Abstract

We propose a polygonal snake segmentation technique adapted to objects that can be composed of several regions with gray-level fluctuations described by a priori unknown probability laws. This approach is based on a histogram equalization and on the minimization of a criterion without parameter to be tuned by the user. We demonstrate the efficiency of this approach, which has low computational cost, on synthetic and real images perturbed by different types of optical noise.

© 2005 Optical Society of America

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  1. M. Kass, A. Witkin, and D. Terzopoulos, Int. J. Comput. Vis. 1, 321 (1988).
    [CrossRef]
  2. R. Ronfard, Int. J. Comput. Vis. 2, 229 (1994).
    [CrossRef]
  3. G. Storvik, IEEE Trans. Pattern Anal. Mach. Intell. 16, 976 (1994).
    [CrossRef]
  4. A. K. Jain, Y. Zhong, and S. Lakshmanan, IEEE Trans. Pattern Anal. Mach. Intell. 18, 268 (1996).
  5. O. Germain and Ph. Réfrégier, Opt. Lett. 21, 1845 (1996).
    [CrossRef] [PubMed]
  6. C. Chesnaud, Ph. Réfrégier, and V. Boulet, IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145 (1999).
    [CrossRef]
  7. M. Figueiredo, J. Leitão, and A. K. Jain, IEEE Trans. Image Process. 9, 1075 (2000).
    [CrossRef]
  8. O. Ruch and Ph. Réfrégier, Opt. Lett. 41, 977 (2001).
    [CrossRef]
  9. J. Rissanen, Stochastic Complexity in Statistical Inquiry (World Scientific, 1989).
  10. Y. G. Leclerc, Int. J. Comput. Vis. 3, 73 (1989).
    [CrossRef]
  11. F. Galland, N. Bertaux, and Ph. Réfrégier, IEEE Trans. Image Process. 12, 995 (2003).
    [CrossRef]
  12. F. Galland and Ph. Réfrégier, Opt. Lett. 29, 1611 (2004).
    [CrossRef] [PubMed]
  13. C. E. Shannon, Bell Syst. Tech. J. 27, 379; 623 (1948).
    [CrossRef]
  14. S. J. Sangwine and R. E.N. Horne, eds. The Colour Image Processing Handbook (Chapman & Hall, 1998).
    [CrossRef]

2004 (1)

2003 (1)

F. Galland, N. Bertaux, and Ph. Réfrégier, IEEE Trans. Image Process. 12, 995 (2003).
[CrossRef]

2001 (1)

O. Ruch and Ph. Réfrégier, Opt. Lett. 41, 977 (2001).
[CrossRef]

2000 (1)

M. Figueiredo, J. Leitão, and A. K. Jain, IEEE Trans. Image Process. 9, 1075 (2000).
[CrossRef]

1999 (1)

C. Chesnaud, Ph. Réfrégier, and V. Boulet, IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145 (1999).
[CrossRef]

1996 (2)

A. K. Jain, Y. Zhong, and S. Lakshmanan, IEEE Trans. Pattern Anal. Mach. Intell. 18, 268 (1996).

O. Germain and Ph. Réfrégier, Opt. Lett. 21, 1845 (1996).
[CrossRef] [PubMed]

1994 (2)

R. Ronfard, Int. J. Comput. Vis. 2, 229 (1994).
[CrossRef]

G. Storvik, IEEE Trans. Pattern Anal. Mach. Intell. 16, 976 (1994).
[CrossRef]

1989 (1)

Y. G. Leclerc, Int. J. Comput. Vis. 3, 73 (1989).
[CrossRef]

1988 (1)

M. Kass, A. Witkin, and D. Terzopoulos, Int. J. Comput. Vis. 1, 321 (1988).
[CrossRef]

1948 (1)

C. E. Shannon, Bell Syst. Tech. J. 27, 379; 623 (1948).
[CrossRef]

Bertaux, N.

F. Galland, N. Bertaux, and Ph. Réfrégier, IEEE Trans. Image Process. 12, 995 (2003).
[CrossRef]

Boulet, V.

C. Chesnaud, Ph. Réfrégier, and V. Boulet, IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145 (1999).
[CrossRef]

Chesnaud, C.

C. Chesnaud, Ph. Réfrégier, and V. Boulet, IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145 (1999).
[CrossRef]

Figueiredo, M.

M. Figueiredo, J. Leitão, and A. K. Jain, IEEE Trans. Image Process. 9, 1075 (2000).
[CrossRef]

Galland, F.

F. Galland and Ph. Réfrégier, Opt. Lett. 29, 1611 (2004).
[CrossRef] [PubMed]

F. Galland, N. Bertaux, and Ph. Réfrégier, IEEE Trans. Image Process. 12, 995 (2003).
[CrossRef]

Germain, O.

Jain, A. K.

M. Figueiredo, J. Leitão, and A. K. Jain, IEEE Trans. Image Process. 9, 1075 (2000).
[CrossRef]

A. K. Jain, Y. Zhong, and S. Lakshmanan, IEEE Trans. Pattern Anal. Mach. Intell. 18, 268 (1996).

Kass, M.

M. Kass, A. Witkin, and D. Terzopoulos, Int. J. Comput. Vis. 1, 321 (1988).
[CrossRef]

Lakshmanan, S.

A. K. Jain, Y. Zhong, and S. Lakshmanan, IEEE Trans. Pattern Anal. Mach. Intell. 18, 268 (1996).

Leclerc, Y. G.

Y. G. Leclerc, Int. J. Comput. Vis. 3, 73 (1989).
[CrossRef]

Leitão, J.

M. Figueiredo, J. Leitão, and A. K. Jain, IEEE Trans. Image Process. 9, 1075 (2000).
[CrossRef]

Réfrégier, Ph.

F. Galland and Ph. Réfrégier, Opt. Lett. 29, 1611 (2004).
[CrossRef] [PubMed]

F. Galland, N. Bertaux, and Ph. Réfrégier, IEEE Trans. Image Process. 12, 995 (2003).
[CrossRef]

O. Ruch and Ph. Réfrégier, Opt. Lett. 41, 977 (2001).
[CrossRef]

C. Chesnaud, Ph. Réfrégier, and V. Boulet, IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145 (1999).
[CrossRef]

O. Germain and Ph. Réfrégier, Opt. Lett. 21, 1845 (1996).
[CrossRef] [PubMed]

Rissanen, J.

J. Rissanen, Stochastic Complexity in Statistical Inquiry (World Scientific, 1989).

Ronfard, R.

R. Ronfard, Int. J. Comput. Vis. 2, 229 (1994).
[CrossRef]

Ruch, O.

O. Ruch and Ph. Réfrégier, Opt. Lett. 41, 977 (2001).
[CrossRef]

Shannon, C. E.

C. E. Shannon, Bell Syst. Tech. J. 27, 379; 623 (1948).
[CrossRef]

Storvik, G.

G. Storvik, IEEE Trans. Pattern Anal. Mach. Intell. 16, 976 (1994).
[CrossRef]

Terzopoulos, D.

M. Kass, A. Witkin, and D. Terzopoulos, Int. J. Comput. Vis. 1, 321 (1988).
[CrossRef]

Witkin, A.

M. Kass, A. Witkin, and D. Terzopoulos, Int. J. Comput. Vis. 1, 321 (1988).
[CrossRef]

Zhong, Y.

A. K. Jain, Y. Zhong, and S. Lakshmanan, IEEE Trans. Pattern Anal. Mach. Intell. 18, 268 (1996).

Bell Syst. Tech. J. (1)

C. E. Shannon, Bell Syst. Tech. J. 27, 379; 623 (1948).
[CrossRef]

IEEE Trans. Image Process. (2)

M. Figueiredo, J. Leitão, and A. K. Jain, IEEE Trans. Image Process. 9, 1075 (2000).
[CrossRef]

F. Galland, N. Bertaux, and Ph. Réfrégier, IEEE Trans. Image Process. 12, 995 (2003).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (3)

G. Storvik, IEEE Trans. Pattern Anal. Mach. Intell. 16, 976 (1994).
[CrossRef]

A. K. Jain, Y. Zhong, and S. Lakshmanan, IEEE Trans. Pattern Anal. Mach. Intell. 18, 268 (1996).

C. Chesnaud, Ph. Réfrégier, and V. Boulet, IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145 (1999).
[CrossRef]

Int. J. Comput. Vis. (3)

M. Kass, A. Witkin, and D. Terzopoulos, Int. J. Comput. Vis. 1, 321 (1988).
[CrossRef]

R. Ronfard, Int. J. Comput. Vis. 2, 229 (1994).
[CrossRef]

Y. G. Leclerc, Int. J. Comput. Vis. 3, 73 (1989).
[CrossRef]

Opt. Lett. (3)

Other (2)

S. J. Sangwine and R. E.N. Horne, eds. The Colour Image Processing Handbook (Chapman & Hall, 1998).
[CrossRef]

J. Rissanen, Stochastic Complexity in Statistical Inquiry (World Scientific, 1989).

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Figures (3)

Fig. 1
Fig. 1

Evolution of the NMP determined on 50 noisy realizations of images (a) and (b). (c), (d) Initial contours superposed on noisy versions of (a) and (b) with a gamma pdf ( L = 1 ). (e) NMP as a function of the ratio of the gray-level mean values in the two regions of image (a). (f) NMP as a function of Q when the contrast ratio is equal to 5 in noisy versions of image (a). (g) NMP as a function of the size of the object (in pixels) in image (b) when the contrast is equal to 5.

Fig. 2
Fig. 2

Segmentation of a synthetic image corrupted with different noises. (a) Image corrupted with a gamma noise ( L = 1 ). (b) Gray levels correspond to the logarithm of image (a). (c) Image corrupted with a gamma noise with different mean and order in each region. The segmentation results are obtained with a gamma criterion with L = 1 (top row) or with the proposed ten-level technique (bottom row). The gray levels have been modified to be better visualized. The computation time is less than 0.8     s . The same initial contours as in Fig. 1(c) have been used.

Fig. 3
Fig. 3

(a) Sonar image ( 212 × 192     pixels ) provided by the Office of Naval Research and the SACLANT Center. (b) Initial contours. (c) Partitioning of (a) with the proposed approach with Q = 10 (computation time 0.8     s ). (d) Red-green-blue image ( 232 × 232   pixels ) with the initial contour displayed on the gray-level image. (e) Segmentation of the gray-level image with the algorithm adapted to Gaussian noise. (f) Segmentation of the hue component of the HSV representation with the proposed approach and Q = 10 (computation time 0.9     s ). (g) Same as (f) but with the algorithm adapted to Gaussian noise.

Equations (3)

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Δ r ( w ) = ( x , y ) Ω r log 2 P r [ q ( x , y ) ] .
Δ Q ( w ) r = 1 R q = 1 Q N r ( q ) log 2 N r ( q ) N r ,
Δ G ( w ) = n ( log 2 N + log 2 p ) + log 2 p + p [ 2 log 2 e + log 2 ( 2 m ̂ x ) + log 2 ( 2 m ̂ y ) ] ,

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