Abstract

A range of applications in visual science rely on accurate tracking of the human pupil’s movement and contraction in response to light. While the literature for independent contour detection and fitting of the iris-pupil boundary is vast, a joint approach, in which it is assumed that the pupil has a given geometric shape has been largely overlooked. We present here a global method for simultaneously finding and fitting of an elliptic or circular contour against a dark interior, which produces consistently accurate results even under non-ideal recording conditions, such as reflections near and over the boundary, droopy eye lids, or the sudden formation of tears. The specific form of the proposed optimization problem allows us to write down closed analytic formulae for the gradient and the Hessian of the objective function. Moreover, both the objective function and its derivatives can be cast into vectorized form, making the proposed algorithm significantly faster than its closest relative in the literature. We compare methods in multiple ways, both analytically and numerically, using real iris images as well as idealizations of the iris for which the ground truth boundary is precisely known. The method proposed here is illustrated under challenging recording conditions and it is shown to be robust.

© 2014 Optical Society of America

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References

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  1. E. S. Maini, “Robust ellipse-specific fitting for real-time machine vision,” in Brain, Vision, and Artificial Intelligence, M. Gregorio, V. Maio, M. Frucci, and C. Musio, eds. (Springer Berlin Heidelberg, 2005) vol. 3704, pp. 318–327.
    [CrossRef]
  2. A. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct least square fitting of ellipses,” IEEE Trans. Pattern Analysis Mach. Intell.21, 476–480 (1999).
    [CrossRef]
  3. K. Kanatani, “Ellipse fitting with hyperaccuracy,” IEICE Trans. Inf. Syst.E89-D, 2653–2660 (2006).
    [CrossRef]
  4. K. Kanatani, “Statistical bias of conic fitting and renormalization,” IEEE Trans. Pattern Analysis Mach. Intell.16, 320–326 (1994).
    [CrossRef]
  5. J. Porrill, “Fitting ellipses and predicting confidence envelopes using a bias corrected kalman filter,” Image Vis. Comput.8, 37–41 (1990).
    [CrossRef]
  6. J. Daugman, “High confidence visual recognition of persons by a test of statistical independence,” IEEE Trans. Pattern Analysis Mach. Intell.15(11), 1148–1161 (1993).
    [CrossRef]
  7. S. A. C. Schuckers, N. A. Schmid, A. Abhyankar, V. Dorairaj, C. K. Boyce, and L. A. Hornak, “On techniques for angle compensation in nonideal iris recognition,” IEEE transactions on systems, man, cybernetics. Part B, Cybern. a publication IEEE Syst. Man, Cybern. Soc., 37, 1176–1190 (2007).
    [CrossRef] [PubMed]
  8. W. Sankowski, K. Grabowski, M. Napieralska, M. Zubert, and A. Napieralski, “Reliable algorithm for iris segmentation in eye image,” Image Vis. Comput., 28, 231–237, 2010.
    [CrossRef]
  9. H. Yuen, J. Princen, J. Illingworth, and J. Kittler, “Comparative study of hough transform methods for circle finding,” Image Vis. Comput.8, 71–77 (1990).
    [CrossRef]
  10. H. Proenca, “Iris recognition: On the segmentation of degraded images acquired in the visible wavelength,” IEEE Trans. Pattern Analysis Mach. Intell.32(8), 1502–1516 (2010).
    [CrossRef]
  11. Z. He, T. Tan, Z. Sun, and X. Qiu, “Toward accurate and fast iris segmentation for iris biometrics,” IEEE Trans. Pattern Analysis Mach. Intell.31(9), 1670–1684 (2009).
    [CrossRef]
  12. Z. He, T. Tan, and Z. Sun, “Iris localization via pulling and pushing,” in 18th International Conference on Pattern Recognition, 2006. ICPR 2006, 4, 366–369, 2006.
  13. T. Camus and R. Wildes, “Reliable and fast eye finding in close-up images,” in 16th International Conference on Pattern Recognition, 2002. Proceedings, 1, 389–394 vol. 1, 2002.
  14. Y. Chen, M. Adjouadi, C. Han, J. Wang, A. Barreto, N. Rishe, and J. Andrian, “A highly accurate and computationally efficient approach for unconstrained iris segmentation,” Image Vis. Comput.28, 261–269 (2010).
    [CrossRef]
  15. A. Bell, A. C. James, M. Kolic, R. W. Essex, and T. Maddess, “Dichoptic multifocal pupillography reveals afferent visual field defects in early type 2 diabetes,” Invest. Ophthalmol. Vis. Sci.51, 602–608 (2010).
    [CrossRef]
  16. C. F. Carle, T. Maddess, and A. C. James, “Contraction anisocoria: Segregation, summation, and saturation in the pupillary pathway,” Invest. Ophthalmol. Vis. Sci.52, 2365–2371 (2011).
    [CrossRef] [PubMed]
  17. J. Miles., www.milesresearch.com . Image use permission kindly granted by owner.
  18. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comp. J.7, 308–313 (1965).
    [CrossRef]
  19. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the nelder-mead simplex method in low dimensions,” SIAM J. Optim.9, 112–147 (1998).
    [CrossRef]
  20. G. Taubin, “Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation,” IEEE Trans. Pattern Analysis Mach. Intell.13, 1115–1138 (1991).
    [CrossRef]
  21. C. Broyden, “The convergence of a class of double-rank minimization algorithms 1. general considerations,” IMA J. Appl. Math.6, 76–90 (1970).
    [CrossRef]
  22. R. Fletcher, “A new approach to variable metric algorithms,” The Comp. J.13, 317–322 (1970).
    [CrossRef]
  23. D. Goldfarb, “A family of variable-metric methods derived by variational means,” Math. Comput.24, 23–26 (1970).
    [CrossRef]
  24. D. F. Shanno, “Conditioning of quasi-newton methods for function minimization,” Math. Comput.24, 647–656 (1970).
    [CrossRef]

2011 (1)

C. F. Carle, T. Maddess, and A. C. James, “Contraction anisocoria: Segregation, summation, and saturation in the pupillary pathway,” Invest. Ophthalmol. Vis. Sci.52, 2365–2371 (2011).
[CrossRef] [PubMed]

2010 (4)

Y. Chen, M. Adjouadi, C. Han, J. Wang, A. Barreto, N. Rishe, and J. Andrian, “A highly accurate and computationally efficient approach for unconstrained iris segmentation,” Image Vis. Comput.28, 261–269 (2010).
[CrossRef]

A. Bell, A. C. James, M. Kolic, R. W. Essex, and T. Maddess, “Dichoptic multifocal pupillography reveals afferent visual field defects in early type 2 diabetes,” Invest. Ophthalmol. Vis. Sci.51, 602–608 (2010).
[CrossRef]

W. Sankowski, K. Grabowski, M. Napieralska, M. Zubert, and A. Napieralski, “Reliable algorithm for iris segmentation in eye image,” Image Vis. Comput., 28, 231–237, 2010.
[CrossRef]

H. Proenca, “Iris recognition: On the segmentation of degraded images acquired in the visible wavelength,” IEEE Trans. Pattern Analysis Mach. Intell.32(8), 1502–1516 (2010).
[CrossRef]

2009 (1)

Z. He, T. Tan, Z. Sun, and X. Qiu, “Toward accurate and fast iris segmentation for iris biometrics,” IEEE Trans. Pattern Analysis Mach. Intell.31(9), 1670–1684 (2009).
[CrossRef]

2007 (1)

S. A. C. Schuckers, N. A. Schmid, A. Abhyankar, V. Dorairaj, C. K. Boyce, and L. A. Hornak, “On techniques for angle compensation in nonideal iris recognition,” IEEE transactions on systems, man, cybernetics. Part B, Cybern. a publication IEEE Syst. Man, Cybern. Soc., 37, 1176–1190 (2007).
[CrossRef] [PubMed]

2006 (1)

K. Kanatani, “Ellipse fitting with hyperaccuracy,” IEICE Trans. Inf. Syst.E89-D, 2653–2660 (2006).
[CrossRef]

1999 (1)

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

1998 (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the nelder-mead simplex method in low dimensions,” SIAM J. Optim.9, 112–147 (1998).
[CrossRef]

1994 (1)

K. Kanatani, “Statistical bias of conic fitting and renormalization,” IEEE Trans. Pattern Analysis Mach. Intell.16, 320–326 (1994).
[CrossRef]

1993 (1)

J. Daugman, “High confidence visual recognition of persons by a test of statistical independence,” IEEE Trans. Pattern Analysis Mach. Intell.15(11), 1148–1161 (1993).
[CrossRef]

1991 (1)

G. Taubin, “Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation,” IEEE Trans. Pattern Analysis Mach. Intell.13, 1115–1138 (1991).
[CrossRef]

1990 (2)

J. Porrill, “Fitting ellipses and predicting confidence envelopes using a bias corrected kalman filter,” Image Vis. Comput.8, 37–41 (1990).
[CrossRef]

H. Yuen, J. Princen, J. Illingworth, and J. Kittler, “Comparative study of hough transform methods for circle finding,” Image Vis. Comput.8, 71–77 (1990).
[CrossRef]

1970 (4)

C. Broyden, “The convergence of a class of double-rank minimization algorithms 1. general considerations,” IMA J. Appl. Math.6, 76–90 (1970).
[CrossRef]

R. Fletcher, “A new approach to variable metric algorithms,” The Comp. J.13, 317–322 (1970).
[CrossRef]

D. Goldfarb, “A family of variable-metric methods derived by variational means,” Math. Comput.24, 23–26 (1970).
[CrossRef]

D. F. Shanno, “Conditioning of quasi-newton methods for function minimization,” Math. Comput.24, 647–656 (1970).
[CrossRef]

1965 (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comp. J.7, 308–313 (1965).
[CrossRef]

Abhyankar, A.

S. A. C. Schuckers, N. A. Schmid, A. Abhyankar, V. Dorairaj, C. K. Boyce, and L. A. Hornak, “On techniques for angle compensation in nonideal iris recognition,” IEEE transactions on systems, man, cybernetics. Part B, Cybern. a publication IEEE Syst. Man, Cybern. Soc., 37, 1176–1190 (2007).
[CrossRef] [PubMed]

Adjouadi, M.

Y. Chen, M. Adjouadi, C. Han, J. Wang, A. Barreto, N. Rishe, and J. Andrian, “A highly accurate and computationally efficient approach for unconstrained iris segmentation,” Image Vis. Comput.28, 261–269 (2010).
[CrossRef]

Andrian, J.

Y. Chen, M. Adjouadi, C. Han, J. Wang, A. Barreto, N. Rishe, and J. Andrian, “A highly accurate and computationally efficient approach for unconstrained iris segmentation,” Image Vis. Comput.28, 261–269 (2010).
[CrossRef]

Barreto, A.

Y. Chen, M. Adjouadi, C. Han, J. Wang, A. Barreto, N. Rishe, and J. Andrian, “A highly accurate and computationally efficient approach for unconstrained iris segmentation,” Image Vis. Comput.28, 261–269 (2010).
[CrossRef]

Bell, A.

A. Bell, A. C. James, M. Kolic, R. W. Essex, and T. Maddess, “Dichoptic multifocal pupillography reveals afferent visual field defects in early type 2 diabetes,” Invest. Ophthalmol. Vis. Sci.51, 602–608 (2010).
[CrossRef]

Boyce, C. K.

S. A. C. Schuckers, N. A. Schmid, A. Abhyankar, V. Dorairaj, C. K. Boyce, and L. A. Hornak, “On techniques for angle compensation in nonideal iris recognition,” IEEE transactions on systems, man, cybernetics. Part B, Cybern. a publication IEEE Syst. Man, Cybern. Soc., 37, 1176–1190 (2007).
[CrossRef] [PubMed]

Broyden, C.

C. Broyden, “The convergence of a class of double-rank minimization algorithms 1. general considerations,” IMA J. Appl. Math.6, 76–90 (1970).
[CrossRef]

Camus, T.

T. Camus and R. Wildes, “Reliable and fast eye finding in close-up images,” in 16th International Conference on Pattern Recognition, 2002. Proceedings, 1, 389–394 vol. 1, 2002.

Carle, C. F.

C. F. Carle, T. Maddess, and A. C. James, “Contraction anisocoria: Segregation, summation, and saturation in the pupillary pathway,” Invest. Ophthalmol. Vis. Sci.52, 2365–2371 (2011).
[CrossRef] [PubMed]

Chen, Y.

Y. Chen, M. Adjouadi, C. Han, J. Wang, A. Barreto, N. Rishe, and J. Andrian, “A highly accurate and computationally efficient approach for unconstrained iris segmentation,” Image Vis. Comput.28, 261–269 (2010).
[CrossRef]

Daugman, J.

J. Daugman, “High confidence visual recognition of persons by a test of statistical independence,” IEEE Trans. Pattern Analysis Mach. Intell.15(11), 1148–1161 (1993).
[CrossRef]

Dorairaj, V.

S. A. C. Schuckers, N. A. Schmid, A. Abhyankar, V. Dorairaj, C. K. Boyce, and L. A. Hornak, “On techniques for angle compensation in nonideal iris recognition,” IEEE transactions on systems, man, cybernetics. Part B, Cybern. a publication IEEE Syst. Man, Cybern. Soc., 37, 1176–1190 (2007).
[CrossRef] [PubMed]

Essex, R. W.

A. Bell, A. C. James, M. Kolic, R. W. Essex, and T. Maddess, “Dichoptic multifocal pupillography reveals afferent visual field defects in early type 2 diabetes,” Invest. Ophthalmol. Vis. Sci.51, 602–608 (2010).
[CrossRef]

Fisher, R. B.

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

Fitzgibbon, A.

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

Fletcher, R.

R. Fletcher, “A new approach to variable metric algorithms,” The Comp. J.13, 317–322 (1970).
[CrossRef]

Goldfarb, D.

D. Goldfarb, “A family of variable-metric methods derived by variational means,” Math. Comput.24, 23–26 (1970).
[CrossRef]

Grabowski, K.

W. Sankowski, K. Grabowski, M. Napieralska, M. Zubert, and A. Napieralski, “Reliable algorithm for iris segmentation in eye image,” Image Vis. Comput., 28, 231–237, 2010.
[CrossRef]

Han, C.

Y. Chen, M. Adjouadi, C. Han, J. Wang, A. Barreto, N. Rishe, and J. Andrian, “A highly accurate and computationally efficient approach for unconstrained iris segmentation,” Image Vis. Comput.28, 261–269 (2010).
[CrossRef]

He, Z.

Z. He, T. Tan, Z. Sun, and X. Qiu, “Toward accurate and fast iris segmentation for iris biometrics,” IEEE Trans. Pattern Analysis Mach. Intell.31(9), 1670–1684 (2009).
[CrossRef]

Z. He, T. Tan, and Z. Sun, “Iris localization via pulling and pushing,” in 18th International Conference on Pattern Recognition, 2006. ICPR 2006, 4, 366–369, 2006.

Hornak, L. A.

S. A. C. Schuckers, N. A. Schmid, A. Abhyankar, V. Dorairaj, C. K. Boyce, and L. A. Hornak, “On techniques for angle compensation in nonideal iris recognition,” IEEE transactions on systems, man, cybernetics. Part B, Cybern. a publication IEEE Syst. Man, Cybern. Soc., 37, 1176–1190 (2007).
[CrossRef] [PubMed]

Illingworth, J.

H. Yuen, J. Princen, J. Illingworth, and J. Kittler, “Comparative study of hough transform methods for circle finding,” Image Vis. Comput.8, 71–77 (1990).
[CrossRef]

James, A. C.

C. F. Carle, T. Maddess, and A. C. James, “Contraction anisocoria: Segregation, summation, and saturation in the pupillary pathway,” Invest. Ophthalmol. Vis. Sci.52, 2365–2371 (2011).
[CrossRef] [PubMed]

A. Bell, A. C. James, M. Kolic, R. W. Essex, and T. Maddess, “Dichoptic multifocal pupillography reveals afferent visual field defects in early type 2 diabetes,” Invest. Ophthalmol. Vis. Sci.51, 602–608 (2010).
[CrossRef]

Kanatani, K.

K. Kanatani, “Ellipse fitting with hyperaccuracy,” IEICE Trans. Inf. Syst.E89-D, 2653–2660 (2006).
[CrossRef]

K. Kanatani, “Statistical bias of conic fitting and renormalization,” IEEE Trans. Pattern Analysis Mach. Intell.16, 320–326 (1994).
[CrossRef]

Kittler, J.

H. Yuen, J. Princen, J. Illingworth, and J. Kittler, “Comparative study of hough transform methods for circle finding,” Image Vis. Comput.8, 71–77 (1990).
[CrossRef]

Kolic, M.

A. Bell, A. C. James, M. Kolic, R. W. Essex, and T. Maddess, “Dichoptic multifocal pupillography reveals afferent visual field defects in early type 2 diabetes,” Invest. Ophthalmol. Vis. Sci.51, 602–608 (2010).
[CrossRef]

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the nelder-mead simplex method in low dimensions,” SIAM J. Optim.9, 112–147 (1998).
[CrossRef]

Maddess, T.

C. F. Carle, T. Maddess, and A. C. James, “Contraction anisocoria: Segregation, summation, and saturation in the pupillary pathway,” Invest. Ophthalmol. Vis. Sci.52, 2365–2371 (2011).
[CrossRef] [PubMed]

A. Bell, A. C. James, M. Kolic, R. W. Essex, and T. Maddess, “Dichoptic multifocal pupillography reveals afferent visual field defects in early type 2 diabetes,” Invest. Ophthalmol. Vis. Sci.51, 602–608 (2010).
[CrossRef]

Maini, E. S.

E. S. Maini, “Robust ellipse-specific fitting for real-time machine vision,” in Brain, Vision, and Artificial Intelligence, M. Gregorio, V. Maio, M. Frucci, and C. Musio, eds. (Springer Berlin Heidelberg, 2005) vol. 3704, pp. 318–327.
[CrossRef]

Mead, R.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comp. J.7, 308–313 (1965).
[CrossRef]

Miles., J.

J. Miles., www.milesresearch.com . Image use permission kindly granted by owner.

Napieralska, M.

W. Sankowski, K. Grabowski, M. Napieralska, M. Zubert, and A. Napieralski, “Reliable algorithm for iris segmentation in eye image,” Image Vis. Comput., 28, 231–237, 2010.
[CrossRef]

Napieralski, A.

W. Sankowski, K. Grabowski, M. Napieralska, M. Zubert, and A. Napieralski, “Reliable algorithm for iris segmentation in eye image,” Image Vis. Comput., 28, 231–237, 2010.
[CrossRef]

Nelder, J. A.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comp. J.7, 308–313 (1965).
[CrossRef]

Pilu, M.

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

Porrill, J.

J. Porrill, “Fitting ellipses and predicting confidence envelopes using a bias corrected kalman filter,” Image Vis. Comput.8, 37–41 (1990).
[CrossRef]

Princen, J.

H. Yuen, J. Princen, J. Illingworth, and J. Kittler, “Comparative study of hough transform methods for circle finding,” Image Vis. Comput.8, 71–77 (1990).
[CrossRef]

Proenca, H.

H. Proenca, “Iris recognition: On the segmentation of degraded images acquired in the visible wavelength,” IEEE Trans. Pattern Analysis Mach. Intell.32(8), 1502–1516 (2010).
[CrossRef]

Qiu, X.

Z. He, T. Tan, Z. Sun, and X. Qiu, “Toward accurate and fast iris segmentation for iris biometrics,” IEEE Trans. Pattern Analysis Mach. Intell.31(9), 1670–1684 (2009).
[CrossRef]

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the nelder-mead simplex method in low dimensions,” SIAM J. Optim.9, 112–147 (1998).
[CrossRef]

Rishe, N.

Y. Chen, M. Adjouadi, C. Han, J. Wang, A. Barreto, N. Rishe, and J. Andrian, “A highly accurate and computationally efficient approach for unconstrained iris segmentation,” Image Vis. Comput.28, 261–269 (2010).
[CrossRef]

Sankowski, W.

W. Sankowski, K. Grabowski, M. Napieralska, M. Zubert, and A. Napieralski, “Reliable algorithm for iris segmentation in eye image,” Image Vis. Comput., 28, 231–237, 2010.
[CrossRef]

Schmid, N. A.

S. A. C. Schuckers, N. A. Schmid, A. Abhyankar, V. Dorairaj, C. K. Boyce, and L. A. Hornak, “On techniques for angle compensation in nonideal iris recognition,” IEEE transactions on systems, man, cybernetics. Part B, Cybern. a publication IEEE Syst. Man, Cybern. Soc., 37, 1176–1190 (2007).
[CrossRef] [PubMed]

Schuckers, S. A. C.

S. A. C. Schuckers, N. A. Schmid, A. Abhyankar, V. Dorairaj, C. K. Boyce, and L. A. Hornak, “On techniques for angle compensation in nonideal iris recognition,” IEEE transactions on systems, man, cybernetics. Part B, Cybern. a publication IEEE Syst. Man, Cybern. Soc., 37, 1176–1190 (2007).
[CrossRef] [PubMed]

Shanno, D. F.

D. F. Shanno, “Conditioning of quasi-newton methods for function minimization,” Math. Comput.24, 647–656 (1970).
[CrossRef]

Sun, Z.

Z. He, T. Tan, Z. Sun, and X. Qiu, “Toward accurate and fast iris segmentation for iris biometrics,” IEEE Trans. Pattern Analysis Mach. Intell.31(9), 1670–1684 (2009).
[CrossRef]

Z. He, T. Tan, and Z. Sun, “Iris localization via pulling and pushing,” in 18th International Conference on Pattern Recognition, 2006. ICPR 2006, 4, 366–369, 2006.

Tan, T.

Z. He, T. Tan, Z. Sun, and X. Qiu, “Toward accurate and fast iris segmentation for iris biometrics,” IEEE Trans. Pattern Analysis Mach. Intell.31(9), 1670–1684 (2009).
[CrossRef]

Z. He, T. Tan, and Z. Sun, “Iris localization via pulling and pushing,” in 18th International Conference on Pattern Recognition, 2006. ICPR 2006, 4, 366–369, 2006.

Taubin, G.

G. Taubin, “Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation,” IEEE Trans. Pattern Analysis Mach. Intell.13, 1115–1138 (1991).
[CrossRef]

Wang, J.

Y. Chen, M. Adjouadi, C. Han, J. Wang, A. Barreto, N. Rishe, and J. Andrian, “A highly accurate and computationally efficient approach for unconstrained iris segmentation,” Image Vis. Comput.28, 261–269 (2010).
[CrossRef]

Wildes, R.

T. Camus and R. Wildes, “Reliable and fast eye finding in close-up images,” in 16th International Conference on Pattern Recognition, 2002. Proceedings, 1, 389–394 vol. 1, 2002.

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the nelder-mead simplex method in low dimensions,” SIAM J. Optim.9, 112–147 (1998).
[CrossRef]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the nelder-mead simplex method in low dimensions,” SIAM J. Optim.9, 112–147 (1998).
[CrossRef]

Yuen, H.

H. Yuen, J. Princen, J. Illingworth, and J. Kittler, “Comparative study of hough transform methods for circle finding,” Image Vis. Comput.8, 71–77 (1990).
[CrossRef]

Zubert, M.

W. Sankowski, K. Grabowski, M. Napieralska, M. Zubert, and A. Napieralski, “Reliable algorithm for iris segmentation in eye image,” Image Vis. Comput., 28, 231–237, 2010.
[CrossRef]

Comp. J. (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comp. J.7, 308–313 (1965).
[CrossRef]

IEEE Trans. Pattern Analysis Mach. Intell. (6)

G. Taubin, “Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation,” IEEE Trans. Pattern Analysis Mach. Intell.13, 1115–1138 (1991).
[CrossRef]

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

K. Kanatani, “Statistical bias of conic fitting and renormalization,” IEEE Trans. Pattern Analysis Mach. Intell.16, 320–326 (1994).
[CrossRef]

J. Daugman, “High confidence visual recognition of persons by a test of statistical independence,” IEEE Trans. Pattern Analysis Mach. Intell.15(11), 1148–1161 (1993).
[CrossRef]

H. Proenca, “Iris recognition: On the segmentation of degraded images acquired in the visible wavelength,” IEEE Trans. Pattern Analysis Mach. Intell.32(8), 1502–1516 (2010).
[CrossRef]

Z. He, T. Tan, Z. Sun, and X. Qiu, “Toward accurate and fast iris segmentation for iris biometrics,” IEEE Trans. Pattern Analysis Mach. Intell.31(9), 1670–1684 (2009).
[CrossRef]

IEEE transactions on systems, man, cybernetics. Part B, Cybern. a publication IEEE Syst. Man, Cybern. Soc. (1)

S. A. C. Schuckers, N. A. Schmid, A. Abhyankar, V. Dorairaj, C. K. Boyce, and L. A. Hornak, “On techniques for angle compensation in nonideal iris recognition,” IEEE transactions on systems, man, cybernetics. Part B, Cybern. a publication IEEE Syst. Man, Cybern. Soc., 37, 1176–1190 (2007).
[CrossRef] [PubMed]

IEICE Trans. Inf. Syst. (1)

K. Kanatani, “Ellipse fitting with hyperaccuracy,” IEICE Trans. Inf. Syst.E89-D, 2653–2660 (2006).
[CrossRef]

IMA J. Appl. Math. (1)

C. Broyden, “The convergence of a class of double-rank minimization algorithms 1. general considerations,” IMA J. Appl. Math.6, 76–90 (1970).
[CrossRef]

Image Vis. Comput. (4)

J. Porrill, “Fitting ellipses and predicting confidence envelopes using a bias corrected kalman filter,” Image Vis. Comput.8, 37–41 (1990).
[CrossRef]

W. Sankowski, K. Grabowski, M. Napieralska, M. Zubert, and A. Napieralski, “Reliable algorithm for iris segmentation in eye image,” Image Vis. Comput., 28, 231–237, 2010.
[CrossRef]

H. Yuen, J. Princen, J. Illingworth, and J. Kittler, “Comparative study of hough transform methods for circle finding,” Image Vis. Comput.8, 71–77 (1990).
[CrossRef]

Y. Chen, M. Adjouadi, C. Han, J. Wang, A. Barreto, N. Rishe, and J. Andrian, “A highly accurate and computationally efficient approach for unconstrained iris segmentation,” Image Vis. Comput.28, 261–269 (2010).
[CrossRef]

Invest. Ophthalmol. Vis. Sci. (2)

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Supplementary Material (3)

» Media 1: AVI (9271 KB)     
» Media 2: AVI (7780 KB)     
» Media 3: AVI (8360 KB)     

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Figures (10)

Fig. 1
Fig. 1

An example of a pupil image showing a candidate fit in dashed line and the (far off) initial circle using a solid line. The parameters for the pupil-iris boundary fit corresponds to the minimum of the proposed objective function in Eq. (6). Image from [17].

Fig. 2
Fig. 2

An approximation of the objective function Z in Eq. (6), as a function of z and for various values of the product . The pupil Heaviside approximation parameter is k while δ is the scale parameter in Eq. (5).

Fig. 3
Fig. 3

The optimization landscape for the idealized pupil of Eq. (10). On the left is the landscape contours for the integro-differential operator method of Eq. (1), and on the right for the objective function of the method proposed in this paper in Eq. (6). Both the radius r and center c in pixels.

Fig. 4
Fig. 4

The optimization landscape for the real pupil image of Fig. 1. On the left is the landscape contours for the integro-differential operator method of Eq. (1) and on the right for the objective function of this paper, Eq. (6). Actual radius at r = 24 and center c = 31, in pixels.

Fig. 5
Fig. 5

Stability analysis of integro-differential (left) and the method proposed here (right), using a simulated pupil and a range of initial radii. In the top row the deviation is shown as a function of the offset angle (averaged by offset), while in the bottom row it is displayed as function of the offset (averaged by offset angle).

Fig. 6
Fig. 6

Stability analysis of integro-differential (left) and the method proposed here (right), using a real pupil and a range of initial radii. In the top row the deviation is shown as a function of the offset angle (averaged by offset), while in the bottom row it is displayed as function of the offset (averaged by offset angle).

Fig. 7
Fig. 7

The computational performance of our algorithm (dashed lines) compared to the integro-differential operator method (solid line). We fit a circle within a tight error margin of the artificial pupil center and radius, stressing both algorithms equally. While the proposed method scales quadratically with image size, it does so gently, and even for very large images it is still considerably faster.

Fig. 8
Fig. 8

An example of tracking with lots of blinks and tear formation, see Media 1.

Fig. 9
Fig. 9

An example of tracking with prominent droopy eye-lid interference, see Media 2.

Fig. 10
Fig. 10

An example of tracking with large dynamic range and uneven shape during dilation, see Media 3.

Equations (10)

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G σ ( r ) * r { c I ( x , y ) π r d s c I ( x , y ) π r d s } .
G σ ( r ) * r ( H ( r ) H ( r a ) ) .
r ( H ( r ) H ( r a ) ) = δ ( r ) H ( r a ) H ( r ) δ ( r a ) H ( r a ) 2 ,
r sq = c ( x a ) 2 + d ( y b ) 2 + 2 e ( x a ) ( y b ) .
z = 1 δ ( r sq 1 ) , w = z exp 1 2 z 2 ,
Z = x , y w ( z ) I ( x , y ) ,
Z ( r sq ) = 1 δ x y w ( r sq ) r sq I ( x , y ) ,
r sq = ( 2 e ( b y ) + 2 c ( a x ) 2 e ( a x ) + 2 d ( b y ) ( a x ) 2 ( b y ) 2 2 ( a x ) ( b y ) ) .
H ( δ z ) = lim k ( 1 + erf ( k δ z ) ) 2 ,
Z = w ( z ) H ( δ z ) d z d θ , = π exp z 2 / 2 ( 1 + erf ( k δ z ) + δ k erf ( α z ) α ,

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