Abstract

A new snake-based segmentation technique of a single object (simply connected) in the presence of inhomogeneous Gaussian noise is proposed, in which the mean in each region is modeled as a polynomial function of the coordinates and which is thus adapted to inhomogeneous illumination. It is shown that the minimization of the stochastic complexity of the image, which can be implemented efficiently, allows one to automatically estimate not only the number and the position of the nodes of the polygonal contour used to describe the object but also the degree of the polynomials that model the variations of the mean.

© 2007 Optical Society of America

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References

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    [CrossRef]

2005 (1)

F. Galland, N. Bertaux, and P. Réfrégier, Pattern Recogn. 38, 1926 (2005).
[CrossRef]

2001 (1)

2000 (1)

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

1999 (1)

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

1998 (1)

C. Xu and J. L. Prince, IEEE Trans. Image Process. 7, 359 (1998).

1996 (1)

1994 (1)

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

1988 (1)

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

IEEE Trans. Image Process. (2)

C. Xu and J. L. Prince, IEEE Trans. Image Process. 7, 359 (1998).

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

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

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

Int. J. Comput. Vis. (2)

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]

Opt. Lett. (2)

Pattern Recogn. (1)

F. Galland, N. Bertaux, and P. Réfrégier, Pattern Recogn. 38, 1926 (2005).
[CrossRef]

Other (5)

J. Rissanen, Stochastic Complexity in Statistical Inquiry, Vol. 15 of Series in Computer Science (World Scientific, 1989).

T. Kanungo, B. Dom, W. Niblack, and D. Steele, in Proceedings of Computer Vision and Pattern Recognition 1994 (IEEE, 1994), p. 609.
[CrossRef]

T. S. Ferguson, Mathematical Statistics, a Decision Theoretic Approach (Academic, 1967).

P. Garthwaite, I. Jolliffe, and B. Jones, Statistical Inference (Prentice-Hall Europe, 1995).

P. Viola and M. Jones, Proceedings of 13th Conference on Computer Vision and Pattern Recognition (IEEE, 2001), p. 511.

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Figures (3)

Fig. 1
Fig. 1

Segmentation of two video images ( 320 × 240   pixels ) : a gray-scale image of a spoon (top row) and a color image of an orange lying on a red fabric (bottom row). Column 1, initial contour; columns 2–4, segmentation results and computation time t using a polynomial for the mean with a degree d = 0 (column 2), d = 1 (column 3), and d = 2 (column 4).

Fig. 2
Fig. 2

Segmentation with estimation of the polynomial degree of the mean inside each region. (a), (c) synthetic images ( 256 × 256   pixels ) corrupted with Gaussian noise ( d a = 0 , d b = 2 ); (b), (d) segmentation results ( d ̂ a = 0 , d ̂ b = 2 ). The initial contours are shown in (a), (c).

Fig. 3
Fig. 3

Variation of the stochastic complexity (a) as a function of ( d ̂ a , d ̂ b ) on the two real images of Fig. 1, with a minimum for ( d ̂ a , d ̂ b ) = ( 2 , 2 ) . (b), (c) reconstructions of the corresponding estimated means.

Equations (7)

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m r d ( x , y ) = 0 i + j d a r d [ i , j ] x i y j with ( i , j ) N 2 ,
Δ P ( w ) = r { a , b } α ( d r ) 2 log N r , α ( d r ) = d r 2 + 3 d r + 4 2 .
L ( Ω r θ r d r ) = N r 2 log 2 π σ r 2 ( x , y ) Ω r [ s ( x , y ) m r d r ( x , y ) ] 2 2 σ r 2 ,
i + j d r S r [ i 0 + i , j 0 + j , 0 ] a ̂ r d r [ i , j ] = S r [ i 0 , j 0 , 1 ] ,
σ ̂ r 2 = 1 N r [ S r [ 0 , 0 , 2 ] 2 i + j d r S r [ i , j , 1 ] a ̂ r d r [ i , j ] + i + j d r k + l d r S r [ i + k , j + l , 0 ] a ̂ r d r [ i , j ] a ̂ r d r [ k , l ] ] .
L ( Ω r θ ̂ r d r ) = N r 2 log ( 2 π σ ̂ r 2 ) N r 2 .
Δ ( w ) = Δ C ( w ) + r min d [ α ( d ) 2 log N r L ( Ω r θ ̂ r d ) ] .

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