Abstract

We describe a segmentation processor that is optimal for tracking the shape of a target with random white Gaussian intensity appearing on a random white Gaussian spatially disjoint background. This algorithm, based on an active contours model (snakes), consists of correlations of binary references with preprocessed versions of the scene image. This result can provide a practical method to adapt the reference image to correlation techniques.

© 1996 Optical Society of America

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References

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  1. A. Vander Lugt, IEEE Trans. Inform. Theory IT-10, 139 (1964).
    [CrossRef]
  2. B. Vijaya, V. K. Kumar, Appl. Opt. 31, 4773 (1992).
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  3. B. Javidi, J. Wang, Opt. Lett. 20, 401 (1994).
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  4. D. L. Flannery, J. Opt. Soc. Am. 12, 66 (1995).
    [CrossRef]
  5. F. Goudail, Ph. Réfrégier, Opt. Lett. 21, 495 (1996).
    [CrossRef] [PubMed]
  6. M. Kass, A. Witkin, D. Terzopoulos, Int. J. Comput. Vision 2, 321 (1988).
    [CrossRef]
  7. R. Ronfard, Comput. Vision 2, 229 (1994).
    [CrossRef]
  8. C. Kervrann, F. Heitz, “A hierarchical statistical framework for the segmentation of deformable objects in image sequences,” in Proceedings of the IEEE Conference on Computer Vision Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 724–728.
    [CrossRef]
  9. R. Deriche, Int. J. Comput. Vision 1, 167 (1987).
    [CrossRef]

1996 (1)

1995 (1)

D. L. Flannery, J. Opt. Soc. Am. 12, 66 (1995).
[CrossRef]

1994 (2)

1992 (1)

1988 (1)

M. Kass, A. Witkin, D. Terzopoulos, Int. J. Comput. Vision 2, 321 (1988).
[CrossRef]

1987 (1)

R. Deriche, Int. J. Comput. Vision 1, 167 (1987).
[CrossRef]

1964 (1)

A. Vander Lugt, IEEE Trans. Inform. Theory IT-10, 139 (1964).
[CrossRef]

Deriche, R.

R. Deriche, Int. J. Comput. Vision 1, 167 (1987).
[CrossRef]

Flannery, D. L.

D. L. Flannery, J. Opt. Soc. Am. 12, 66 (1995).
[CrossRef]

Goudail, F.

Heitz, F.

C. Kervrann, F. Heitz, “A hierarchical statistical framework for the segmentation of deformable objects in image sequences,” in Proceedings of the IEEE Conference on Computer Vision Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 724–728.
[CrossRef]

Javidi, B.

Kass, M.

M. Kass, A. Witkin, D. Terzopoulos, Int. J. Comput. Vision 2, 321 (1988).
[CrossRef]

Kervrann, C.

C. Kervrann, F. Heitz, “A hierarchical statistical framework for the segmentation of deformable objects in image sequences,” in Proceedings of the IEEE Conference on Computer Vision Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 724–728.
[CrossRef]

Kumar, V. K.

Réfrégier, Ph.

Ronfard, R.

R. Ronfard, Comput. Vision 2, 229 (1994).
[CrossRef]

Terzopoulos, D.

M. Kass, A. Witkin, D. Terzopoulos, Int. J. Comput. Vision 2, 321 (1988).
[CrossRef]

Vander Lugt, A.

A. Vander Lugt, IEEE Trans. Inform. Theory IT-10, 139 (1964).
[CrossRef]

Vijaya, B.

Wang, J.

Witkin, A.

M. Kass, A. Witkin, D. Terzopoulos, Int. J. Comput. Vision 2, 321 (1988).
[CrossRef]

Appl. Opt. (1)

Comput. Vision (1)

R. Ronfard, Comput. Vision 2, 229 (1994).
[CrossRef]

IEEE Trans. Inform. Theory (1)

A. Vander Lugt, IEEE Trans. Inform. Theory IT-10, 139 (1964).
[CrossRef]

Int. J. Comput. Vision (2)

M. Kass, A. Witkin, D. Terzopoulos, Int. J. Comput. Vision 2, 321 (1988).
[CrossRef]

R. Deriche, Int. J. Comput. Vision 1, 167 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

D. L. Flannery, J. Opt. Soc. Am. 12, 66 (1995).
[CrossRef]

Opt. Lett. (2)

Other (1)

C. Kervrann, F. Heitz, “A hierarchical statistical framework for the segmentation of deformable objects in image sequences,” in Proceedings of the IEEE Conference on Computer Vision Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 724–728.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Scene in the white Gaussian case, a realization of the image model of Eq. (1) with statistical parameters ma = 0.57, mb = 0.43, and σa = σb = 0.14. (b) Initialization of the snake. (c) Threshold result of the processing of (a) with a Canny–Deriche filter. (d) Final state of the snake after optimization of J(w, s).

Fig. 2
Fig. 2

(a) Scene in the white Gaussian case, a realization of the image model of Eq. (1) with statistical parameters ma = mb = 0.5, σa = 0.05, and σb = 0.15. ( b) Final state of the snake after optimization of J(w, s). The snake initialization is the same as in Fig. 1.

Fig. 3
Fig. 3

(a) Realistic scene. (b) Final state of the snake after optimization of J(w, s). The snake initialization is the same as in Fig. 1.

Equations (11)

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s i = a i w i + b i [ 1 - w i ] .
P [ s H w ] = A exp [ - 1 2 σ a 2 i = 1 N ( s i - m a ) 2 w i - 1 2 σ b 2 i = 1 N ( s i - m b ) 2 ( 1 - w i ) ] ,
A - 1 = ( 2 π ) N / 2 σ a N a ( w ) σ b N b ( w ) ,
m ^ a ( w ) = 1 N a ( w ) { i w i = 1 } s i ,
m ^ b ( w ) = 1 N b ( w ) { i w i = 1 } s i ,
σ ^ a 2 ( w ) = 1 N a ( w ) { i w i = 1 } [ s i - m ^ a ( w ) ] 2 ,
σ ^ b 2 ( w ) = 1 N b ( w ) { i w i = 0 } [ s i - m ^ b ( w ) ] 2 .
F ( w , s ) = - N 2 log ( 2 π ) - N a ( w ) log [ σ ^ a ( w ) ] - N b ( w ) log [ σ ^ b ( w ) ] - 1 2 σ ^ a 2 ( w ) i = 1 N [ s i - m a ( w ) ] 2 × w i - 1 2 σ ^ b 2 ( w ) i = 1 N [ s i - m ^ b ( w ) ] 2 ( 1 - w i ) .
J ( w , s ) = N a ( w ) log [ σ ^ a 2 ( w ) ] + N b ( w ) log [ σ ^ b 2 ( w ) ] .
σ ^ a 2 ( w ) = 1 N a ( w ) ( s 2 w ) 0 - 1 N a 2 ( w ) ( s w ) 0 2 ,
σ ^ b 2 ( w ) = 1 N b ( w ) [ i = 1 N s i 2 - ( s 2 w ) 0 ] - 1 N b 2 ( w ) [ i = 1 N s i - ( s w ) 0 ] 2 .

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