Fast Segmentation for the Piecewise Smooth Mumford-Shah Functional
This paper is concerned with an improved algorithm
based on the piecewise-smooth Mumford and Shah (MS) functional
for an efficient and reliable segmentation. In order to speed up
convergence, an additional force, at each time step, is introduced
further to drive the evolution of the curves instead of only driven by
the extensions of the complementary functions u + and u - . In our
scheme, furthermore, the piecewise-constant MS functional is
integrated to generate the extra force based on a temporary image that
is dynamically created by computing the union of u + and u - during
segmenting. Therefore, some drawbacks of the original algorithm,
such as smaller objects generated by noise and local minimal problem
also are eliminated or improved. The resulting algorithm has been
implemented in Matlab and Visual Cµ, and demonstrated efficiently
by several cases.
[1] D. Mumford and J. Shah, "Optimal approximation by piecewise smooth
functions and associated variational problems," Comm. Pure Appl. Math.,
vol. 42, pp. 577-685, 1989.
[2] M. Kass, A. Witkin, and D. Terzopoulos, "Snakes: Active contour
models," International Journal of Computer Vision, vol. 1, no. 4, 1988,
pp. 321-331.
[3] P. Perona and J. Malik, "Scale space and edge detection using anisotropic
diffusion," IEEE Transactions on Pattern Analysis and Machine
Intelligence, vol. 12, no. 8, 1990, pp. 629-639.
[4] S. Osher and J. A. Sethian, "Fronts propagating with curvature dependent
speed: Algorithms based on hamilton-jacobi formulation," Journal of
Computational Physics, vol. 79, 1988, pp. 12-49.
[5] T. Chan and L. Vese, "Active Contours without edges," IEEE
Transactions on Image Processing, vol. 10, no. 2, 2001, pp. 266-277.
[6] T. Chan and L. Vese, "A level set algorithm for minimizing the
Mumford-Shah functional in image processing," IEEE Computer Society,
Proceeding of the First IEEE Workshop on Variational and Level Set
Methods in Computer Vision, 2001, pp. 161-168.
[7] L. Vese and T. Chan, "A multiphase level set framework for image
segmentation using the Mumford and Shah model," International Journal
of Computer Vision, vol. 50, no. 3, 2002, pp. 271-293.
[8] Choi, G. Kim, P. Park, G. Wang, and S. Kim, "Efficient PDE-based
segmentation algorithms and their application to CT images," Journal.
Korean Institute of Plant Engineering, 2003, pp. 1-17.
[9] S. Teboul, L. B. Féraud, G. Aubert, and M. Barlaud, "Variational
approach for edge-preserving regularization using coupled PDE-s," IEEE
Transaction Image Processing, vol.7, no.3, 1998, pp. 387-397.
[10] S. Geman and D. E. Mcclure, "Bayesian image analysis: An application to
single photon emission tomography," Proc. Stat. Comput. Sect.
Washington, DC: Amer. Stat. Assoc. 1985, pp. 12-18.
[11] S. Ji and H. Park, "Image segmentation of color image based on region
coherency," in Proceedings 1998 International Conference on Image
Processing, vol. 1, Chicago, IL, USA, 1998, pp. 80-83.
[12] C. Xu and J. Prince, "Snakes, shapes, and gradient vector flow," IEEE
Transactions on Image Processing, vol. 7, 1998, pp. 359-369.
[13] J. Haddon and J. Boyce, "Image segmentation by unifying region and
boundary information," IEEE Transactions on Pattern Analysis and
Machine Intelligence, vol. 12, no. 10, 1990, pp. 929-948.
[14] K. Haris, S. Efstratiadis, N. Maglaveras, and A.Katsaggelos, "Hybeid
image segmentation using watersheds and fast region merging," IEEE
Transactions on Image Processing, vol. 7, no. 12, 1998, pp. 1684-1699.
[15] S. Kichenassamy, A. Kumar, P. Oliver, A. Tannenbaum, and A. Yezzi,
"Conformal curvature flows: from phase transitions to active vision,"
Archive for Rational Mech. and Anal., vol. 134, 1996, pp. 275-301.
[16] D. Marr and E. Hildreth, "Theory of edge detection," Proc. R. Soc. Lond.,
vol. B207, 1980, pp. 187-217.
[17] P. Hill, C. Canagarajah, and D. Bull, "Texture gradient based watershed
segmentation," IEEE International Conference on Acoustics, Speech, and
Signal Processing, vol. 4, Orlando, FL, USA, 2002, pp. 3381-3384.
[18] S. Osher and R. Fedkiw, Level set methods and dynamic implicit
surfaces, New York, Springer-Verlag, 2003.
[1] D. Mumford and J. Shah, "Optimal approximation by piecewise smooth
functions and associated variational problems," Comm. Pure Appl. Math.,
vol. 42, pp. 577-685, 1989.
[2] M. Kass, A. Witkin, and D. Terzopoulos, "Snakes: Active contour
models," International Journal of Computer Vision, vol. 1, no. 4, 1988,
pp. 321-331.
[3] P. Perona and J. Malik, "Scale space and edge detection using anisotropic
diffusion," IEEE Transactions on Pattern Analysis and Machine
Intelligence, vol. 12, no. 8, 1990, pp. 629-639.
[4] S. Osher and J. A. Sethian, "Fronts propagating with curvature dependent
speed: Algorithms based on hamilton-jacobi formulation," Journal of
Computational Physics, vol. 79, 1988, pp. 12-49.
[5] T. Chan and L. Vese, "Active Contours without edges," IEEE
Transactions on Image Processing, vol. 10, no. 2, 2001, pp. 266-277.
[6] T. Chan and L. Vese, "A level set algorithm for minimizing the
Mumford-Shah functional in image processing," IEEE Computer Society,
Proceeding of the First IEEE Workshop on Variational and Level Set
Methods in Computer Vision, 2001, pp. 161-168.
[7] L. Vese and T. Chan, "A multiphase level set framework for image
segmentation using the Mumford and Shah model," International Journal
of Computer Vision, vol. 50, no. 3, 2002, pp. 271-293.
[8] Choi, G. Kim, P. Park, G. Wang, and S. Kim, "Efficient PDE-based
segmentation algorithms and their application to CT images," Journal.
Korean Institute of Plant Engineering, 2003, pp. 1-17.
[9] S. Teboul, L. B. Féraud, G. Aubert, and M. Barlaud, "Variational
approach for edge-preserving regularization using coupled PDE-s," IEEE
Transaction Image Processing, vol.7, no.3, 1998, pp. 387-397.
[10] S. Geman and D. E. Mcclure, "Bayesian image analysis: An application to
single photon emission tomography," Proc. Stat. Comput. Sect.
Washington, DC: Amer. Stat. Assoc. 1985, pp. 12-18.
[11] S. Ji and H. Park, "Image segmentation of color image based on region
coherency," in Proceedings 1998 International Conference on Image
Processing, vol. 1, Chicago, IL, USA, 1998, pp. 80-83.
[12] C. Xu and J. Prince, "Snakes, shapes, and gradient vector flow," IEEE
Transactions on Image Processing, vol. 7, 1998, pp. 359-369.
[13] J. Haddon and J. Boyce, "Image segmentation by unifying region and
boundary information," IEEE Transactions on Pattern Analysis and
Machine Intelligence, vol. 12, no. 10, 1990, pp. 929-948.
[14] K. Haris, S. Efstratiadis, N. Maglaveras, and A.Katsaggelos, "Hybeid
image segmentation using watersheds and fast region merging," IEEE
Transactions on Image Processing, vol. 7, no. 12, 1998, pp. 1684-1699.
[15] S. Kichenassamy, A. Kumar, P. Oliver, A. Tannenbaum, and A. Yezzi,
"Conformal curvature flows: from phase transitions to active vision,"
Archive for Rational Mech. and Anal., vol. 134, 1996, pp. 275-301.
[16] D. Marr and E. Hildreth, "Theory of edge detection," Proc. R. Soc. Lond.,
vol. B207, 1980, pp. 187-217.
[17] P. Hill, C. Canagarajah, and D. Bull, "Texture gradient based watershed
segmentation," IEEE International Conference on Acoustics, Speech, and
Signal Processing, vol. 4, Orlando, FL, USA, 2002, pp. 3381-3384.
[18] S. Osher and R. Fedkiw, Level set methods and dynamic implicit
surfaces, New York, Springer-Verlag, 2003.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:50523", author = "Yingjie Zhang", title = "Fast Segmentation for the Piecewise Smooth Mumford-Shah Functional", abstract = "This paper is concerned with an improved algorithm
based on the piecewise-smooth Mumford and Shah (MS) functional
for an efficient and reliable segmentation. In order to speed up
convergence, an additional force, at each time step, is introduced
further to drive the evolution of the curves instead of only driven by
the extensions of the complementary functions u + and u - . In our
scheme, furthermore, the piecewise-constant MS functional is
integrated to generate the extra force based on a temporary image that
is dynamically created by computing the union of u + and u - during
segmenting. Therefore, some drawbacks of the original algorithm,
such as smaller objects generated by noise and local minimal problem
also are eliminated or improved. The resulting algorithm has been
implemented in Matlab and Visual Cµ, and demonstrated efficiently
by several cases.", keywords = "Active contours, energy minimization, image
segmentation, level sets.", volume = "2", number = "9", pages = "620-6", }
