fitting, and the localization of the fovea. The proposed method can be extended to establish a foveal coordinate system to facilitate grading lesions based on the spatial relationships between lesions and landmark features. The proposed method was evaluated on publicly available image databases, and the results have demonstrated a significant improvement over the current state-of-the-art methods.

© 2011 Optical Society of America

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  30. M. Foracchia and E. A. R. Grisan, “Detection of optic disc in retinal images by means of a geometrical model of vessel structure,” IEEE Trans. Med. Imaging 23, 1189–1195 (2004).
    [CrossRef] [PubMed]
  31. A. Osareh, M. Mirmehdi, B. Thomas, and R. Markham, “Classification and localisation of diabetic-related eye disease,” in 7th European Conference on Computer Vision (ECCV), Vol. 2353 ( 2002), 502–516.
  32. A. Osareh, M. Mirmehdi, B. Thomas, and R. Markham, “Automated identification of diabetic retinal exudates in digital colour images,” Br. J. Ophthalmol. 87, 1220–1223(2003).
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    [CrossRef] [PubMed]
  37. A. Youssif, A. Ghalwash, and A. Ghoneim, “Optic disc detection from normalized digital fundus images by means of a vessels’ direction matched filter,” IEEE Trans. Med. Imaging 27, 11–18 (2008).
    [CrossRef] [PubMed]
  38. H. Ying, M. Zhang, and J. Liu, “Fractal-based automatic localization and segmentation of optic disc in retinal images,” in Proceedings of the IEEE 29th Annual International Conference of the Engineering in Medicine and Biology Society (IEEE, 2007), pp. 4139–4141.
    [CrossRef] [PubMed]

2008 (1)

A. Youssif, A. Ghalwash, and A. Ghoneim, “Optic disc detection from normalized digital fundus images by means of a vessels’ direction matched filter,” IEEE Trans. Med. Imaging 27, 11–18 (2008).
[CrossRef] [PubMed]

2007 (1)

S. A. Salem, S. A. Nancy, and A. K. Nandi, “Segmentation of retinal blood vessels using a novel clustering algorithm (racal) with a partial supervision strategy,” Med. Biol. Eng. Comput. 45, 261–273 (2007).
[CrossRef] [PubMed]

2006 (1)

D. Wu, M. Zhang, J. Liu, and W. Bauman, “On the adaptive detection of blood vessels in retinal images,” IEEE Trans. Biomed. Eng. 53, 341–343 (2006).
[CrossRef] [PubMed]

2004 (5)

M. Foracchia and E. A. R. Grisan, “Detection of optic disc in retinal images by means of a geometrical model of vessel structure,” IEEE Trans. Med. Imaging 23, 1189–1195 (2004).
[CrossRef] [PubMed]

J. Staal, M. D. Abramoff, M. Niemeijer, M. A. Viergever, and B. V. Ginneken, “Ridge based vessel segmentation in color images of the retina,” IEEE Trans. Med. Imaging 23, 501–509 (2004).
[CrossRef] [PubMed]

H. Li and O. Chutatape, “Automated feature extraction in colour retinal images by a model based approach,” IEEE Trans. Biomed. Eng. 51, 246–254 (2004).
[CrossRef] [PubMed]

M. Niemeijer, J. Staal, B. Van Ginneken, M. Long, and M. Abramoff, “Comparative study of retinal vessel segmentation methods on a new publicly available database,” in Proc. SPIE 5370, 648–656 (2004).
[CrossRef]

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

2003 (3)

A. Hoover and M. Goldbaum, “Locating the optic nerve in a retinal image using the fuzzy convergence of the blood vessels,” IEEE Trans. Biomed. Eng. 22, 951–958 (2003).

A. Osareh, M. Mirmehdi, B. Thomas, and R. Markham, “Automated identification of diabetic retinal exudates in digital colour images,” Br. J. Ophthalmol. 87, 1220–1223(2003).
[CrossRef] [PubMed]

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

2001 (1)

M. Lalonde, M. Beaulieu, and L. Gagnon, “Fast and robust optic disc detection using pyramidal decomposition and Hausdorff-based template matching,” IEEE Trans. Med. Imaging 20, 1193–1200 (2001).
[CrossRef] [PubMed]

2000 (1)

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 (2)

C. Sinthanayothin, J. F. Boyce, H. L. Cook, and T. H. Williamson, “Automatic localisation of the optic disc, fovea, and retinal blood vessels from digital colour fundus images,” Br.J. Ophthalmol. 83, 902–910 (1999).
[CrossRef] [PubMed]

M. Ibanez and A. Simo, “Bayesian detection of the fovea in eye fundus angiographies,” Pattern Recogn. Lett. 20, 229–240(1999).
[CrossRef]

1994 (1)

L. Zhou, M. Rzeszotarski, L. Singerman, and J. Chokreff, “The detection and quantification of retinopathy using digital angiograms,” IEEE Trans. Med. Imaging 13, 619–626(1994).
[CrossRef] [PubMed]

1992 (1)

J. Serra and L. Vincent, “An overview of morphological filtering,” Circuits Syst. Signal Process. 11, 47–108(1992).
[CrossRef]

1989 (1)

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]

1982 (1)

K. Akita and H. Kuga, “A computer method of understanding ocular fundus images,” Pattern Recognition 15, 431–443(1982).
[CrossRef]

1972 (1)

R. O. Duda and P. E. Hart, “Use of Hough transformation to detect lines and curves in pictures,” Commun. ACM 15, 11–15 (1972).
[CrossRef]

Abramoff, M.

M. Niemeijer, J. Staal, B. Van Ginneken, M. Long, and M. Abramoff, “Comparative study of retinal vessel segmentation methods on a new publicly available database,” in Proc. SPIE 5370, 648–656 (2004).
[CrossRef]

Abramoff, M. D.

J. Staal, M. D. Abramoff, M. Niemeijer, M. A. Viergever, and B. V. Ginneken, “Ridge based vessel segmentation in color images of the retina,” IEEE Trans. Med. Imaging 23, 501–509 (2004).
[CrossRef] [PubMed]

Akita, K.

K. Akita and H. Kuga, “A computer method of understanding ocular fundus images,” Pattern Recognition 15, 431–443(1982).
[CrossRef]

Basu, A.

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

Bauman, W.

D. Wu, M. Zhang, J. Liu, and W. Bauman, “On the adaptive detection of blood vessels in retinal images,” IEEE Trans. Biomed. Eng. 53, 341–343 (2006).
[CrossRef] [PubMed]

Beaulieu, M.

M. Lalonde, M. Beaulieu, and L. Gagnon, “Fast and robust optic disc detection using pyramidal decomposition and Hausdorff-based template matching,” IEEE Trans. Med. Imaging 20, 1193–1200 (2001).
[CrossRef] [PubMed]

Bharath, A. A.

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, A. A. Bharath, and K. H. Parker, “Scale-space analysis for the characterization of retinal blood vessels,” in BMVA Proceedings of Medical Image Understanding and Analysis (Oxford, 1999), pp. 57–60.

Boyce, J. F.

C. Sinthanayothin, J. F. Boyce, H. L. Cook, and T. H. Williamson, “Automatic localisation of the optic disc, fovea, and retinal blood vessels from digital colour fundus images,” Br.J. Ophthalmol. 83, 902–910 (1999).
[CrossRef] [PubMed]

Boyd, J.

M. Goldbaum, S. Moezzi, S. Taylor, S. Chatterjee, J. Boyd, E. Hunter, and R. Jain, “Automated diagnosis and image understanding with object extraction, object classification, and inferencing in retinal images,” in Proceedings of the IEEE International Conference on Image Processing, Vol. 3 (IEEE, 1996), pp. 695–698.
[CrossRef]

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]

M. Goldbaum, S. Moezzi, S. Taylor, S. Chatterjee, J. Boyd, E. Hunter, and R. Jain, “Automated diagnosis and image understanding with object extraction, object classification, and inferencing in retinal images,” in Proceedings of the IEEE International Conference on Image Processing, Vol. 3 (IEEE, 1996), pp. 695–698.
[CrossRef]

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]

Chokreff, J.

L. Zhou, M. Rzeszotarski, L. Singerman, and J. Chokreff, “The detection and quantification of retinopathy using digital angiograms,” IEEE Trans. Med. Imaging 13, 619–626(1994).
[CrossRef] [PubMed]

Chrastek, R.

R. Chrastek, M. Wolf, K. Donath, G. Michelson, and H. Niemann, “Optic disc segmentation in retinal images,” in Bildverarbeitung für die Medizin (Springer-Verlag, 2002), pp. 263–266.

Chutatape, O.

H. Li and O. Chutatape, “Automated feature extraction in colour retinal images by a model based approach,” IEEE Trans. Biomed. Eng. 51, 246–254 (2004).
[CrossRef] [PubMed]

O. Chutatape, L. Zheng, and S. Krishnan, “Retinal blood vessel detection and tracking by matched Gaussian and Kalman filters,” in Proceedings of the 20th Annual International Conference of the IEEE on Engineering in Medicine and Biology Society, Vol. 6 (IEEE, 1998), pp. 3144–3149.

O. Chutatape, “Fundus foveal localization based on vessel model,” in Proceedings of the IEEE International Conference of the Engineering in Medicine and Biology Society (IEEE, 2006), pp. 4440–4444.
[CrossRef]

Cook, H. L.

C. Sinthanayothin, J. F. Boyce, H. L. Cook, and T. H. Williamson, “Automatic localisation of the optic disc, fovea, and retinal blood vessels from digital colour fundus images,” Br.J. Ophthalmol. 83, 902–910 (1999).
[CrossRef] [PubMed]

Damian, F.

F. Damian, ARIA (2006), http://www.eyecharity.com/aria_online/.

Donath, K.

R. Chrastek, M. Wolf, K. Donath, G. Michelson, and H. Niemann, “Optic disc segmentation in retinal images,” in Bildverarbeitung für die Medizin (Springer-Verlag, 2002), pp. 263–266.

Duda, R. O.

R. O. Duda and P. E. Hart, “Use of Hough transformation to detect lines and curves in pictures,” Commun. ACM 15, 11–15 (1972).
[CrossRef]

Fletcher, E.

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

Foracchia, M.

M. Foracchia and E. A. R. Grisan, “Detection of optic disc in retinal images by means of a geometrical model of vessel structure,” IEEE Trans. Med. Imaging 23, 1189–1195 (2004).
[CrossRef] [PubMed]

Gagnon, L.

M. Lalonde, M. Beaulieu, and L. Gagnon, “Fast and robust optic disc detection using pyramidal decomposition and Hausdorff-based template matching,” IEEE Trans. Med. Imaging 20, 1193–1200 (2001).
[CrossRef] [PubMed]

Ghalwash, A.

A. Youssif, A. Ghalwash, and A. Ghoneim, “Optic disc detection from normalized digital fundus images by means of a vessels’ direction matched filter,” IEEE Trans. Med. Imaging 27, 11–18 (2008).
[CrossRef] [PubMed]

Ghoneim, A.

A. Youssif, A. Ghalwash, and A. Ghoneim, “Optic disc detection from normalized digital fundus images by means of a vessels’ direction matched filter,” IEEE Trans. Med. Imaging 27, 11–18 (2008).
[CrossRef] [PubMed]

Ginneken, B. V.

J. Staal, M. D. Abramoff, M. Niemeijer, M. A. Viergever, and B. V. Ginneken, “Ridge based vessel segmentation in color images of the retina,” IEEE Trans. Med. Imaging 23, 501–509 (2004).
[CrossRef] [PubMed]

Goldbaum, M.

A. Hoover and M. Goldbaum, “Locating the optic nerve in a retinal image using the fuzzy convergence of the blood vessels,” IEEE Trans. Biomed. Eng. 22, 951–958 (2003).

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]

M. Goldbaum, S. Moezzi, S. Taylor, S. Chatterjee, J. Boyd, E. Hunter, and R. Jain, “Automated diagnosis and image understanding with object extraction, object classification, and inferencing in retinal images,” in Proceedings of the IEEE International Conference on Image Processing, Vol. 3 (IEEE, 1996), pp. 695–698.
[CrossRef]

Grisan, E. A. R.

M. Foracchia and E. A. R. Grisan, “Detection of optic disc in retinal images by means of a geometrical model of vessel structure,” IEEE Trans. Med. Imaging 23, 1189–1195 (2004).
[CrossRef] [PubMed]

Hart, P. E.

R. O. Duda and P. E. Hart, “Use of Hough transformation to detect lines and curves in pictures,” Commun. ACM 15, 11–15 (1972).
[CrossRef]

Heneghan, C.

F. Mendels, C. Heneghan, and J. P. Thiran, “Identification of the optic disc boundary in retinal images using active contours,” in IEEE Proceedings of the Irish Machine Vision Image Processing Conference (IMVIP) (IEEE, 1999), pp. 103–115.

Hoover, A.

A. Hoover and M. Goldbaum, “Locating the optic nerve in a retinal image using the fuzzy convergence of the blood vessels,” IEEE Trans. Biomed. Eng. 22, 951–958 (2003).

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]

Hughes, A. D.

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, A. A. Bharath, and K. H. Parker, “Scale-space analysis for the characterization of retinal blood vessels,” in BMVA Proceedings of Medical Image Understanding and Analysis (Oxford, 1999), pp. 57–60.

Hunter, A.

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

Hunter, E.

M. Goldbaum, S. Moezzi, S. Taylor, S. Chatterjee, J. Boyd, E. Hunter, and R. Jain, “Automated diagnosis and image understanding with object extraction, object classification, and inferencing in retinal images,” in Proceedings of the IEEE International Conference on Image Processing, Vol. 3 (IEEE, 1996), pp. 695–698.
[CrossRef]

Ibanez, M.

M. Ibanez and A. Simo, “Bayesian detection of the fovea in eye fundus angiographies,” Pattern Recogn. Lett. 20, 229–240(1999).
[CrossRef]

Jain, R.

M. Goldbaum, S. Moezzi, S. Taylor, S. Chatterjee, J. Boyd, E. Hunter, and R. Jain, “Automated diagnosis and image understanding with object extraction, object classification, and inferencing in retinal images,” in Proceedings of the IEEE International Conference on Image Processing, Vol. 3 (IEEE, 1996), pp. 695–698.
[CrossRef]

Jiang, X.

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

Jin, J. S.

M. Park, J. S. Jin, and S. Luo, “Locating the optic disc in retinal images,” in Proceedings of the IEEE International Conference on Computer Graphics, Imaging and Visualization (IEEE, 2006), pp. 141–145.
[PubMed]

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]

Kennedy, L.

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

Klein, J. C.

T. Walter and J. C. Klein, “Segmentation of colour fundus images of the human retina: Detection of the optic disc and the vascular tree using morphological techniques,” in Proceedings of the 2nd International Symposium on Medical Data Analysis, LNCS 2199 (Springer-Verlag, 2001), pp. 282–287.

F. Zana, I. Meunier, and J. C. Klein, “A region merging algorithm using mathematical morphology: application to macula detection,” in International Symposium on Mathematical Morphology (Kluwer, 1998), pp. 423–430.

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.

O. Chutatape, L. Zheng, and S. Krishnan, “Retinal blood vessel detection and tracking by matched Gaussian and Kalman filters,” in Proceedings of the 20th Annual International Conference of the IEEE on Engineering in Medicine and Biology Society, Vol. 6 (IEEE, 1998), pp. 3144–3149.

Kuga, H.

K. Akita and H. Kuga, “A computer method of understanding ocular fundus images,” Pattern Recognition 15, 431–443(1982).
[CrossRef]

Lalonde, M.

M. Lalonde, M. Beaulieu, and L. Gagnon, “Fast and robust optic disc detection using pyramidal decomposition and Hausdorff-based template matching,” IEEE Trans. Med. Imaging 20, 1193–1200 (2001).
[CrossRef] [PubMed]

Li, H.

H. Li and O. Chutatape, “Automated feature extraction in colour retinal images by a model based approach,” IEEE Trans. Biomed. Eng. 51, 246–254 (2004).
[CrossRef] [PubMed]

Lindeberg, T.

T. Lindeberg, Scale-Space Theory in Computer Vision (Kluwer, 1994).

Liu, J.

D. Wu, M. Zhang, J. Liu, and W. Bauman, “On the adaptive detection of blood vessels in retinal images,” IEEE Trans. Biomed. Eng. 53, 341–343 (2006).
[CrossRef] [PubMed]

H. Ying, M. Zhang, and J. Liu, “Fractal-based automatic localization and segmentation of optic disc in retinal images,” in Proceedings of the IEEE 29th Annual International Conference of the Engineering in Medicine and Biology Society (IEEE, 2007), pp. 4139–4141.
[CrossRef] [PubMed]

Long, M.

M. Niemeijer, J. Staal, B. Van Ginneken, M. Long, and M. Abramoff, “Comparative study of retinal vessel segmentation methods on a new publicly available database,” in Proc. SPIE 5370, 648–656 (2004).
[CrossRef]

Lowell, J.

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

Luo, S.

M. Park, J. S. Jin, and S. Luo, “Locating the optic disc in retinal images,” in Proceedings of the IEEE International Conference on Computer Graphics, Imaging and Visualization (IEEE, 2006), pp. 141–145.
[PubMed]

Markham, R.

A. Osareh, M. Mirmehdi, B. Thomas, and R. Markham, “Automated identification of diabetic retinal exudates in digital colour images,” Br. J. Ophthalmol. 87, 1220–1223(2003).
[CrossRef] [PubMed]

A. Osareh, M. Mirmehdi, B. Thomas, and R. Markham, “Classification and localisation of diabetic-related eye disease,” in 7th European Conference on Computer Vision (ECCV), Vol. 2353 ( 2002), 502–516.

Martinez-Perez, M. E.

M. E. Martinez-Perez, “Computer analysis of the geometry of the retinal vasculature,” Ph.D. thesis (Imperial College of Science, Technology and Medicine, 2000).

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, A. A. Bharath, and K. H. Parker, “Scale-space analysis for the characterization of retinal blood vessels,” in BMVA Proceedings of Medical Image Understanding and Analysis (Oxford, 1999), pp. 57–60.

Mendels, F.

F. Mendels, C. Heneghan, and J. P. Thiran, “Identification of the optic disc boundary in retinal images using active contours,” in IEEE Proceedings of the Irish Machine Vision Image Processing Conference (IMVIP) (IEEE, 1999), pp. 103–115.

Meunier, I.

F. Zana, I. Meunier, and J. C. Klein, “A region merging algorithm using mathematical morphology: application to macula detection,” in International Symposium on Mathematical Morphology (Kluwer, 1998), pp. 423–430.

Michelson, G.

R. Chrastek, M. Wolf, K. Donath, G. Michelson, and H. Niemann, “Optic disc segmentation in retinal images,” in Bildverarbeitung für die Medizin (Springer-Verlag, 2002), pp. 263–266.

Mirmehdi, M.

A. Osareh, M. Mirmehdi, B. Thomas, and R. Markham, “Automated identification of diabetic retinal exudates in digital colour images,” Br. J. Ophthalmol. 87, 1220–1223(2003).
[CrossRef] [PubMed]

A. Osareh, M. Mirmehdi, B. Thomas, and R. Markham, “Classification and localisation of diabetic-related eye disease,” in 7th European Conference on Computer Vision (ECCV), Vol. 2353 ( 2002), 502–516.

Moezzi, S.

M. Goldbaum, S. Moezzi, S. Taylor, S. Chatterjee, J. Boyd, E. Hunter, and R. Jain, “Automated diagnosis and image understanding with object extraction, object classification, and inferencing in retinal images,” in Proceedings of the IEEE International Conference on Image Processing, Vol. 3 (IEEE, 1996), pp. 695–698.
[CrossRef]

Mojon, D.

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

Nancy, S. A.

S. A. Salem, S. A. Nancy, and A. K. Nandi, “Segmentation of retinal blood vessels using a novel clustering algorithm (racal) with a partial supervision strategy,” Med. Biol. Eng. Comput. 45, 261–273 (2007).
[CrossRef] [PubMed]

S. A. Nancy, S. A. Salem, and A. K. Nandi, “Segmentation of retinal blood vessels based on analysis of the Hessian matrix and clustering algorithm,” in Proceedings of the European Signal Processing Conference (EUSIPCO) (EURASIP, 2007), pp. 428–432.

Nandi, A. K.

S. A. Salem, S. A. Nancy, and A. K. Nandi, “Segmentation of retinal blood vessels using a novel clustering algorithm (racal) with a partial supervision strategy,” Med. Biol. Eng. Comput. 45, 261–273 (2007).
[CrossRef] [PubMed]

S. A. Nancy, S. A. Salem, and A. K. Nandi, “Segmentation of retinal blood vessels based on analysis of the Hessian matrix and clustering algorithm,” in Proceedings of the European Signal Processing Conference (EUSIPCO) (EURASIP, 2007), pp. 428–432.

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]

Niemann, H.

R. Chrastek, M. Wolf, K. Donath, G. Michelson, and H. Niemann, “Optic disc segmentation in retinal images,” in Bildverarbeitung für die Medizin (Springer-Verlag, 2002), pp. 263–266.

Niemeijer, M.

M. Niemeijer, J. Staal, B. Van Ginneken, M. Long, and M. Abramoff, “Comparative study of retinal vessel segmentation methods on a new publicly available database,” in Proc. SPIE 5370, 648–656 (2004).
[CrossRef]

J. Staal, M. D. Abramoff, M. Niemeijer, M. A. Viergever, and B. V. Ginneken, “Ridge based vessel segmentation in color images of the retina,” IEEE Trans. Med. Imaging 23, 501–509 (2004).
[CrossRef] [PubMed]

Osareh, A.

A. Osareh, M. Mirmehdi, B. Thomas, and R. Markham, “Automated identification of diabetic retinal exudates in digital colour images,” Br. J. Ophthalmol. 87, 1220–1223(2003).
[CrossRef] [PubMed]

A. Osareh, “Automated identification of diabetic retinal exudates and the optic disc,” Ph.D. thesis (University of Bristol, 2004).

A. Osareh, M. Mirmehdi, B. Thomas, and R. Markham, “Classification and localisation of diabetic-related eye disease,” in 7th European Conference on Computer Vision (ECCV), Vol. 2353 ( 2002), 502–516.

Park, M.

M. Park, J. S. Jin, and S. Luo, “Locating the optic disc in retinal images,” in Proceedings of the IEEE International Conference on Computer Graphics, Imaging and Visualization (IEEE, 2006), pp. 141–145.
[PubMed]

Parker, K. H.

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, A. A. Bharath, and K. H. Parker, “Scale-space analysis for the characterization of retinal blood vessels,” in BMVA Proceedings of Medical Image Understanding and Analysis (Oxford, 1999), pp. 57–60.

Ryder, R.

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

Rzeszotarski, M.

L. Zhou, M. Rzeszotarski, L. Singerman, and J. Chokreff, “The detection and quantification of retinopathy using digital angiograms,” IEEE Trans. Med. Imaging 13, 619–626(1994).
[CrossRef] [PubMed]

Salem, S. A.

S. A. Salem, S. A. Nancy, and A. K. Nandi, “Segmentation of retinal blood vessels using a novel clustering algorithm (racal) with a partial supervision strategy,” Med. Biol. Eng. Comput. 45, 261–273 (2007).
[CrossRef] [PubMed]

S. A. Nancy, S. A. Salem, and A. K. Nandi, “Segmentation of retinal blood vessels based on analysis of the Hessian matrix and clustering algorithm,” in Proceedings of the European Signal Processing Conference (EUSIPCO) (EURASIP, 2007), pp. 428–432.

Serra, J.

J. Serra and L. Vincent, “An overview of morphological filtering,” Circuits Syst. Signal Process. 11, 47–108(1992).
[CrossRef]

Simo, A.

M. Ibanez and A. Simo, “Bayesian detection of the fovea in eye fundus angiographies,” Pattern Recogn. Lett. 20, 229–240(1999).
[CrossRef]

Singerman, L.

L. Zhou, M. Rzeszotarski, L. Singerman, and J. Chokreff, “The detection and quantification of retinopathy using digital angiograms,” IEEE Trans. Med. Imaging 13, 619–626(1994).
[CrossRef] [PubMed]

Sinthanayothin, C.

C. Sinthanayothin, J. F. Boyce, H. L. Cook, and T. H. Williamson, “Automatic localisation of the optic disc, fovea, and retinal blood vessels from digital colour fundus images,” Br.J. Ophthalmol. 83, 902–910 (1999).
[CrossRef] [PubMed]

Staal, J.

M. Niemeijer, J. Staal, B. Van Ginneken, M. Long, and M. Abramoff, “Comparative study of retinal vessel segmentation methods on a new publicly available database,” in Proc. SPIE 5370, 648–656 (2004).
[CrossRef]

J. Staal, M. D. Abramoff, M. Niemeijer, M. A. Viergever, and B. V. Ginneken, “Ridge based vessel segmentation in color images of the retina,” IEEE Trans. Med. Imaging 23, 501–509 (2004).
[CrossRef] [PubMed]

Stanton, A. V.

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, A. A. Bharath, and K. H. Parker, “Scale-space analysis for the characterization of retinal blood vessels,” in BMVA Proceedings of Medical Image Understanding and Analysis (Oxford, 1999), pp. 57–60.

Steel, D.

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

Taylor, S.

M. Goldbaum, S. Moezzi, S. Taylor, S. Chatterjee, J. Boyd, E. Hunter, and R. Jain, “Automated diagnosis and image understanding with object extraction, object classification, and inferencing in retinal images,” in Proceedings of the IEEE International Conference on Image Processing, Vol. 3 (IEEE, 1996), pp. 695–698.
[CrossRef]

ter Haar, F.

F. ter Haar, “Automatic localization of the optic disc in digital colour images of the human retina,” Master’s thesis (Utrecht University, 2005).

Thiran, J. P.

F. Mendels, C. Heneghan, and J. P. Thiran, “Identification of the optic disc boundary in retinal images using active contours,” in IEEE Proceedings of the Irish Machine Vision Image Processing Conference (IMVIP) (IEEE, 1999), pp. 103–115.

Thom, S. A.

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, A. A. Bharath, and K. H. Parker, “Scale-space analysis for the characterization of retinal blood vessels,” in BMVA Proceedings of Medical Image Understanding and Analysis (Oxford, 1999), pp. 57–60.

Thomas, B.

A. Osareh, M. Mirmehdi, B. Thomas, and R. Markham, “Automated identification of diabetic retinal exudates in digital colour images,” Br. J. Ophthalmol. 87, 1220–1223(2003).
[CrossRef] [PubMed]

A. Osareh, M. Mirmehdi, B. Thomas, and R. Markham, “Classification and localisation of diabetic-related eye disease,” in 7th European Conference on Computer Vision (ECCV), Vol. 2353 ( 2002), 502–516.

Van Ginneken, B.

M. Niemeijer, J. Staal, B. Van Ginneken, M. Long, and M. Abramoff, “Comparative study of retinal vessel segmentation methods on a new publicly available database,” in Proc. SPIE 5370, 648–656 (2004).
[CrossRef]

Viergever, M. A.

J. Staal, M. D. Abramoff, M. Niemeijer, M. A. Viergever, and B. V. Ginneken, “Ridge based vessel segmentation in color images of the retina,” IEEE Trans. Med. Imaging 23, 501–509 (2004).
[CrossRef] [PubMed]

Vincent, L.

J. Serra and L. Vincent, “An overview of morphological filtering,” Circuits Syst. Signal Process. 11, 47–108(1992).
[CrossRef]

Walter, T.

T. Walter and J. C. Klein, “Segmentation of colour fundus images of the human retina: Detection of the optic disc and the vascular tree using morphological techniques,” in Proceedings of the 2nd International Symposium on Medical Data Analysis, LNCS 2199 (Springer-Verlag, 2001), pp. 282–287.

Williamson, T. H.

C. Sinthanayothin, J. F. Boyce, H. L. Cook, and T. H. Williamson, “Automatic localisation of the optic disc, fovea, and retinal blood vessels from digital colour fundus images,” Br.J. Ophthalmol. 83, 902–910 (1999).
[CrossRef] [PubMed]

Wolf, M.

R. Chrastek, M. Wolf, K. Donath, G. Michelson, and H. Niemann, “Optic disc segmentation in retinal images,” in Bildverarbeitung für die Medizin (Springer-Verlag, 2002), pp. 263–266.

Wu, D.

D. Wu, M. Zhang, J. Liu, and W. Bauman, “On the adaptive detection of blood vessels in retinal images,” IEEE Trans. Biomed. Eng. 53, 341–343 (2006).
[CrossRef] [PubMed]

Ying, H.

H. Ying, M. Zhang, and J. Liu, “Fractal-based automatic localization and segmentation of optic disc in retinal images,” in Proceedings of the IEEE 29th Annual International Conference of the Engineering in Medicine and Biology Society (IEEE, 2007), pp. 4139–4141.
[CrossRef] [PubMed]

Youssif, A.

A. Youssif, A. Ghalwash, and A. Ghoneim, “Optic disc detection from normalized digital fundus images by means of a vessels’ direction matched filter,” IEEE Trans. Med. Imaging 27, 11–18 (2008).
[CrossRef] [PubMed]

Zana, F.

F. Zana, I. Meunier, and J. C. Klein, “A region merging algorithm using mathematical morphology: application to macula detection,” in International Symposium on Mathematical Morphology (Kluwer, 1998), pp. 423–430.

Zhang, M.

D. Wu, M. Zhang, J. Liu, and W. Bauman, “On the adaptive detection of blood vessels in retinal images,” IEEE Trans. Biomed. Eng. 53, 341–343 (2006).
[CrossRef] [PubMed]

H. Ying, M. Zhang, and J. Liu, “Fractal-based automatic localization and segmentation of optic disc in retinal images,” in Proceedings of the IEEE 29th Annual International Conference of the Engineering in Medicine and Biology Society (IEEE, 2007), pp. 4139–4141.
[CrossRef] [PubMed]

Zheng, L.

O. Chutatape, L. Zheng, and S. Krishnan, “Retinal blood vessel detection and tracking by matched Gaussian and Kalman filters,” in Proceedings of the 20th Annual International Conference of the IEEE on Engineering in Medicine and Biology Society, Vol. 6 (IEEE, 1998), pp. 3144–3149.

Zhou, L.

L. Zhou, M. Rzeszotarski, L. Singerman, and J. Chokreff, “The detection and quantification of retinopathy using digital angiograms,” IEEE Trans. Med. Imaging 13, 619–626(1994).
[CrossRef] [PubMed]

Br. J. Ophthalmol. (1)

A. Osareh, M. Mirmehdi, B. Thomas, and R. Markham, “Automated identification of diabetic retinal exudates in digital colour images,” Br. J. Ophthalmol. 87, 1220–1223(2003).
[CrossRef] [PubMed]

Br.J. Ophthalmol. (1)

C. Sinthanayothin, J. F. Boyce, H. L. Cook, and T. H. Williamson, “Automatic localisation of the optic disc, fovea, and retinal blood vessels from digital colour fundus images,” Br.J. Ophthalmol. 83, 902–910 (1999).
[CrossRef] [PubMed]

Circuits Syst. Signal Process. (1)

J. Serra and L. Vincent, “An overview of morphological filtering,” Circuits Syst. Signal Process. 11, 47–108(1992).
[CrossRef]

Commun. ACM (1)

R. O. Duda and P. E. Hart, “Use of Hough transformation to detect lines and curves in pictures,” Commun. ACM 15, 11–15 (1972).
[CrossRef]

IEEE Trans. Biomed. Eng. (3)

D. Wu, M. Zhang, J. Liu, and W. Bauman, “On the adaptive detection of blood vessels in retinal images,” IEEE Trans. Biomed. Eng. 53, 341–343 (2006).
[CrossRef] [PubMed]

A. Hoover and M. Goldbaum, “Locating the optic nerve in a retinal image using the fuzzy convergence of the blood vessels,” IEEE Trans. Biomed. Eng. 22, 951–958 (2003).

H. Li and O. Chutatape, “Automated feature extraction in colour retinal images by a model based approach,” IEEE Trans. Biomed. Eng. 51, 246–254 (2004).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging (8)

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]

L. Zhou, M. Rzeszotarski, L. Singerman, and J. Chokreff, “The detection and quantification of retinopathy using digital angiograms,” IEEE Trans. Med. Imaging 13, 619–626(1994).
[CrossRef] [PubMed]

M. Foracchia and E. A. R. Grisan, “Detection of optic disc in retinal images by means of a geometrical model of vessel structure,” IEEE Trans. Med. Imaging 23, 1189–1195 (2004).
[CrossRef] [PubMed]

J. Staal, M. D. Abramoff, M. Niemeijer, M. A. Viergever, and B. V. Ginneken, “Ridge based vessel segmentation in color images of the retina,” IEEE Trans. Med. Imaging 23, 501–509 (2004).
[CrossRef] [PubMed]

M. Lalonde, M. Beaulieu, and L. Gagnon, “Fast and robust optic disc detection using pyramidal decomposition and Hausdorff-based template matching,” IEEE Trans. Med. Imaging 20, 1193–1200 (2001).
[CrossRef] [PubMed]

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

A. Youssif, A. Ghalwash, and A. Ghoneim, “Optic disc detection from normalized digital fundus images by means of a vessels’ direction matched filter,” IEEE Trans. Med. Imaging 27, 11–18 (2008).
[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]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

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

Med. Biol. Eng. Comput. (1)

S. A. Salem, S. A. Nancy, and A. K. Nandi, “Segmentation of retinal blood vessels using a novel clustering algorithm (racal) with a partial supervision strategy,” Med. Biol. Eng. Comput. 45, 261–273 (2007).
[CrossRef] [PubMed]

Pattern Recogn. Lett. (1)

M. Ibanez and A. Simo, “Bayesian detection of the fovea in eye fundus angiographies,” Pattern Recogn. Lett. 20, 229–240(1999).
[CrossRef]

Pattern Recognition (1)

K. Akita and H. Kuga, “A computer method of understanding ocular fundus images,” Pattern Recognition 15, 431–443(1982).
[CrossRef]

Proc. SPIE (1)

M. Niemeijer, J. Staal, B. Van Ginneken, M. Long, and M. Abramoff, “Comparative study of retinal vessel segmentation methods on a new publicly available database,” in Proc. SPIE 5370, 648–656 (2004).
[CrossRef]

Other (18)

F. Zana, I. Meunier, and J. C. Klein, “A region merging algorithm using mathematical morphology: application to macula detection,” in International Symposium on Mathematical Morphology (Kluwer, 1998), pp. 423–430.

F. Mendels, C. Heneghan, and J. P. Thiran, “Identification of the optic disc boundary in retinal images using active contours,” in IEEE Proceedings of the Irish Machine Vision Image Processing Conference (IMVIP) (IEEE, 1999), pp. 103–115.

T. Walter and J. C. Klein, “Segmentation of colour fundus images of the human retina: Detection of the optic disc and the vascular tree using morphological techniques,” in Proceedings of the 2nd International Symposium on Medical Data Analysis, LNCS 2199 (Springer-Verlag, 2001), pp. 282–287.

O. Chutatape, L. Zheng, and S. Krishnan, “Retinal blood vessel detection and tracking by matched Gaussian and Kalman filters,” in Proceedings of the 20th Annual International Conference of the IEEE on Engineering in Medicine and Biology Society, Vol. 6 (IEEE, 1998), pp. 3144–3149.

M. Goldbaum, S. Moezzi, S. Taylor, S. Chatterjee, J. Boyd, E. Hunter, and R. Jain, “Automated diagnosis and image understanding with object extraction, object classification, and inferencing in retinal images,” in Proceedings of the IEEE International Conference on Image Processing, Vol. 3 (IEEE, 1996), pp. 695–698.
[CrossRef]

M. Park, J. S. Jin, and S. Luo, “Locating the optic disc in retinal images,” in Proceedings of the IEEE International Conference on Computer Graphics, Imaging and Visualization (IEEE, 2006), pp. 141–145.
[PubMed]

M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, A. A. Bharath, and K. H. Parker, “Scale-space analysis for the characterization of retinal blood vessels,” in BMVA Proceedings of Medical Image Understanding and Analysis (Oxford, 1999), pp. 57–60.

M. E. Martinez-Perez, “Computer analysis of the geometry of the retinal vasculature,” Ph.D. thesis (Imperial College of Science, Technology and Medicine, 2000).

T. Lindeberg, Scale-Space Theory in Computer Vision (Kluwer, 1994).

S. A. Nancy, S. A. Salem, and A. K. Nandi, “Segmentation of retinal blood vessels based on analysis of the Hessian matrix and clustering algorithm,” in Proceedings of the European Signal Processing Conference (EUSIPCO) (EURASIP, 2007), pp. 428–432.

R. Chrastek, M. Wolf, K. Donath, G. Michelson, and H. Niemann, “Optic disc segmentation in retinal images,” in Bildverarbeitung für die Medizin (Springer-Verlag, 2002), pp. 263–266.

A. Osareh, M. Mirmehdi, B. Thomas, and R. Markham, “Classification and localisation of diabetic-related eye disease,” in 7th European Conference on Computer Vision (ECCV), Vol. 2353 ( 2002), 502–516.

Clemson University: STARE project. http://www.ces.clemson.edu/ ahoover/stare.

F. Damian, ARIA (2006), http://www.eyecharity.com/aria_online/.

O. Chutatape, “Fundus foveal localization based on vessel model,” in Proceedings of the IEEE International Conference of the Engineering in Medicine and Biology Society (IEEE, 2006), pp. 4440–4444.
[CrossRef]

H. Ying, M. Zhang, and J. Liu, “Fractal-based automatic localization and segmentation of optic disc in retinal images,” in Proceedings of the IEEE 29th Annual International Conference of the Engineering in Medicine and Biology Society (IEEE, 2007), pp. 4139–4141.
[CrossRef] [PubMed]

F. ter Haar, “Automatic localization of the optic disc in digital colour images of the human retina,” Master’s thesis (Utrecht University, 2005).

A. Osareh, “Automated identification of diabetic retinal exudates and the optic disc,” Ph.D. thesis (University of Bristol, 2004).

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Figures (15)

Fig. 1
Fig. 1

Fundus image showing main retinal features.

Fig. 2
Fig. 2

Preprocessing of retinal image: (a) Green channel. (b) Shade corrected.

Fig. 3
Fig. 3

(a) Red channel. (b) Pixel intensities corresponding to the white line crossing the image. (c) Intensity gradient. (d) Ridge strength. (e) Magnitude of λ + . (f) Green channel. (g) Pixel intensities corresponding to the white line crossing the image. (h) Intensity gradient. (i) Ridge strength. (j) Magnitude of λ + .

Fig. 4
Fig. 4

(a) Retinal image with different width blood vessels. (b) Magnitude of λ + of the Hessian at all image pixels with s = 1 . (c) Magnitude of λ + of the Hessian at all image pixels with s = 6 . (d) Point-by-point maximum over all λ + images for all scales.

Fig. 5
Fig. 5

Eigenvectors at different scales. (a)  s = 1 , (b)  s = 6 .

Fig. 6
Fig. 6

uv coordinate system. The u-axis is the tangent direction to the vessel curve, the v-axis is orthogonal to the vessel direction.

Fig. 7
Fig. 7

Thin blood vessels removal. (a) Filtered blood vessel overlaid on original blood vessels, (b) skeletonized blood vessels after the removal of thin blood vessels.

Fig. 8
Fig. 8

Detected parabola overlaid on blood vessels.

Fig. 9
Fig. 9

Initial traced boundary using morphological operators. Plus sign denotes the center of the ROI.

Fig. 10
Fig. 10

Gradient of the ROI surrounding the optic disc.

Fig. 11
Fig. 11

Optic disc detected using two methods: by PHT in white and by CHT in black. Overlap percentage = 88 % .

Fig. 12
Fig. 12

Localized optic disc—Image 34 from the DRIVE dataset. (a) Entry and exit points. (b) Detected optic disc using Hedgehog processing with average disc diameter from the DRIVE database.

Fig. 13
Fig. 13

Geometry of the macula in the candidate ROI shown in white.

Fig. 14
Fig. 14

Foveal localization. (a) ROI for macula localization, (b) ROI after applying adaptive threshold, (c) smoothed macula, (d) identified macula and fovea.

Fig. 15
Fig. 15

Blood vessel segmentation. (a) Original image, (b) hand-labeled by human observer, (c) proposed method, and (d) Jiang [28].

Tables (2)

Tables Icon

Table 1 Performance of Blood Vessel Segmentation Using Subset of 20 Images of STARE

Tables Icon

Table 2 Result Comparison of Optic Disc Detection

Equations (23)

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

L eq ( x , y ) = L ( x , y ) + m A ( x , y ) ,
G ( x , y ; s ) = 1 2 π s 2 e ( x 2 + y 2 2 s 2 ) ,
L x = L ( x , y ) s G x ,
L y = L ( x , y ) s G y ,
L x x = L ( x , y ) s 2 G x x ,
L x y = L ( x , y ) s 2 G x y ,
L y y = L ( x , y ) s 2 G y y ,
H = L x x L x y L y x L y y .
| L x x λ L x y L y x L y y λ | = 0 ,
λ 2 ( L x x + L y y ) λ + ( L x x L y y L x y 2 ) = 0.
λ + = L x x + L y y + α 2 ,
λ = L x x + L y y α 2 ,
α = ( L x x L y y ) 2 + 4 L x y 2 .
e ^ + = 1 N | 2 L x y L y y L x x + α | ,
e ^ = 1 N | L y y L x x + α 2 L x y | ,
θ + = tan 1 L y y L x x + α 2 L x y ,
θ = tan 1 2 L x y L y y L x x + α .
V = λ max θ std + ϵ ,
u = x sin θ y cos θ ,
v = x sin θ + y cos θ ,
( y y c ) 2 = 2 a ( x x c ) ,
x = x 0 + r sin θ ,
y = y 0 + r cos θ ,

Metrics