Abstract

Proliferative diabetic retinopathy can lead to blindness. However, early recognition allows appropriate, timely intervention. Fluorescein-labeled retinal blood vessels of 27 digital images were automatically segmented using the Gabor wavelet transform and classified using traditional features such as area, perimeter, and an additional five morphological features based on the derivatives-of-Gaussian wavelet-derived data. Discriminant analysis indicated that traditional features do not detect early proliferative retinopathy. The best single feature for discrimination was the wavelet curvature with an area under the curve (AUC) of 0.76. Linear discriminant analysis with a selection of six features achieved an AUC of 0.90 (0.73–0.97, 95% confidence interval). The wavelet method was able to segment retinal blood vessels and classify the images according to the presence or absence of proliferative retinopathy.

© 2007 Optical Society of America

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2006 (1)

J. V. B. Soares, J. J. G. Leandro, R. M. Cesar, Jr., H. F. Jelinek, and M. J. Cree, "Retinal vessel segmentation using the 2-D Gabor wavelet and supervised classification," IEEE Trans. Med. Imaging 25, 1214-1222 (2006).
[CrossRef] [PubMed]

2005 (1)

D. Cornforth, H. F. Jelinek, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., M. J. Cree, P. Mitchell, and T. Bossomaier, "Evolution of retinal blood vessel segmentation using wavelet transform in assessment of diabetic retinopathy," Complexity International 11, 50-60 (2005), http://www.complexity.org.au/ci/vol11/.

2004 (4)

J. J. Staal, M. D. Abràmoff, M. Niemeijer, M. A. 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]

K. S. Coyne, M. K. Margolis, T. Kennedy-Martin, T. M. Baker, R. Klein, M. D. Paul, and D. A. Revicki, "The impact of diabetic retinopathy: perspectives from patient focus groups," Fam. Pract. 21, 447-453 (2004).
[CrossRef] [PubMed]

C. Kirbas and F. Quek, "A review of vessel extraction techniques and algorithms," ACM Comput. Surv. 32, 81-121 (2004).
[CrossRef]

B. R. Masters, "Fractal analysis of the vascular tree in the human retina," Annu. Rev. Biomed. Eng. 6, 427-452 (2004).
[CrossRef] [PubMed]

2003 (5)

H. F. Jelinek, R. M. Cesar, Jr., and J. J. G. Leandro, "Exploring wavelet transforms for morphological differentiation between functionally different cat retinal ganglion cells," Brain and Mind 4, 67-90 (2003).
[CrossRef]

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

L. F. Estrozi, L. G. Rios, A. G. Campos, R. M. Cesar, Jr., and L. d. F. Costa, "1D and 2D Fourier-based approaches to numeric curvature estimation and their comparative performance assessment," Digit. Signal Process. 13, 172-197 (2003).
[CrossRef]

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]

K. Viswanath and M. McGavin, "Diabetic retinopathy: clinical findings and management," Community Eye Health 16, 21-24 (2003).

2002 (1)

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

2001 (3)

H. R. Taylor and J. E. Keeffe, "World blindness: a 21st century perspective," Br. J. Ophthamol. 85, 261-266 (2001).
[CrossRef]

U. Freudentsin and J. Verne, "A national screening programme for diabetic retinopathy," BMJ 323, 4-5 (2001).
[CrossRef]

S. J. Lee, C. Sicari, C. A. Harper, H. R. Taylor, and J. E. Keeffe, "Program for the early detection of diabetic retinopathy: a two-year follow-up," Clin. Exp. Optom. 29, 12-25 (2001).

2000 (5)

X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thorn, "Quantification and characterisation of arteries in retinal images," Comput. Methods Programs Biomed. 63, 133-146 (2000).
[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]

F. Zana and J.-C. Klein, "Segmentation of vessel-like patterns using mathematical morphology and curvature evaluation," IEEE Trans. Image Process. 10, 1010-1019 (2000).
[CrossRef]

A. Arnéodo, N. Decoster, and S. G. Roux, "A wavelet-based method for multifractal image analysis. I. Methodology and test applications on isotropic and anisotropic random rough surfaces," Eur. Phys. J. B 15, 567-600 (2000).
[CrossRef]

J. Hipwell, F. Strachan, J. Olson, K. McHardy, P. Sharp, and J. Forrester, "Automated detection of microaneurysms in digital red-free photographs: a diabetic retinopathy screening tool," Diabetic Med. 17, 588-594 (2000).
[CrossRef] [PubMed]

1999 (2)

C. Sinthanayothin, J. F. Boyce, H. Cook, and T. Williamson, "Automated localisation of the optic disc, fovea and retinal blood vessels from digital colour fundus images" Br. J. Ophthamol. 83, 902-912 (1999).
[CrossRef]

P. Asvestas, G. K. Matsopoulos, and K. S. Nikita, "Estimating of fractal dimension of images using fixed mass approach," Pattern Recogn. Lett. 20, 347-354 (1999).
[CrossRef]

1997 (2)

M. J. Cree, J. A. Olson, K. McHardy, P. Sharp, and J. Forrester, "A fully automated comparative microaneurysm digital detection system," Eye 11, 622-628 (1997).
[CrossRef] [PubMed]

J. P. Antoine, D. Barache, R. M. Cesar, Jr., and L. F. Costa, "Shape characterization with the wavelet transform," Signal Process. 62, 265-290 (1997).
[CrossRef]

1993 (1)

J. P. Antoine, P. Carette, R. Murenzi, and B. Piette, "Image analysis with two-dimensional wavelet transform," Signal Process. 3l, 241-272 (1993).
[CrossRef]

1986 (1)

C. E. Metz, "ROC methodology in radiologic imaging," Invest. Radiol. 21, 720-733 (1986).
[CrossRef]

1984 (1)

J. P. Rigaut, "An empirical formulation relating boundary lengths to resolution in specimens showing 'non-ideally fractal' dimensions," J. Microsc. 133, 41-54 (1984).
[CrossRef]

1982 (1)

E. Sussman, W. Tsiaras, and K. Soper, "Diagnosis of diabetic eye disease," J. Am. Med. Assoc. 247, 3231-3234 (1982).
[CrossRef]

Abràmoff, M. D.

J. J. Staal, M. D. Abràmoff, M. Niemeijer, M. A. 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]

Antoine, J. P.

J. P. Antoine, D. Barache, R. M. Cesar, Jr., and L. F. Costa, "Shape characterization with the wavelet transform," Signal Process. 62, 265-290 (1997).
[CrossRef]

J. P. Antoine, P. Carette, R. Murenzi, and B. Piette, "Image analysis with two-dimensional wavelet transform," Signal Process. 3l, 241-272 (1993).
[CrossRef]

Arnéodo, A.

A. Arnéodo, N. Decoster, and S. G. Roux, "A wavelet-based method for multifractal image analysis. I. Methodology and test applications on isotropic and anisotropic random rough surfaces," Eur. Phys. J. B 15, 567-600 (2000).
[CrossRef]

Asvestas, P.

P. Asvestas, G. K. Matsopoulos, and K. S. Nikita, "Estimating of fractal dimension of images using fixed mass approach," Pattern Recogn. Lett. 20, 347-354 (1999).
[CrossRef]

Baker, T. M.

K. S. Coyne, M. K. Margolis, T. Kennedy-Martin, T. M. Baker, R. Klein, M. D. Paul, and D. A. Revicki, "The impact of diabetic retinopathy: perspectives from patient focus groups," Fam. Pract. 21, 447-453 (2004).
[CrossRef] [PubMed]

Barache, D.

J. P. Antoine, D. Barache, R. M. Cesar, Jr., and L. F. Costa, "Shape characterization with the wavelet transform," Signal Process. 62, 265-290 (1997).
[CrossRef]

Bharath, A.

X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thorn, "Quantification and characterisation of arteries in retinal images," Comput. Methods Programs Biomed. 63, 133-146 (2000).
[CrossRef] [PubMed]

Bharath, A. A.

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

Bossomaier, T.

D. Cornforth, H. F. Jelinek, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., M. J. Cree, P. Mitchell, and T. Bossomaier, "Evolution of retinal blood vessel segmentation using wavelet transform in assessment of diabetic retinopathy," Complexity International 11, 50-60 (2005), http://www.complexity.org.au/ci/vol11/.

Boyce, J. F.

C. Sinthanayothin, J. F. Boyce, H. Cook, and T. Williamson, "Automated localisation of the optic disc, fovea and retinal blood vessels from digital colour fundus images" Br. J. Ophthamol. 83, 902-912 (1999).
[CrossRef]

Campos, A. G.

L. F. Estrozi, L. G. Rios, A. G. Campos, R. M. Cesar, Jr., and L. d. F. Costa, "1D and 2D Fourier-based approaches to numeric curvature estimation and their comparative performance assessment," Digit. Signal Process. 13, 172-197 (2003).
[CrossRef]

Carette, P.

J. P. Antoine, P. Carette, R. Murenzi, and B. Piette, "Image analysis with two-dimensional wavelet transform," Signal Process. 3l, 241-272 (1993).
[CrossRef]

Cesar, R. M.

J. V. B. Soares, J. J. G. Leandro, R. M. Cesar, Jr., H. F. Jelinek, and M. J. Cree, "Retinal vessel segmentation using the 2-D Gabor wavelet and supervised classification," IEEE Trans. Med. Imaging 25, 1214-1222 (2006).
[CrossRef] [PubMed]

D. Cornforth, H. F. Jelinek, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., M. J. Cree, P. Mitchell, and T. Bossomaier, "Evolution of retinal blood vessel segmentation using wavelet transform in assessment of diabetic retinopathy," Complexity International 11, 50-60 (2005), http://www.complexity.org.au/ci/vol11/.

L. F. Estrozi, L. G. Rios, A. G. Campos, R. M. Cesar, Jr., and L. d. F. Costa, "1D and 2D Fourier-based approaches to numeric curvature estimation and their comparative performance assessment," Digit. Signal Process. 13, 172-197 (2003).
[CrossRef]

H. F. Jelinek, R. M. Cesar, Jr., and J. J. G. Leandro, "Exploring wavelet transforms for morphological differentiation between functionally different cat retinal ganglion cells," Brain and Mind 4, 67-90 (2003).
[CrossRef]

J. P. Antoine, D. Barache, R. M. Cesar, Jr., and L. F. Costa, "Shape characterization with the wavelet transform," Signal Process. 62, 265-290 (1997).
[CrossRef]

J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., and H. F. Jelinek, "Blood vessel segmentation of non-mydriatic images using wavelets and statistical classifiers," in Proceedings of the Brazilian Conference on Computer Graphics, Image Processing and Vision (Sibgrapi, 2003), pp. 262-269.

R. M. Cesar, Jr., and H. F. Jelinek, "Segmentation of retinal fundus vasculature in nonmydriatic camera images using wavelets," in Angiography and Plaque Imaging, J.S.Suri and S.Laxminarayan, eds. (CRC, 2003), pp. 193-224.

L. F. Costa and R. M. Cesar, Jr., Shape Analysis and Classification: Theory and Practice (CRC, 2001).

M. J. Cree, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., H. F. Jelinek, and D. Cornforth, "Comparison of various methods to delineate blood vessels in retinal images," in Proceedings of the 16th Australian Institute of Physics Congress (AIP, 2005) pp. 1-4.

Chapman, N.

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

X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thorn, "Quantification and characterisation of arteries in retinal images," Comput. Methods Programs Biomed. 63, 133-146 (2000).
[CrossRef] [PubMed]

Cook, H.

C. Sinthanayothin, J. F. Boyce, H. Cook, and T. Williamson, "Automated localisation of the optic disc, fovea and retinal blood vessels from digital colour fundus images" Br. J. Ophthamol. 83, 902-912 (1999).
[CrossRef]

Cornforth, D.

D. Cornforth, H. F. Jelinek, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., M. J. Cree, P. Mitchell, and T. Bossomaier, "Evolution of retinal blood vessel segmentation using wavelet transform in assessment of diabetic retinopathy," Complexity International 11, 50-60 (2005), http://www.complexity.org.au/ci/vol11/.

M. J. Cree, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., H. F. Jelinek, and D. Cornforth, "Comparison of various methods to delineate blood vessels in retinal images," in Proceedings of the 16th Australian Institute of Physics Congress (AIP, 2005) pp. 1-4.

Costa, L. d. F.

L. F. Estrozi, L. G. Rios, A. G. Campos, R. M. Cesar, Jr., and L. d. F. Costa, "1D and 2D Fourier-based approaches to numeric curvature estimation and their comparative performance assessment," Digit. Signal Process. 13, 172-197 (2003).
[CrossRef]

Costa, L. F.

J. P. Antoine, D. Barache, R. M. Cesar, Jr., and L. F. Costa, "Shape characterization with the wavelet transform," Signal Process. 62, 265-290 (1997).
[CrossRef]

L. F. Costa and R. M. Cesar, Jr., Shape Analysis and Classification: Theory and Practice (CRC, 2001).

Coyne, K. S.

K. S. Coyne, M. K. Margolis, T. Kennedy-Martin, T. M. Baker, R. Klein, M. D. Paul, and D. A. Revicki, "The impact of diabetic retinopathy: perspectives from patient focus groups," Fam. Pract. 21, 447-453 (2004).
[CrossRef] [PubMed]

Cree, M. J.

J. V. B. Soares, J. J. G. Leandro, R. M. Cesar, Jr., H. F. Jelinek, and M. J. Cree, "Retinal vessel segmentation using the 2-D Gabor wavelet and supervised classification," IEEE Trans. Med. Imaging 25, 1214-1222 (2006).
[CrossRef] [PubMed]

D. Cornforth, H. F. Jelinek, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., M. J. Cree, P. Mitchell, and T. Bossomaier, "Evolution of retinal blood vessel segmentation using wavelet transform in assessment of diabetic retinopathy," Complexity International 11, 50-60 (2005), http://www.complexity.org.au/ci/vol11/.

M. J. Cree, J. A. Olson, K. McHardy, P. Sharp, and J. Forrester, "A fully automated comparative microaneurysm digital detection system," Eye 11, 622-628 (1997).
[CrossRef] [PubMed]

M. J. Cree, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., H. F. Jelinek, and D. Cornforth, "Comparison of various methods to delineate blood vessels in retinal images," in Proceedings of the 16th Australian Institute of Physics Congress (AIP, 2005) pp. 1-4.

Decoster, N.

A. Arnéodo, N. Decoster, and S. G. Roux, "A wavelet-based method for multifractal image analysis. I. Methodology and test applications on isotropic and anisotropic random rough surfaces," Eur. Phys. J. B 15, 567-600 (2000).
[CrossRef]

Duda, R. O.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, 2nd ed. (Wiley Interscience, 2001).

Eddins, S. L.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using Matlab (Pearson Prentice Hall, 2004).

Estrozi, L. F.

L. F. Estrozi, L. G. Rios, A. G. Campos, R. M. Cesar, Jr., and L. d. F. Costa, "1D and 2D Fourier-based approaches to numeric curvature estimation and their comparative performance assessment," Digit. Signal Process. 13, 172-197 (2003).
[CrossRef]

Forrester, J.

J. Hipwell, F. Strachan, J. Olson, K. McHardy, P. Sharp, and J. Forrester, "Automated detection of microaneurysms in digital red-free photographs: a diabetic retinopathy screening tool," Diabetic Med. 17, 588-594 (2000).
[CrossRef] [PubMed]

M. J. Cree, J. A. Olson, K. McHardy, P. Sharp, and J. Forrester, "A fully automated comparative microaneurysm digital detection system," Eye 11, 622-628 (1997).
[CrossRef] [PubMed]

Freudentsin, U.

U. Freudentsin and J. Verne, "A national screening programme for diabetic retinopathy," BMJ 323, 4-5 (2001).
[CrossRef]

Gao, X. W.

X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thorn, "Quantification and characterisation of arteries in retinal images," Comput. Methods Programs Biomed. 63, 133-146 (2000).
[CrossRef] [PubMed]

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]

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using Matlab (Pearson Prentice Hall, 2004).

Grossmann, A.

A. Grossmann, "Wavelet transforms and edge detection," in Stochastic Processes in Physics and Engineering, S.Albeverio, P.Blanchard, M.Hazewinkel, and L.Streit, eds. (Reidel, 1988), pp. 149-157.
[CrossRef]

Harper, C. A.

S. J. Lee, C. Sicari, C. A. Harper, H. R. Taylor, and J. E. Keeffe, "Program for the early detection of diabetic retinopathy: a two-year follow-up," Clin. Exp. Optom. 29, 12-25 (2001).

Hart, P. E.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, 2nd ed. (Wiley Interscience, 2001).

Hipwell, J.

J. Hipwell, F. Strachan, J. Olson, K. McHardy, P. Sharp, and J. Forrester, "Automated detection of microaneurysms in digital red-free photographs: a diabetic retinopathy screening tool," Diabetic Med. 17, 588-594 (2000).
[CrossRef] [PubMed]

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]

Hughes, A.

X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thorn, "Quantification and characterisation of arteries in retinal images," Comput. Methods Programs Biomed. 63, 133-146 (2000).
[CrossRef] [PubMed]

Hughes, A. D.

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

Jelinek, H. F.

J. V. B. Soares, J. J. G. Leandro, R. M. Cesar, Jr., H. F. Jelinek, and M. J. Cree, "Retinal vessel segmentation using the 2-D Gabor wavelet and supervised classification," IEEE Trans. Med. Imaging 25, 1214-1222 (2006).
[CrossRef] [PubMed]

D. Cornforth, H. F. Jelinek, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., M. J. Cree, P. Mitchell, and T. Bossomaier, "Evolution of retinal blood vessel segmentation using wavelet transform in assessment of diabetic retinopathy," Complexity International 11, 50-60 (2005), http://www.complexity.org.au/ci/vol11/.

H. F. Jelinek, R. M. Cesar, Jr., and J. J. G. Leandro, "Exploring wavelet transforms for morphological differentiation between functionally different cat retinal ganglion cells," Brain and Mind 4, 67-90 (2003).
[CrossRef]

R. M. Cesar, Jr., and H. F. Jelinek, "Segmentation of retinal fundus vasculature in nonmydriatic camera images using wavelets," in Angiography and Plaque Imaging, J.S.Suri and S.Laxminarayan, eds. (CRC, 2003), pp. 193-224.

J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., and H. F. Jelinek, "Blood vessel segmentation of non-mydriatic images using wavelets and statistical classifiers," in Proceedings of the Brazilian Conference on Computer Graphics, Image Processing and Vision (Sibgrapi, 2003), pp. 262-269.

M. J. Cree, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., H. F. Jelinek, and D. Cornforth, "Comparison of various methods to delineate blood vessels in retinal images," in Proceedings of the 16th Australian Institute of Physics Congress (AIP, 2005) pp. 1-4.

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]

Kanski, J.

J. Kanski, Clinical Ophthalmology: A Systematic Approach (Butterworth-Heinemann, 1989), pp. 465-479.

Keeffe, J. E.

H. R. Taylor and J. E. Keeffe, "World blindness: a 21st century perspective," Br. J. Ophthamol. 85, 261-266 (2001).
[CrossRef]

S. J. Lee, C. Sicari, C. A. Harper, H. R. Taylor, and J. E. Keeffe, "Program for the early detection of diabetic retinopathy: a two-year follow-up," Clin. Exp. Optom. 29, 12-25 (2001).

Kennedy-Martin, T.

K. S. Coyne, M. K. Margolis, T. Kennedy-Martin, T. M. Baker, R. Klein, M. D. Paul, and D. A. Revicki, "The impact of diabetic retinopathy: perspectives from patient focus groups," Fam. Pract. 21, 447-453 (2004).
[CrossRef] [PubMed]

Kirbas, C.

C. Kirbas and F. Quek, "A review of vessel extraction techniques and algorithms," ACM Comput. Surv. 32, 81-121 (2004).
[CrossRef]

Klein, J. C.

T. Walter, J. C. Klein, P. Massin, and F. Zana, "Automated segmentation and registration of retinal fluorescein angiographies," presented at the First International Workshop on Computer Assisted Fundus Image Analysis (CAFIA), Herlev Hospital, Copenhagen, Denmark, May 29-30, 2000.

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 (2000).
[CrossRef]

Klein, R.

K. S. Coyne, M. K. Margolis, T. Kennedy-Martin, T. M. Baker, R. Klein, M. D. Paul, and D. A. Revicki, "The impact of diabetic retinopathy: perspectives from patient focus groups," Fam. Pract. 21, 447-453 (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]

Leandro, J. J. G.

J. V. B. Soares, J. J. G. Leandro, R. M. Cesar, Jr., H. F. Jelinek, and M. J. Cree, "Retinal vessel segmentation using the 2-D Gabor wavelet and supervised classification," IEEE Trans. Med. Imaging 25, 1214-1222 (2006).
[CrossRef] [PubMed]

D. Cornforth, H. F. Jelinek, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., M. J. Cree, P. Mitchell, and T. Bossomaier, "Evolution of retinal blood vessel segmentation using wavelet transform in assessment of diabetic retinopathy," Complexity International 11, 50-60 (2005), http://www.complexity.org.au/ci/vol11/.

H. F. Jelinek, R. M. Cesar, Jr., and J. J. G. Leandro, "Exploring wavelet transforms for morphological differentiation between functionally different cat retinal ganglion cells," Brain and Mind 4, 67-90 (2003).
[CrossRef]

J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., and H. F. Jelinek, "Blood vessel segmentation of non-mydriatic images using wavelets and statistical classifiers," in Proceedings of the Brazilian Conference on Computer Graphics, Image Processing and Vision (Sibgrapi, 2003), pp. 262-269.

M. J. Cree, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., H. F. Jelinek, and D. Cornforth, "Comparison of various methods to delineate blood vessels in retinal images," in Proceedings of the 16th Australian Institute of Physics Congress (AIP, 2005) pp. 1-4.

Lee, S. J.

S. J. Lee, C. Sicari, C. A. Harper, H. R. Taylor, and J. E. Keeffe, "Program for the early detection of diabetic retinopathy: a two-year follow-up," Clin. Exp. Optom. 29, 12-25 (2001).

Margolis, M. K.

K. S. Coyne, M. K. Margolis, T. Kennedy-Martin, T. M. Baker, R. Klein, M. D. Paul, and D. A. Revicki, "The impact of diabetic retinopathy: perspectives from patient focus groups," Fam. Pract. 21, 447-453 (2004).
[CrossRef] [PubMed]

Markham, R.

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

Martinez-Perez, M. E.

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

Massin, P.

T. Walter, J. C. Klein, P. Massin, and F. Zana, "Automated segmentation and registration of retinal fluorescein angiographies," presented at the First International Workshop on Computer Assisted Fundus Image Analysis (CAFIA), Herlev Hospital, Copenhagen, Denmark, May 29-30, 2000.

Masters, B. R.

B. R. Masters, "Fractal analysis of the vascular tree in the human retina," Annu. Rev. Biomed. Eng. 6, 427-452 (2004).
[CrossRef] [PubMed]

Matsopoulos, G. K.

P. Asvestas, G. K. Matsopoulos, and K. S. Nikita, "Estimating of fractal dimension of images using fixed mass approach," Pattern Recogn. Lett. 20, 347-354 (1999).
[CrossRef]

McGavin, M.

K. Viswanath and M. McGavin, "Diabetic retinopathy: clinical findings and management," Community Eye Health 16, 21-24 (2003).

McHardy, K.

J. Hipwell, F. Strachan, J. Olson, K. McHardy, P. Sharp, and J. Forrester, "Automated detection of microaneurysms in digital red-free photographs: a diabetic retinopathy screening tool," Diabetic Med. 17, 588-594 (2000).
[CrossRef] [PubMed]

M. J. Cree, J. A. Olson, K. McHardy, P. Sharp, and J. Forrester, "A fully automated comparative microaneurysm digital detection system," Eye 11, 622-628 (1997).
[CrossRef] [PubMed]

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C. E. Metz, "ROC methodology in radiologic imaging," Invest. Radiol. 21, 720-733 (1986).
[CrossRef]

Mirmehdi, M.

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

Mitchell, P.

D. Cornforth, H. F. Jelinek, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., M. J. Cree, P. Mitchell, and T. Bossomaier, "Evolution of retinal blood vessel segmentation using wavelet transform in assessment of diabetic retinopathy," Complexity International 11, 50-60 (2005), http://www.complexity.org.au/ci/vol11/.

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]

Murenzi, R.

J. P. Antoine, P. Carette, R. Murenzi, and B. Piette, "Image analysis with two-dimensional wavelet transform," Signal Process. 3l, 241-272 (1993).
[CrossRef]

Nain, D.

D. Nain, A. Yezzi, and G. Turk, "Vessel segmentation using a shape driven flow," presented at the Medical Image Computing and Computer-Assisted Intervention (MICCAI), Saint-Malo, France, September 26-29, 2004.

Niemeijer, M.

J. J. Staal, M. D. Abràmoff, M. Niemeijer, M. A. 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]

Nikita, K. S.

P. Asvestas, G. K. Matsopoulos, and K. S. Nikita, "Estimating of fractal dimension of images using fixed mass approach," Pattern Recogn. Lett. 20, 347-354 (1999).
[CrossRef]

Olson, J.

J. Hipwell, F. Strachan, J. Olson, K. McHardy, P. Sharp, and J. Forrester, "Automated detection of microaneurysms in digital red-free photographs: a diabetic retinopathy screening tool," Diabetic Med. 17, 588-594 (2000).
[CrossRef] [PubMed]

Olson, J. A.

M. J. Cree, J. A. Olson, K. McHardy, P. Sharp, and J. Forrester, "A fully automated comparative microaneurysm digital detection system," Eye 11, 622-628 (1997).
[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. Ophthamol. 87, 1220-1223 (2003).
[CrossRef]

Parker, K. H.

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

Paul, M. D.

K. S. Coyne, M. K. Margolis, T. Kennedy-Martin, T. M. Baker, R. Klein, M. D. Paul, and D. A. Revicki, "The impact of diabetic retinopathy: perspectives from patient focus groups," Fam. Pract. 21, 447-453 (2004).
[CrossRef] [PubMed]

Piette, B.

J. P. Antoine, P. Carette, R. Murenzi, and B. Piette, "Image analysis with two-dimensional wavelet transform," Signal Process. 3l, 241-272 (1993).
[CrossRef]

Quek, F.

C. Kirbas and F. Quek, "A review of vessel extraction techniques and algorithms," ACM Comput. Surv. 32, 81-121 (2004).
[CrossRef]

Revicki, D. A.

K. S. Coyne, M. K. Margolis, T. Kennedy-Martin, T. M. Baker, R. Klein, M. D. Paul, and D. A. Revicki, "The impact of diabetic retinopathy: perspectives from patient focus groups," Fam. Pract. 21, 447-453 (2004).
[CrossRef] [PubMed]

Rigaut, J. P.

J. P. Rigaut, "An empirical formulation relating boundary lengths to resolution in specimens showing 'non-ideally fractal' dimensions," J. Microsc. 133, 41-54 (1984).
[CrossRef]

Rios, L. G.

L. F. Estrozi, L. G. Rios, A. G. Campos, R. M. Cesar, Jr., and L. d. F. Costa, "1D and 2D Fourier-based approaches to numeric curvature estimation and their comparative performance assessment," Digit. Signal Process. 13, 172-197 (2003).
[CrossRef]

Roux, S. G.

A. Arnéodo, N. Decoster, and S. G. Roux, "A wavelet-based method for multifractal image analysis. I. Methodology and test applications on isotropic and anisotropic random rough surfaces," Eur. Phys. J. B 15, 567-600 (2000).
[CrossRef]

Sharp, P.

J. Hipwell, F. Strachan, J. Olson, K. McHardy, P. Sharp, and J. Forrester, "Automated detection of microaneurysms in digital red-free photographs: a diabetic retinopathy screening tool," Diabetic Med. 17, 588-594 (2000).
[CrossRef] [PubMed]

M. J. Cree, J. A. Olson, K. McHardy, P. Sharp, and J. Forrester, "A fully automated comparative microaneurysm digital detection system," Eye 11, 622-628 (1997).
[CrossRef] [PubMed]

Sicari, C.

S. J. Lee, C. Sicari, C. A. Harper, H. R. Taylor, and J. E. Keeffe, "Program for the early detection of diabetic retinopathy: a two-year follow-up," Clin. Exp. Optom. 29, 12-25 (2001).

Sinthanayothin, C.

C. Sinthanayothin, J. F. Boyce, H. Cook, and T. Williamson, "Automated localisation of the optic disc, fovea and retinal blood vessels from digital colour fundus images" Br. J. Ophthamol. 83, 902-912 (1999).
[CrossRef]

Soares, J. V. B.

J. V. B. Soares, J. J. G. Leandro, R. M. Cesar, Jr., H. F. Jelinek, and M. J. Cree, "Retinal vessel segmentation using the 2-D Gabor wavelet and supervised classification," IEEE Trans. Med. Imaging 25, 1214-1222 (2006).
[CrossRef] [PubMed]

D. Cornforth, H. F. Jelinek, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., M. J. Cree, P. Mitchell, and T. Bossomaier, "Evolution of retinal blood vessel segmentation using wavelet transform in assessment of diabetic retinopathy," Complexity International 11, 50-60 (2005), http://www.complexity.org.au/ci/vol11/.

J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., and H. F. Jelinek, "Blood vessel segmentation of non-mydriatic images using wavelets and statistical classifiers," in Proceedings of the Brazilian Conference on Computer Graphics, Image Processing and Vision (Sibgrapi, 2003), pp. 262-269.

M. J. Cree, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., H. F. Jelinek, and D. Cornforth, "Comparison of various methods to delineate blood vessels in retinal images," in Proceedings of the 16th Australian Institute of Physics Congress (AIP, 2005) pp. 1-4.

Soper, K.

E. Sussman, W. Tsiaras, and K. Soper, "Diagnosis of diabetic eye disease," J. Am. Med. Assoc. 247, 3231-3234 (1982).
[CrossRef]

Staal, J. J.

J. J. Staal, M. D. Abràmoff, M. Niemeijer, M. A. 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]

Stanton, A.

X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thorn, "Quantification and characterisation of arteries in retinal images," Comput. Methods Programs Biomed. 63, 133-146 (2000).
[CrossRef] [PubMed]

Stanton, A. V.

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

Stork, D. G.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, 2nd ed. (Wiley Interscience, 2001).

Strachan, F.

J. Hipwell, F. Strachan, J. Olson, K. McHardy, P. Sharp, and J. Forrester, "Automated detection of microaneurysms in digital red-free photographs: a diabetic retinopathy screening tool," Diabetic Med. 17, 588-594 (2000).
[CrossRef] [PubMed]

Sussman, E.

E. Sussman, W. Tsiaras, and K. Soper, "Diagnosis of diabetic eye disease," J. Am. Med. Assoc. 247, 3231-3234 (1982).
[CrossRef]

Taylor, H. R.

S. J. Lee, C. Sicari, C. A. Harper, H. R. Taylor, and J. E. Keeffe, "Program for the early detection of diabetic retinopathy: a two-year follow-up," Clin. Exp. Optom. 29, 12-25 (2001).

H. R. Taylor and J. E. Keeffe, "World blindness: a 21st century perspective," Br. J. Ophthamol. 85, 261-266 (2001).
[CrossRef]

Thom, S. A.

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

Thomas, B.

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

Thorn, S.

X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thorn, "Quantification and characterisation of arteries in retinal images," Comput. Methods Programs Biomed. 63, 133-146 (2000).
[CrossRef] [PubMed]

Tsiaras, W.

E. Sussman, W. Tsiaras, and K. Soper, "Diagnosis of diabetic eye disease," J. Am. Med. Assoc. 247, 3231-3234 (1982).
[CrossRef]

Turk, G.

D. Nain, A. Yezzi, and G. Turk, "Vessel segmentation using a shape driven flow," presented at the Medical Image Computing and Computer-Assisted Intervention (MICCAI), Saint-Malo, France, September 26-29, 2004.

van Ginneken, B.

J. J. Staal, M. D. Abràmoff, M. Niemeijer, M. A. 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]

Verne, J.

U. Freudentsin and J. Verne, "A national screening programme for diabetic retinopathy," BMJ 323, 4-5 (2001).
[CrossRef]

Viergever, M. A.

J. J. Staal, M. D. Abràmoff, M. Niemeijer, M. A. 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]

Viswanath, K.

K. Viswanath and M. McGavin, "Diabetic retinopathy: clinical findings and management," Community Eye Health 16, 21-24 (2003).

Walter, T.

T. Walter, J. C. Klein, P. Massin, and F. Zana, "Automated segmentation and registration of retinal fluorescein angiographies," presented at the First International Workshop on Computer Assisted Fundus Image Analysis (CAFIA), Herlev Hospital, Copenhagen, Denmark, May 29-30, 2000.

Williamson, T.

C. Sinthanayothin, J. F. Boyce, H. Cook, and T. Williamson, "Automated localisation of the optic disc, fovea and retinal blood vessels from digital colour fundus images" Br. J. Ophthamol. 83, 902-912 (1999).
[CrossRef]

Woods, R. E.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using Matlab (Pearson Prentice Hall, 2004).

Yezzi, A.

D. Nain, A. Yezzi, and G. Turk, "Vessel segmentation using a shape driven flow," presented at the Medical Image Computing and Computer-Assisted Intervention (MICCAI), Saint-Malo, France, September 26-29, 2004.

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 (2000).
[CrossRef]

T. Walter, J. C. Klein, P. Massin, and F. Zana, "Automated segmentation and registration of retinal fluorescein angiographies," presented at the First International Workshop on Computer Assisted Fundus Image Analysis (CAFIA), Herlev Hospital, Copenhagen, Denmark, May 29-30, 2000.

ACM Comput. Surv. (1)

C. Kirbas and F. Quek, "A review of vessel extraction techniques and algorithms," ACM Comput. Surv. 32, 81-121 (2004).
[CrossRef]

Annu. Rev. Biomed. Eng. (1)

B. R. Masters, "Fractal analysis of the vascular tree in the human retina," Annu. Rev. Biomed. Eng. 6, 427-452 (2004).
[CrossRef] [PubMed]

BMJ (1)

U. Freudentsin and J. Verne, "A national screening programme for diabetic retinopathy," BMJ 323, 4-5 (2001).
[CrossRef]

Br. J. Ophthamol. (3)

H. R. Taylor and J. E. Keeffe, "World blindness: a 21st century perspective," Br. J. Ophthamol. 85, 261-266 (2001).
[CrossRef]

C. Sinthanayothin, J. F. Boyce, H. Cook, and T. Williamson, "Automated localisation of the optic disc, fovea and retinal blood vessels from digital colour fundus images" Br. J. Ophthamol. 83, 902-912 (1999).
[CrossRef]

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

Brain and Mind (1)

H. F. Jelinek, R. M. Cesar, Jr., and J. J. G. Leandro, "Exploring wavelet transforms for morphological differentiation between functionally different cat retinal ganglion cells," Brain and Mind 4, 67-90 (2003).
[CrossRef]

Clin. Exp. Optom. (1)

S. J. Lee, C. Sicari, C. A. Harper, H. R. Taylor, and J. E. Keeffe, "Program for the early detection of diabetic retinopathy: a two-year follow-up," Clin. Exp. Optom. 29, 12-25 (2001).

Community Eye Health (1)

K. Viswanath and M. McGavin, "Diabetic retinopathy: clinical findings and management," Community Eye Health 16, 21-24 (2003).

Complexity International (1)

D. Cornforth, H. F. Jelinek, J. J. G. Leandro, J. V. B. Soares, R. M. Cesar, Jr., M. J. Cree, P. Mitchell, and T. Bossomaier, "Evolution of retinal blood vessel segmentation using wavelet transform in assessment of diabetic retinopathy," Complexity International 11, 50-60 (2005), http://www.complexity.org.au/ci/vol11/.

Comput. Methods Programs Biomed. (1)

X. W. Gao, A. Bharath, A. Stanton, A. Hughes, N. Chapman, and S. Thorn, "Quantification and characterisation of arteries in retinal images," Comput. Methods Programs Biomed. 63, 133-146 (2000).
[CrossRef] [PubMed]

Diabetic Med. (1)

J. Hipwell, F. Strachan, J. Olson, K. McHardy, P. Sharp, and J. Forrester, "Automated detection of microaneurysms in digital red-free photographs: a diabetic retinopathy screening tool," Diabetic Med. 17, 588-594 (2000).
[CrossRef] [PubMed]

Digit. Signal Process. (1)

L. F. Estrozi, L. G. Rios, A. G. Campos, R. M. Cesar, Jr., and L. d. F. Costa, "1D and 2D Fourier-based approaches to numeric curvature estimation and their comparative performance assessment," Digit. Signal Process. 13, 172-197 (2003).
[CrossRef]

Eur. Phys. J. B (1)

A. Arnéodo, N. Decoster, and S. G. Roux, "A wavelet-based method for multifractal image analysis. I. Methodology and test applications on isotropic and anisotropic random rough surfaces," Eur. Phys. J. B 15, 567-600 (2000).
[CrossRef]

Eye (1)

M. J. Cree, J. A. Olson, K. McHardy, P. Sharp, and J. Forrester, "A fully automated comparative microaneurysm digital detection system," Eye 11, 622-628 (1997).
[CrossRef] [PubMed]

Fam. Pract. (1)

K. S. Coyne, M. K. Margolis, T. Kennedy-Martin, T. M. Baker, R. Klein, M. D. Paul, and D. A. Revicki, "The impact of diabetic retinopathy: perspectives from patient focus groups," Fam. Pract. 21, 447-453 (2004).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Eng. (1)

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

IEEE Trans. Image Process. (1)

F. Zana and J.-C. Klein, "Segmentation of vessel-like patterns using mathematical morphology and curvature evaluation," IEEE Trans. Image Process. 10, 1010-1019 (2000).
[CrossRef]

IEEE Trans. Med. Imaging (3)

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]

J. V. B. Soares, J. J. G. Leandro, R. M. Cesar, Jr., H. F. Jelinek, and M. J. Cree, "Retinal vessel segmentation using the 2-D Gabor wavelet and supervised classification," IEEE Trans. Med. Imaging 25, 1214-1222 (2006).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Illustration of outcome for automated and manual segmentation compared with the original gray-scale image. Overlaid skeletonized vessel pattern, in black, has been dilated for easier viewing.

Fig. 2
Fig. 2

Correlation integral and wavelet histogram for the orientation entropy comparing an image with neovascularization to one with no neovascularization. Vertical lines on the log–log correlation integral indicate upper and lower bounds (see text).

Fig. 3
Fig. 3

ROC curve for automated segmented images using the set of six best features. ∗ is operating point; solid curve, six best features; dashed curve, traditional features (area, perimeter, circularity).

Tables (2)

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Table 1 Feature Comparison for Manual and Automated Segmented Images

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Table 2 Comparison between Nonproliferative and Proliferative Diabetic Retinopathy for the Eight Features Analyzed

Equations (9)

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

ν ̂ i = ν i μ i σ i ,
ψ 1 ( x , y ) = g ( x , y ) x , ψ 2 ( x , y ) g ( x , y ) y ,
T ψ [ f ] ( b , a ) = ( T ψ 1 [ f ] ( b , a ) T ψ 2 [ f ] ( b , a ) ) ,
k = f f = f x x f y 2 2 f x f y f x y + f y y f x 2 ( f x 2 + f y 2 ) 3 2 ,
E = i K p i ln ( p i ) ,
m 2 = i K i 2 q i .
C ( ε ) = lim N 2 N 2 i = 1 , j > i N θ ( ε X i X j ) ,
θ ( ε X i X j ) = { 1 if ( ε X i X j 0 ) 0 if ( ε X i X j < 0 ) ,
C ( ε ) ε C D .

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