Abstract

Image and sequence segmentation of a the segmentation task are discussed from the point of view of optimizing the segmentation criterion. Such a segmentation criterion involves so-called (boundary and region) descriptors, which, in general, may depend on their respective boundaries or regions. This dependency must be taken into account when one is computing the criterion derivative with respect to the unknown object domain (defined by its boundary). If this dependency not considered, some correctional terms may be omitted. Computing the derivative of the segmentation criterion with a dynamic scheme is described. The scheme is general enough to provide a framework for a wide variety of applications in segmentation. It also provides a theoretical meaning to the philosophy of active contours.

© 2004 Optical Society of America

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2003 (3)

S. Jehan-Besson, M. Barlaud, and G. Aubert, “DREAM2S: deformable regions driven by an Eulerian accurate minimization method for image and video segmentation,” Int. J. Comput. Vision 53, 45–70 (2003).
[CrossRef]

G. Aubert, M. Barlaud, O. Faugeras, and S. Jehan-Besson, “Image segmentation using active contours: calculus of variations or shape gradients?” SIAM J. Appl. Math. 63, 2128–2154 (2003).
[CrossRef]

F. Precioso, M. Barlaud, T. Blu, and M. Unser, “Smoothing b-spline active contour for fast and robust image and video segmentation,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003).

2002 (8)

F. Precioso and M. Barlaud, “B-spline active contours with handling of topological changes for fast video segmentation,” EURASIP J. Appl. Signal Process. 2002, 555–560 (2002).
[CrossRef]

F. Precioso and M. Barlaud, “Regular spatial b-spline active contours for fast video segmentation,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 761–764.

M. Gastaud and M. Barlaud, “Video segmentation using region based active contours on a group of pictures,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), Vol. II, pp. 81–84.

S. Soatto and A. J. Yezzi, “Deformation: deforming motion, shape average and the joint registration and segmentation of images,” in Proceedings of European Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 32–47.

M. Gastaud, M. Barlaud, and G. Aubert, “Tracking video objects using active contours,” in Proceedings of Workshop on Motion and Video Computing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 90–95.

Y. Chen, H. D. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. S. Gopinath, R. W. Briggs, and E. A. Geiser, “Using prior shapes in geometric active contours in a variational framework,” Int. J. Comput. Vision 50, 315–328 (2002).
[CrossRef]

D. Cremers, F. Tischhäuser, J. Weickert, and C. Schnörr, “Diffusion snakes: introducing statistical shape knowledge into the Mumford-Shah functional,” Int. J. Comput. Vision 50, 295–313 (2002).
[CrossRef]

N. Paragios and R. Deriche, “Geodesic active regions and level set methods for supervised texture segmentation,” Int. J. Comput. Vision 46, 223–247 (2002).
[CrossRef]

2001 (4)

T. Chan and L. Vese, “Active contours without edges,” IEEE Trans. Image Process. 10, 266–277 (2001).
[CrossRef]

E. Debreuve, M. Barlaud, G. Aubert, and J. Darcourt, “Space time segmentation using level set active contours applied to myocardial gated SPECT,” IEEE Trans. Med. Imaging 20, 643–659 (2001).
[CrossRef] [PubMed]

S. Jehan-Besson, M. Barlaud, and G. Aubert, “Video object segmentation using Eulerian region-based active contours,” in Proceedings of International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2001), pp. 353–361.

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus and Optimization (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001).

2000 (3)

C. Samson, L. Blanc-Féraud, G. Aubert, and J. Zerubia, “A level set model for image classification,” Int. J. Comput. Vision 40, 187–197 (2000).
[CrossRef]

J. Gomes and O. D. Faugeras, “Reconciling distance functions and level sets,” J. Visual Commun. Image Represent. 11, 209–223 (2000).
[CrossRef]

P. Thevenaz, T. Blu, and M. Unser, “Interpolation revisited [medical images application],” IEEE Trans. Med. Imaging 19, 739–758 (2000).
[CrossRef]

1999 (5)

D. Adalsteinsson and J. A. Sethian, “The fast construction of extension velocities in level set methods,” J. Comput. Phys. 148, 2–22 (1999).
[CrossRef]

N. Paragios and R. Deriche, “Geodesic active regions for motion estimation and tracking,” in Proceedings of International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 688–694.
[CrossRef]

O. Amadieu, E. Debreuve, M. Barlaud, and G. Aubert, “Inward and outward curve evolution using level set method,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 188–192.

A. J. Yezzi, A. Tsai, and A. Willsky, “A statistical approach to snakes for bimodal and trimodal imagery,” in Proceedings of International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 898–903.
[CrossRef]

C. Chesnaud, P. Réfrégier, and V. Boulet, “Statistical region snake-based segmentation adapted to different physical noise models,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145–1156 (1999).
[CrossRef]

1997 (1)

V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,” Int. J. Comput. Vision 22, 61–79 (1997).
[CrossRef]

1996 (3)

A. Chakraborty, L. Staib, and J. Duncan, “Deformable boundary finding in medical images by integrating gradient and region information,” IEEE Trans. Med. Imaging 15, 859–870 (1996).
[CrossRef] [PubMed]

S. Zhu and A. Yuille, “Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 884–900 (1996).
[CrossRef]

J. A. Sethian, “A fast marching level set method for monotonically advancing fronts,” Proc. Natl. Acad. Sci. (USA) 93, 1591–1595 (1996).
[CrossRef]

1995 (1)

D. Adalsteinsson and J. A. Sethian, “A fast level set method for propagating interfaces,” J. Comput. Phys. 118, 269–277 (1995).
[CrossRef]

1994 (1)

R. Ronfard, “Region-based strategies for active contour models,” Int. J. Comput. Vision 13, 229–251 (1994).
[CrossRef]

1993 (2)

D. L. Chopp, “Computing minimal surfaces via level set curvature flow,” J. Comput. Phys. 106, 77–91 (1993).
[CrossRef]

M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing. I. Theory,” IEEE Trans. Signal Process. 41, 821–833 (1993).
[CrossRef]

1992 (1)

J. Sokolowski and J.-P. Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis (Springer-Verlag, Berlin, 1992).
[CrossRef]

1991 (1)

L. Cohen, “On active contour models and balloons,” Comput. Vis. Graph. Image Process. 53, 211–218 (1991).

1988 (2)

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vision 1, 321–332 (1988).
[CrossRef]

S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys. 79, 12–49 (1988).
[CrossRef]

Adalsteinsson, D.

D. Adalsteinsson and J. A. Sethian, “The fast construction of extension velocities in level set methods,” J. Comput. Phys. 148, 2–22 (1999).
[CrossRef]

D. Adalsteinsson and J. A. Sethian, “A fast level set method for propagating interfaces,” J. Comput. Phys. 118, 269–277 (1995).
[CrossRef]

Aldroubi, A.

M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing. I. Theory,” IEEE Trans. Signal Process. 41, 821–833 (1993).
[CrossRef]

Amadieu, O.

O. Amadieu, E. Debreuve, M. Barlaud, and G. Aubert, “Inward and outward curve evolution using level set method,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 188–192.

Aubert, G.

S. Jehan-Besson, M. Barlaud, and G. Aubert, “DREAM2S: deformable regions driven by an Eulerian accurate minimization method for image and video segmentation,” Int. J. Comput. Vision 53, 45–70 (2003).
[CrossRef]

G. Aubert, M. Barlaud, O. Faugeras, and S. Jehan-Besson, “Image segmentation using active contours: calculus of variations or shape gradients?” SIAM J. Appl. Math. 63, 2128–2154 (2003).
[CrossRef]

M. Gastaud, M. Barlaud, and G. Aubert, “Tracking video objects using active contours,” in Proceedings of Workshop on Motion and Video Computing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 90–95.

E. Debreuve, M. Barlaud, G. Aubert, and J. Darcourt, “Space time segmentation using level set active contours applied to myocardial gated SPECT,” IEEE Trans. Med. Imaging 20, 643–659 (2001).
[CrossRef] [PubMed]

S. Jehan-Besson, M. Barlaud, and G. Aubert, “Video object segmentation using Eulerian region-based active contours,” in Proceedings of International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2001), pp. 353–361.

C. Samson, L. Blanc-Féraud, G. Aubert, and J. Zerubia, “A level set model for image classification,” Int. J. Comput. Vision 40, 187–197 (2000).
[CrossRef]

O. Amadieu, E. Debreuve, M. Barlaud, and G. Aubert, “Inward and outward curve evolution using level set method,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 188–192.

Barlaud, M.

G. Aubert, M. Barlaud, O. Faugeras, and S. Jehan-Besson, “Image segmentation using active contours: calculus of variations or shape gradients?” SIAM J. Appl. Math. 63, 2128–2154 (2003).
[CrossRef]

S. Jehan-Besson, M. Barlaud, and G. Aubert, “DREAM2S: deformable regions driven by an Eulerian accurate minimization method for image and video segmentation,” Int. J. Comput. Vision 53, 45–70 (2003).
[CrossRef]

F. Precioso, M. Barlaud, T. Blu, and M. Unser, “Smoothing b-spline active contour for fast and robust image and video segmentation,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003).

F. Precioso and M. Barlaud, “B-spline active contours with handling of topological changes for fast video segmentation,” EURASIP J. Appl. Signal Process. 2002, 555–560 (2002).
[CrossRef]

F. Precioso and M. Barlaud, “Regular spatial b-spline active contours for fast video segmentation,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 761–764.

M. Gastaud and M. Barlaud, “Video segmentation using region based active contours on a group of pictures,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), Vol. II, pp. 81–84.

M. Gastaud, M. Barlaud, and G. Aubert, “Tracking video objects using active contours,” in Proceedings of Workshop on Motion and Video Computing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 90–95.

E. Debreuve, M. Barlaud, G. Aubert, and J. Darcourt, “Space time segmentation using level set active contours applied to myocardial gated SPECT,” IEEE Trans. Med. Imaging 20, 643–659 (2001).
[CrossRef] [PubMed]

S. Jehan-Besson, M. Barlaud, and G. Aubert, “Video object segmentation using Eulerian region-based active contours,” in Proceedings of International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2001), pp. 353–361.

O. Amadieu, E. Debreuve, M. Barlaud, and G. Aubert, “Inward and outward curve evolution using level set method,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 188–192.

Blanc-Féraud, L.

C. Samson, L. Blanc-Féraud, G. Aubert, and J. Zerubia, “A level set model for image classification,” Int. J. Comput. Vision 40, 187–197 (2000).
[CrossRef]

Blu, T.

F. Precioso, M. Barlaud, T. Blu, and M. Unser, “Smoothing b-spline active contour for fast and robust image and video segmentation,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003).

P. Thevenaz, T. Blu, and M. Unser, “Interpolation revisited [medical images application],” IEEE Trans. Med. Imaging 19, 739–758 (2000).
[CrossRef]

Boulet, V.

C. Chesnaud, P. Réfrégier, and V. Boulet, “Statistical region snake-based segmentation adapted to different physical noise models,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145–1156 (1999).
[CrossRef]

Briggs, R. W.

Y. Chen, H. D. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. S. Gopinath, R. W. Briggs, and E. A. Geiser, “Using prior shapes in geometric active contours in a variational framework,” Int. J. Comput. Vision 50, 315–328 (2002).
[CrossRef]

Caselles, V.

V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,” Int. J. Comput. Vision 22, 61–79 (1997).
[CrossRef]

Chakraborty, A.

A. Chakraborty, L. Staib, and J. Duncan, “Deformable boundary finding in medical images by integrating gradient and region information,” IEEE Trans. Med. Imaging 15, 859–870 (1996).
[CrossRef] [PubMed]

Chan, T.

T. Chan and L. Vese, “Active contours without edges,” IEEE Trans. Image Process. 10, 266–277 (2001).
[CrossRef]

Chen, Y.

Y. Chen, H. D. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. S. Gopinath, R. W. Briggs, and E. A. Geiser, “Using prior shapes in geometric active contours in a variational framework,” Int. J. Comput. Vision 50, 315–328 (2002).
[CrossRef]

Chesnaud, C.

C. Chesnaud, P. Réfrégier, and V. Boulet, “Statistical region snake-based segmentation adapted to different physical noise models,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145–1156 (1999).
[CrossRef]

Chopp, D. L.

D. L. Chopp, “Computing minimal surfaces via level set curvature flow,” J. Comput. Phys. 106, 77–91 (1993).
[CrossRef]

Cohen, L.

L. Cohen, “On active contour models and balloons,” Comput. Vis. Graph. Image Process. 53, 211–218 (1991).

Cremers, D.

D. Cremers, F. Tischhäuser, J. Weickert, and C. Schnörr, “Diffusion snakes: introducing statistical shape knowledge into the Mumford-Shah functional,” Int. J. Comput. Vision 50, 295–313 (2002).
[CrossRef]

Darcourt, J.

E. Debreuve, M. Barlaud, G. Aubert, and J. Darcourt, “Space time segmentation using level set active contours applied to myocardial gated SPECT,” IEEE Trans. Med. Imaging 20, 643–659 (2001).
[CrossRef] [PubMed]

Debreuve, E.

E. Debreuve, M. Barlaud, G. Aubert, and J. Darcourt, “Space time segmentation using level set active contours applied to myocardial gated SPECT,” IEEE Trans. Med. Imaging 20, 643–659 (2001).
[CrossRef] [PubMed]

O. Amadieu, E. Debreuve, M. Barlaud, and G. Aubert, “Inward and outward curve evolution using level set method,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 188–192.

Delfour, M. C.

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus and Optimization (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001).

Deriche, R.

N. Paragios and R. Deriche, “Geodesic active regions and level set methods for supervised texture segmentation,” Int. J. Comput. Vision 46, 223–247 (2002).
[CrossRef]

N. Paragios and R. Deriche, “Geodesic active regions for motion estimation and tracking,” in Proceedings of International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 688–694.
[CrossRef]

Duncan, J.

A. Chakraborty, L. Staib, and J. Duncan, “Deformable boundary finding in medical images by integrating gradient and region information,” IEEE Trans. Med. Imaging 15, 859–870 (1996).
[CrossRef] [PubMed]

Eden, M.

M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing. I. Theory,” IEEE Trans. Signal Process. 41, 821–833 (1993).
[CrossRef]

Faugeras, O.

G. Aubert, M. Barlaud, O. Faugeras, and S. Jehan-Besson, “Image segmentation using active contours: calculus of variations or shape gradients?” SIAM J. Appl. Math. 63, 2128–2154 (2003).
[CrossRef]

Faugeras, O. D.

J. Gomes and O. D. Faugeras, “Reconciling distance functions and level sets,” J. Visual Commun. Image Represent. 11, 209–223 (2000).
[CrossRef]

Gastaud, M.

M. Gastaud and M. Barlaud, “Video segmentation using region based active contours on a group of pictures,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), Vol. II, pp. 81–84.

M. Gastaud, M. Barlaud, and G. Aubert, “Tracking video objects using active contours,” in Proceedings of Workshop on Motion and Video Computing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 90–95.

Geiser, E. A.

Y. Chen, H. D. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. S. Gopinath, R. W. Briggs, and E. A. Geiser, “Using prior shapes in geometric active contours in a variational framework,” Int. J. Comput. Vision 50, 315–328 (2002).
[CrossRef]

Gomes, J.

J. Gomes and O. D. Faugeras, “Reconciling distance functions and level sets,” J. Visual Commun. Image Represent. 11, 209–223 (2000).
[CrossRef]

Gopinath, K. S.

Y. Chen, H. D. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. S. Gopinath, R. W. Briggs, and E. A. Geiser, “Using prior shapes in geometric active contours in a variational framework,” Int. J. Comput. Vision 50, 315–328 (2002).
[CrossRef]

Huang, F.

Y. Chen, H. D. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. S. Gopinath, R. W. Briggs, and E. A. Geiser, “Using prior shapes in geometric active contours in a variational framework,” Int. J. Comput. Vision 50, 315–328 (2002).
[CrossRef]

Jehan-Besson, S.

G. Aubert, M. Barlaud, O. Faugeras, and S. Jehan-Besson, “Image segmentation using active contours: calculus of variations or shape gradients?” SIAM J. Appl. Math. 63, 2128–2154 (2003).
[CrossRef]

S. Jehan-Besson, M. Barlaud, and G. Aubert, “DREAM2S: deformable regions driven by an Eulerian accurate minimization method for image and video segmentation,” Int. J. Comput. Vision 53, 45–70 (2003).
[CrossRef]

S. Jehan-Besson, M. Barlaud, and G. Aubert, “Video object segmentation using Eulerian region-based active contours,” in Proceedings of International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2001), pp. 353–361.

Kass, M.

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vision 1, 321–332 (1988).
[CrossRef]

Kimmel, R.

V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,” Int. J. Comput. Vision 22, 61–79 (1997).
[CrossRef]

Osher, S.

S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys. 79, 12–49 (1988).
[CrossRef]

Paragios, N.

N. Paragios and R. Deriche, “Geodesic active regions and level set methods for supervised texture segmentation,” Int. J. Comput. Vision 46, 223–247 (2002).
[CrossRef]

N. Paragios and R. Deriche, “Geodesic active regions for motion estimation and tracking,” in Proceedings of International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 688–694.
[CrossRef]

Precioso, F.

F. Precioso, M. Barlaud, T. Blu, and M. Unser, “Smoothing b-spline active contour for fast and robust image and video segmentation,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003).

F. Precioso and M. Barlaud, “Regular spatial b-spline active contours for fast video segmentation,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 761–764.

F. Precioso and M. Barlaud, “B-spline active contours with handling of topological changes for fast video segmentation,” EURASIP J. Appl. Signal Process. 2002, 555–560 (2002).
[CrossRef]

Réfrégier, P.

C. Chesnaud, P. Réfrégier, and V. Boulet, “Statistical region snake-based segmentation adapted to different physical noise models,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145–1156 (1999).
[CrossRef]

Ronfard, R.

R. Ronfard, “Region-based strategies for active contour models,” Int. J. Comput. Vision 13, 229–251 (1994).
[CrossRef]

Samson, C.

C. Samson, L. Blanc-Féraud, G. Aubert, and J. Zerubia, “A level set model for image classification,” Int. J. Comput. Vision 40, 187–197 (2000).
[CrossRef]

Sapiro, G.

V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,” Int. J. Comput. Vision 22, 61–79 (1997).
[CrossRef]

Schnörr, C.

D. Cremers, F. Tischhäuser, J. Weickert, and C. Schnörr, “Diffusion snakes: introducing statistical shape knowledge into the Mumford-Shah functional,” Int. J. Comput. Vision 50, 295–313 (2002).
[CrossRef]

Sethian, J. A.

D. Adalsteinsson and J. A. Sethian, “The fast construction of extension velocities in level set methods,” J. Comput. Phys. 148, 2–22 (1999).
[CrossRef]

J. A. Sethian, “A fast marching level set method for monotonically advancing fronts,” Proc. Natl. Acad. Sci. (USA) 93, 1591–1595 (1996).
[CrossRef]

D. Adalsteinsson and J. A. Sethian, “A fast level set method for propagating interfaces,” J. Comput. Phys. 118, 269–277 (1995).
[CrossRef]

S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys. 79, 12–49 (1988).
[CrossRef]

Soatto, S.

S. Soatto and A. J. Yezzi, “Deformation: deforming motion, shape average and the joint registration and segmentation of images,” in Proceedings of European Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 32–47.

Sokolowski, J.

J. Sokolowski and J.-P. Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis (Springer-Verlag, Berlin, 1992).
[CrossRef]

Staib, L.

A. Chakraborty, L. Staib, and J. Duncan, “Deformable boundary finding in medical images by integrating gradient and region information,” IEEE Trans. Med. Imaging 15, 859–870 (1996).
[CrossRef] [PubMed]

Tagare, H. D.

Y. Chen, H. D. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. S. Gopinath, R. W. Briggs, and E. A. Geiser, “Using prior shapes in geometric active contours in a variational framework,” Int. J. Comput. Vision 50, 315–328 (2002).
[CrossRef]

Terzopoulos, D.

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vision 1, 321–332 (1988).
[CrossRef]

Thevenaz, P.

P. Thevenaz, T. Blu, and M. Unser, “Interpolation revisited [medical images application],” IEEE Trans. Med. Imaging 19, 739–758 (2000).
[CrossRef]

Thiruvenkadam, S.

Y. Chen, H. D. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. S. Gopinath, R. W. Briggs, and E. A. Geiser, “Using prior shapes in geometric active contours in a variational framework,” Int. J. Comput. Vision 50, 315–328 (2002).
[CrossRef]

Tischhäuser, F.

D. Cremers, F. Tischhäuser, J. Weickert, and C. Schnörr, “Diffusion snakes: introducing statistical shape knowledge into the Mumford-Shah functional,” Int. J. Comput. Vision 50, 295–313 (2002).
[CrossRef]

Tsai, A.

A. J. Yezzi, A. Tsai, and A. Willsky, “A statistical approach to snakes for bimodal and trimodal imagery,” in Proceedings of International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 898–903.
[CrossRef]

Unser, M.

F. Precioso, M. Barlaud, T. Blu, and M. Unser, “Smoothing b-spline active contour for fast and robust image and video segmentation,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003).

P. Thevenaz, T. Blu, and M. Unser, “Interpolation revisited [medical images application],” IEEE Trans. Med. Imaging 19, 739–758 (2000).
[CrossRef]

M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing. I. Theory,” IEEE Trans. Signal Process. 41, 821–833 (1993).
[CrossRef]

Vese, L.

T. Chan and L. Vese, “Active contours without edges,” IEEE Trans. Image Process. 10, 266–277 (2001).
[CrossRef]

Weickert, J.

D. Cremers, F. Tischhäuser, J. Weickert, and C. Schnörr, “Diffusion snakes: introducing statistical shape knowledge into the Mumford-Shah functional,” Int. J. Comput. Vision 50, 295–313 (2002).
[CrossRef]

Willsky, A.

A. J. Yezzi, A. Tsai, and A. Willsky, “A statistical approach to snakes for bimodal and trimodal imagery,” in Proceedings of International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 898–903.
[CrossRef]

Wilson, D.

Y. Chen, H. D. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. S. Gopinath, R. W. Briggs, and E. A. Geiser, “Using prior shapes in geometric active contours in a variational framework,” Int. J. Comput. Vision 50, 315–328 (2002).
[CrossRef]

Witkin, A.

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vision 1, 321–332 (1988).
[CrossRef]

Yezzi, A. J.

S. Soatto and A. J. Yezzi, “Deformation: deforming motion, shape average and the joint registration and segmentation of images,” in Proceedings of European Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 32–47.

A. J. Yezzi, A. Tsai, and A. Willsky, “A statistical approach to snakes for bimodal and trimodal imagery,” in Proceedings of International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 898–903.
[CrossRef]

Yuille, A.

S. Zhu and A. Yuille, “Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 884–900 (1996).
[CrossRef]

Zerubia, J.

C. Samson, L. Blanc-Féraud, G. Aubert, and J. Zerubia, “A level set model for image classification,” Int. J. Comput. Vision 40, 187–197 (2000).
[CrossRef]

Zhu, S.

S. Zhu and A. Yuille, “Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 884–900 (1996).
[CrossRef]

Zolésio, J.-P.

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus and Optimization (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001).

J. Sokolowski and J.-P. Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis (Springer-Verlag, Berlin, 1992).
[CrossRef]

Comput. Vis. Graph. Image Process. (1)

L. Cohen, “On active contour models and balloons,” Comput. Vis. Graph. Image Process. 53, 211–218 (1991).

EURASIP J. Appl. Signal Process. (1)

F. Precioso and M. Barlaud, “B-spline active contours with handling of topological changes for fast video segmentation,” EURASIP J. Appl. Signal Process. 2002, 555–560 (2002).
[CrossRef]

IEEE Trans. Image Process. (1)

T. Chan and L. Vese, “Active contours without edges,” IEEE Trans. Image Process. 10, 266–277 (2001).
[CrossRef]

IEEE Trans. Med. Imaging (3)

E. Debreuve, M. Barlaud, G. Aubert, and J. Darcourt, “Space time segmentation using level set active contours applied to myocardial gated SPECT,” IEEE Trans. Med. Imaging 20, 643–659 (2001).
[CrossRef] [PubMed]

A. Chakraborty, L. Staib, and J. Duncan, “Deformable boundary finding in medical images by integrating gradient and region information,” IEEE Trans. Med. Imaging 15, 859–870 (1996).
[CrossRef] [PubMed]

P. Thevenaz, T. Blu, and M. Unser, “Interpolation revisited [medical images application],” IEEE Trans. Med. Imaging 19, 739–758 (2000).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

S. Zhu and A. Yuille, “Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 884–900 (1996).
[CrossRef]

C. Chesnaud, P. Réfrégier, and V. Boulet, “Statistical region snake-based segmentation adapted to different physical noise models,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145–1156 (1999).
[CrossRef]

IEEE Trans. Signal Process. (1)

M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing. I. Theory,” IEEE Trans. Signal Process. 41, 821–833 (1993).
[CrossRef]

Int. J. Comput. Vision (8)

Y. Chen, H. D. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. S. Gopinath, R. W. Briggs, and E. A. Geiser, “Using prior shapes in geometric active contours in a variational framework,” Int. J. Comput. Vision 50, 315–328 (2002).
[CrossRef]

D. Cremers, F. Tischhäuser, J. Weickert, and C. Schnörr, “Diffusion snakes: introducing statistical shape knowledge into the Mumford-Shah functional,” Int. J. Comput. Vision 50, 295–313 (2002).
[CrossRef]

C. Samson, L. Blanc-Féraud, G. Aubert, and J. Zerubia, “A level set model for image classification,” Int. J. Comput. Vision 40, 187–197 (2000).
[CrossRef]

S. Jehan-Besson, M. Barlaud, and G. Aubert, “DREAM2S: deformable regions driven by an Eulerian accurate minimization method for image and video segmentation,” Int. J. Comput. Vision 53, 45–70 (2003).
[CrossRef]

N. Paragios and R. Deriche, “Geodesic active regions and level set methods for supervised texture segmentation,” Int. J. Comput. Vision 46, 223–247 (2002).
[CrossRef]

V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,” Int. J. Comput. Vision 22, 61–79 (1997).
[CrossRef]

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vision 1, 321–332 (1988).
[CrossRef]

R. Ronfard, “Region-based strategies for active contour models,” Int. J. Comput. Vision 13, 229–251 (1994).
[CrossRef]

J. Comput. Phys. (4)

S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys. 79, 12–49 (1988).
[CrossRef]

D. L. Chopp, “Computing minimal surfaces via level set curvature flow,” J. Comput. Phys. 106, 77–91 (1993).
[CrossRef]

D. Adalsteinsson and J. A. Sethian, “A fast level set method for propagating interfaces,” J. Comput. Phys. 118, 269–277 (1995).
[CrossRef]

D. Adalsteinsson and J. A. Sethian, “The fast construction of extension velocities in level set methods,” J. Comput. Phys. 148, 2–22 (1999).
[CrossRef]

J. Visual Commun. Image Represent. (1)

J. Gomes and O. D. Faugeras, “Reconciling distance functions and level sets,” J. Visual Commun. Image Represent. 11, 209–223 (2000).
[CrossRef]

Proc. Natl. Acad. Sci. (USA) (1)

J. A. Sethian, “A fast marching level set method for monotonically advancing fronts,” Proc. Natl. Acad. Sci. (USA) 93, 1591–1595 (1996).
[CrossRef]

SIAM J. Appl. Math. (1)

G. Aubert, M. Barlaud, O. Faugeras, and S. Jehan-Besson, “Image segmentation using active contours: calculus of variations or shape gradients?” SIAM J. Appl. Math. 63, 2128–2154 (2003).
[CrossRef]

Other (15)

M. Gastaud, M. Barlaud, and G. Aubert, “Tracking video objects using active contours,” in Proceedings of Workshop on Motion and Video Computing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 90–95.

O. Amadieu, E. Debreuve, M. Barlaud, and G. Aubert, “Inward and outward curve evolution using level set method,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 188–192.

J. Sokolowski and J.-P. Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis (Springer-Verlag, Berlin, 1992).
[CrossRef]

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus and Optimization (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001).

L. Cohen, E. Bardinet, and N. Ayache, “Reconstruction of digital terrain model with a lake,” in Conference on Geometric Methods in Computer Vision II, Medical Imaging 2001, M. Sonka and K. M. Hanson, eds., Proc. SPIE 2031, 38–50 (1993).
[CrossRef]

A. J. Yezzi, A. Tsai, and A. Willsky, “A statistical approach to snakes for bimodal and trimodal imagery,” in Proceedings of International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 898–903.
[CrossRef]

N. Paragios and R. Deriche, “Geodesic active regions for motion estimation and tracking,” in Proceedings of International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 688–694.
[CrossRef]

G. Barles, “Remarks on a flame propagation model,” Tech. Rep. 464, Project Sinus (Institute National de Recherche en Informatique et en Automatique, Sophia Antipolis, France, 1985).

P. Charbonnier and O. Cuisenaire, “Une étude des contours actifs: modèles classique, géométrique et géodésique,” Tech. Rep. 163 (Laboratoire de Télécommunications et Télédétection, Université Catholique de Louvain, Louvain, France, 1996).

F. Precioso, M. Barlaud, T. Blu, and M. Unser, “Smoothing b-spline active contour for fast and robust image and video segmentation,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003).

S. Jehan-Besson, M. Barlaud, and G. Aubert, “Video object segmentation using Eulerian region-based active contours,” in Proceedings of International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2001), pp. 353–361.

M. Gastaud and M. Barlaud, “Video segmentation using region based active contours on a group of pictures,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), Vol. II, pp. 81–84.

S. Soatto and A. J. Yezzi, “Deformation: deforming motion, shape average and the joint registration and segmentation of images,” in Proceedings of European Conference on Computer Vision (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 32–47.

M. Jacob, T. Blu, and M. Unser, “A unifying approach and interface for spline-based snakes,” in International Symposium on Medical Imaging: Image Processing, Medical Imaging 2001, M. Sonka and K. M. Hanson, eds., Proc. SPIE 4322, 340–347 (2001).
[CrossRef]

F. Precioso and M. Barlaud, “Regular spatial b-spline active contours for fast video segmentation,” in Proceedings of International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), pp. 761–764.

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Figures (6)

Fig. 1
Fig. 1

Segmentation of the image “X-ray” by use of a smoothing spline with (top left) 128 sampling points, (top middle) 256 sampling points and (top right) 512 sampling points with λ equal to 0.005; (bottom) segmentation by use of level sets.

Fig. 2
Fig. 2

Segmentation of “X-ray” by use of a smoothing spline with 256 sampling points and λ equal to 0.005 (left) and 0.025 (right).

Fig. 3
Fig. 3

Image “Phone”: unconstrained segmentation (left), the shape of reference (middle), and segmentation with the shape of reference constraint (right).

Fig. 4
Fig. 4

Segmentation of “Akiyo” image 2, evolution from the initial contour until convergence.

Fig. 5
Fig. 5

Segmentation of “Akiyo” images 5, 15, 25, 40, 55, and 70 of 300.

Fig. 6
Fig. 6

Tracking of “Erik” face, images 1, 10, 27, 40, 50 of 50.

Equations (39)

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

JΓ=Γ kxdx,
Γτ=κk-k · NN,
kx=11+|Ix|,
JΩin=Ωin kinΩin, xdx+η Γ kbxdx,
kinΩin, x=μΩin-Ix2,
μΩin=Ωin IxdxΩindx.
JΩin=α Ωin kinΩin, xdx+β Ωout koutΩout, xdx+Γ kbxdx,
ΩinΓΩout=image domain D,Γ=Ωin=Ωout,Ωin=the domain inside Γ;
kin is minimum in the object, e.g., function 5,kout is minimum in the background, e.g., kout =0, meaning that no specific information is available for the background,kb is minimum on the object boundary, e.g., function 3.
JΩinτ=α Ωinτ kinΩinτ, xdx+β Ωoutτ koutΩoutτ, xdx+Γτ kbxdx,
Jτ=α Ωinτ kinτ, xdx+β Ωoutτ koutτ, xdx+Γτ kbxdx.
J1τ=Ωτ kτ, xdx region integral,
J2τ=Γτ kbxdx contour integral.
dJ1dττ=J1τ=Ωτkτdx-Ωτv·Nkdx,
J˜1τ=Ωinτ kinτ, xdx+Ωoutτ koutτ, xdx.
dJ˜1dττ=J˜1τ=DKτdx-Γτv·NKdx-Ωoutτ/Γτw·NΩoutKdx,
dJ2dττ=J2τ=Γτkb · N-kbκv · Ndx,
Jτ=α Ωinτkinτdx+β Ωoutτkoutτdx+Γτβkout-αkin+kb · N-kbκv·Ndx.
α Ωinτkinτdx+β Ωoutτkoutτdx=Γτ Hkin, koutv · Ndx,
Jτ=Γτ ρv · Ndx,
ρ=Hkin, kout+βkout-αkin+kb · N-kbκ.
Jτ=- Γτ ρ2dx.
Γ0=Γ0,Γττ, x=-ρτ, xNτ, xτ0, xΓτ.
kinτ, x=ϕδ-Ix,
μτ=Ωinτ IxdxΩinτdx.
Hkin, koutτ, x=-μτ-IxΩinτdyΩinτ ϕμτ-Iydy.
σ2τ=Ωinτμτ-Ix2dxΩinτdx,
Hkin, koutτ, x=ϕσ2τσ2τ-μτ-Ix2.
J2τ=Γτ kbτ, xdx=Γτ ϕdx, Γrefdx,
dx, Γref=+|x-yx| x is outside Γref-|x-yx| otherwise,
J2τ=-ΓτϕdNref · N+ϕdκv · Ndx,
JCubicSΓ=Γ Γx2dx,
JSmoothSΓ=λ Γ Γx2dx+ipi-Γi2,
J˜1τ=Ωinτkinτdx-Γτv · Nkindx+Ωoutτkoutτdx-Ωoutτw · NΩoutkoutdx.
J˜1τ=Ωinτkinτdx-Γτv · Nkindx+Ωoutτkoutτdx-Γτv · -Nkoutdx-Ωoutτ\Γτw · NΩoutkoutdx
=Ωinτkinτdx-Γτv · Nkin-koutdx+Ωoutτkoutτdx-Ωoutτ\Γτw · NΩoutkoutdx.
J˜1τ=Ωinτkinτdx+Ωoutτkoutτdx-Γτv · Nkin-koutdx-Ωoutτ\Γτw · NΩoutkoutdx
=ΩinτKτdx+ΩoutτKτdx-Γτv · NKdx-Ωoutτ\Γτw · NΩoutKdx
=DKτdx-Γτv · NKdx-Ωoutτ\Γτw · NΩoutKdx.

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