Abstract

We propose a novel image segmentation model which incorporates subspace clustering techniques into a Mumford-Shah model to solve texture segmentation problems. While the natural unsupervised approach to learn a feature subspace can easily be trapped in a local solution, we propose a novel semi-supervised optimization algorithm that makes use of information derived from both the intermediate segmentation results and the regions-of-interest (ROI) selected by the user to determine the optimal subspaces of the target regions. Meanwhile, these subspaces are embedded into a Mumford-Shah objective function so that each segment of the optimal partition is homogeneous in its own subspace. The method outperforms standard Mumford-Shah models since it can separate textures which are less separated in the full feature space. Experimental results are presented to confirm the usefulness of subspace clustering in texture segmentation.

© 2010 Optical Society of America

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  6. B. Manjunath and W. Ma, “Texture features for browsing and retrieval of image data,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 837–842 (1996).
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  8. M. Datar, D. Padfield, and H. Cline, “Color and texture based segmentation of molecular pathology images using HSOMs,” Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro pp. 292–295 (2008).
    [CrossRef]
  9. S. Belongie, C. Carson, H. Greenspan, and J. Malik, “Color- and texture-based image segmentation using EM and its application to content-based image retrieval,” Proc. IEEE Conf. on Computer Vision p. 675 (1998).
  10. A. Barbu and S. Zhu, “Multigrid and multi-level Swendsen-Wang cuts for hierarchic graph partition,” Proc. IEEE Conf. on Computer Vision and Pattern Recognition 2, 731–738 (2004).
  11. 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]
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    [CrossRef]
  16. M. Haindl and S. Mikeš, “Unsupervised texture segmentation,” in “Patten Recognition Techniques, Technology and Applications,” , P.-Y. Yin, ed. (I-Tech, 2008), pp. 227–248.
  17. D. Mumford and J. Shah, “Optimal approximation by piecewise smooth functions and associated variational problems,” Commun. Pure Appl. Math. 42, 577–685 (1989).
    [CrossRef]
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    [PubMed]
  19. M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contour models,” International Journal of Computer Vision 1, 321–331 (1988).
    [CrossRef]
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  21. C.-H. Cheng, A. W. Fu, and Y. Zhang, “Entropy-based subspace clustering for mining numerical data,” Proc. of the 5th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining pp. 84–93 (1999).
    [CrossRef]
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  25. L. Jing, M. Ng, and J. Huang, “An entropy weighting k-means algorithm for subspace clustering of high-dimensional sparse data,” IEEE Trans. Knowledge and Data Engineering 19, 1026–1041 (2007).
    [CrossRef]
  26. C. Yap and H. Lee, “Identification of cell nucleus using a Mumford-Shah ellipse detector,” Proc. of ISVC 1, 582–593 (2008).
  27. W. Yu, H. Lee, S. Hariharan, W. Bu, and S. Ahmed, “Level set segmentation of cellular images based on topological dependence,” Proc. of ISVC 1, 540–551 (2008).
  28. W. Zhu, T. Chan, and S. Esedog¯lu, “Segmentation with depth: A level set approach,” SIAM J. Sci. Comput. 28, 1957–1973 (2006).
    [CrossRef]
  29. T. F. Chan and L. A. Vese, “Active contours without edges,” IEEE Tran. Image Process. 10, 266–277 (2001).
    [CrossRef]
  30. L. A. Vese and T. F. Chan, “A multiphase level set framework for image segmentation using the Mumford and Shah model,” Int. J. Comput. Vision 50, 271–293 (2002).
    [CrossRef]
  31. T. Chan, B. Sandberg, and L. Vese, “Active contours without edges for vector-valued images,” Journal of Visual Communication and Image Representation 11, 130–141 (2000).
    [CrossRef]
  32. P. Brodatz, Textures: A photographic album for artists and designers (Dover, New York, 1996).
  33. Y. Law, H. Lee, and A. Yip, “A multiresolution stochastic level set method for Mumford-Shah image segmentation,” IEEE Transactions on Image Processing 17, 2289–2300 (2008).
    [CrossRef] [PubMed]
  34. L. Roberts, J. Redan, and H. Reich, “Extraperitoneal endometriosis with catamenial pneumothoraces: A review of the literature,” Journal of the Society of Laparoendoscopic Surgeons 7, 371–375 (2003).
    [PubMed]
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    [CrossRef]
  36. C. Chiao and R. Hanlon, “Cuttlefish camouflage: Visual perception of size, contrast and number of white squares on artificial checkerboard substrata initiates disruptive coloration,” The Journal of Experimental Biology 204, 2119–2125 (2001).
    [PubMed]
  37. M. Rachidi, A. Chappard, C. Marchadier, C. Gadois, E. Lespessailles, and C. L. Benhamou, “Application of Laws’ masks to bone texture analysis: An innovative image analysis tool in osteoporosis,” Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro pp. 1191–1194 (2008).
    [CrossRef]
  38. T. F. Chan, S. Esedoglu, and M. Nikolova, “Algorithms for finding global minimizers of denoising and segmentation models,” SIAM J. Appl. Math. 66, 1632–1648 (2006).
    [CrossRef]
  39. A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89–97 (2004).
    [CrossRef]
  40. J.-F. Aujol, G. Gilboa, T. Chan, and S. Osher, “Structure-texture image decomposition - modeling, algorithms, and parameter selection,” Int. J. Comput. Vis. 67, 111–136 (2006).
    [CrossRef]
  41. X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thiran, and S. Osher, “Fast global minimization of the active contour/snake model,” J. Math. Imaging Vis. 28, 151–167 (2007).
    [CrossRef]

2008 (5)

M. Datar, D. Padfield, and H. Cline, “Color and texture based segmentation of molecular pathology images using HSOMs,” Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro pp. 292–295 (2008).
[CrossRef]

C. Yap and H. Lee, “Identification of cell nucleus using a Mumford-Shah ellipse detector,” Proc. of ISVC 1, 582–593 (2008).

W. Yu, H. Lee, S. Hariharan, W. Bu, and S. Ahmed, “Level set segmentation of cellular images based on topological dependence,” Proc. of ISVC 1, 540–551 (2008).

Y. Law, H. Lee, and A. Yip, “A multiresolution stochastic level set method for Mumford-Shah image segmentation,” IEEE Transactions on Image Processing 17, 2289–2300 (2008).
[CrossRef] [PubMed]

M. Rachidi, A. Chappard, C. Marchadier, C. Gadois, E. Lespessailles, and C. L. Benhamou, “Application of Laws’ masks to bone texture analysis: An innovative image analysis tool in osteoporosis,” Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro pp. 1191–1194 (2008).
[CrossRef]

2007 (2)

L. Jing, M. Ng, and J. Huang, “An entropy weighting k-means algorithm for subspace clustering of high-dimensional sparse data,” IEEE Trans. Knowledge and Data Engineering 19, 1026–1041 (2007).
[CrossRef]

X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thiran, and S. Osher, “Fast global minimization of the active contour/snake model,” J. Math. Imaging Vis. 28, 151–167 (2007).
[CrossRef]

2006 (3)

J.-F. Aujol, G. Gilboa, T. Chan, and S. Osher, “Structure-texture image decomposition - modeling, algorithms, and parameter selection,” Int. J. Comput. Vis. 67, 111–136 (2006).
[CrossRef]

T. F. Chan, S. Esedoglu, and M. Nikolova, “Algorithms for finding global minimizers of denoising and segmentation models,” SIAM J. Appl. Math. 66, 1632–1648 (2006).
[CrossRef]

W. Zhu, T. Chan, and S. Esedog¯lu, “Segmentation with depth: A level set approach,” SIAM J. Sci. Comput. 28, 1957–1973 (2006).
[CrossRef]

2005 (1)

B. Sharif, A. Ahmadian, M. Oghabian, and N. Izadi, “Texture segmentation of endometrial images for aiding diagnosis of hyperplasia,” Proceedings of the International Conference on Computer as a Tool 2, 983–986 (2005).

2004 (3)

J. Friedman and J. Meulman, “Clustering objects on subsets of attributes,” J. R. Statist. Soc. B 66, 815–849 (2004).
[CrossRef]

A. Barbu and S. Zhu, “Multigrid and multi-level Swendsen-Wang cuts for hierarchic graph partition,” Proc. IEEE Conf. on Computer Vision and Pattern Recognition 2, 731–738 (2004).

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89–97 (2004).
[CrossRef]

2003 (2)

L. Roberts, J. Redan, and H. Reich, “Extraperitoneal endometriosis with catamenial pneumothoraces: A review of the literature,” Journal of the Society of Laparoendoscopic Surgeons 7, 371–375 (2003).
[PubMed]

T. Brox, M. Rousson, R. Deriche, and J. Weickert, “Unsupervised segmentation incorporating color, texture, and motion,” Proc. of the Intl. Conf. on Computer Analysis of Images and Patterns pp. 353–360 (2003).
[CrossRef]

2002 (4)

L. A. Vese and T. F. Chan, “A multiphase level set framework for image segmentation using the Mumford and Shah model,” Int. J. Comput. Vision 50, 271–293 (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]

B. Sandberg, T. Chan, and L. Vese, “A level-set and Gabor-based active contour algorithm for segmentation textured images,” Cam report, UCLA (2002).

J. Yang, W. Wang, H. Wang, and P. Yu, “δ-clusters: capturing subspace correlation in a large data set,” Proc. of 18th Interational Conference on Data Engineering pp. 517–528 (2002).
[CrossRef]

2001 (2)

C. Chiao and R. Hanlon, “Cuttlefish camouflage: Visual perception of size, contrast and number of white squares on artificial checkerboard substrata initiates disruptive coloration,” The Journal of Experimental Biology 204, 2119–2125 (2001).
[PubMed]

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

2000 (2)

T. Chan, B. Sandberg, and L. Vese, “Active contours without edges for vector-valued images,” Journal of Visual Communication and Image Representation 11, 130–141 (2000).
[CrossRef]

C. Aggarwal and P. Yu, “Finding generalized projected clusters in high dimensional spaces,” Proc. of the 2000 ACM SIGMOD International Conference on Management of Data pp. 70–81 (2000).

1999 (1)

C.-H. Cheng, A. W. Fu, and Y. Zhang, “Entropy-based subspace clustering for mining numerical data,” Proc. of the 5th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining pp. 84–93 (1999).
[CrossRef]

1998 (2)

R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan, “Automatic subspace clustering of high dimensional data for data mining applications,” Proc. of the 1998 ACM SIGMOD International Conference on Management of Data pp. 94–105 (1998).

S. Belongie, C. Carson, H. Greenspan, and J. Malik, “Color- and texture-based image segmentation using EM and its application to content-based image retrieval,” Proc. IEEE Conf. on Computer Vision p. 675 (1998).

1996 (1)

B. Manjunath and W. Ma, “Texture features for browsing and retrieval of image data,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 837–842 (1996).
[CrossRef]

1993 (1)

T. Reed, “A review of recent texture segmentation and feature extraction techniques,” CVGIP: Image Understanding 57, 359–372 (1993).
[CrossRef]

1991 (1)

A. Jain and F. Farrokhnia, “Unsupervised texture segmentation using Gabor filters,” Pattern. Recog. 24, 1167–1186 (1991).
[CrossRef]

1989 (1)

D. Mumford and J. Shah, “Optimal approximation by piecewise smooth functions and associated variational problems,” Commun. Pure Appl. Math. 42, 577–685 (1989).
[CrossRef]

1988 (1)

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contour models,” International Journal of Computer Vision 1, 321–331 (1988).
[CrossRef]

1979 (1)

K. Laws, “Texture energy measures,” Proc. of Image Understanding Workshop pp. 47–51 (1979).

1946 (1)

D. Gabor, “Theory of communication,” J. IEEE 93, 429–459 (1946).

Aggarwal, C.

C. Aggarwal and P. Yu, “Finding generalized projected clusters in high dimensional spaces,” Proc. of the 2000 ACM SIGMOD International Conference on Management of Data pp. 70–81 (2000).

Agrawal, R.

R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan, “Automatic subspace clustering of high dimensional data for data mining applications,” Proc. of the 1998 ACM SIGMOD International Conference on Management of Data pp. 94–105 (1998).

Ahmadian, A.

B. Sharif, A. Ahmadian, M. Oghabian, and N. Izadi, “Texture segmentation of endometrial images for aiding diagnosis of hyperplasia,” Proceedings of the International Conference on Computer as a Tool 2, 983–986 (2005).

Ahmed, S.

W. Yu, H. Lee, S. Hariharan, W. Bu, and S. Ahmed, “Level set segmentation of cellular images based on topological dependence,” Proc. of ISVC 1, 540–551 (2008).

Aujol, J.-F.

J.-F. Aujol, G. Gilboa, T. Chan, and S. Osher, “Structure-texture image decomposition - modeling, algorithms, and parameter selection,” Int. J. Comput. Vis. 67, 111–136 (2006).
[CrossRef]

Barbu, A.

A. Barbu and S. Zhu, “Multigrid and multi-level Swendsen-Wang cuts for hierarchic graph partition,” Proc. IEEE Conf. on Computer Vision and Pattern Recognition 2, 731–738 (2004).

Belongie, S.

S. Belongie, C. Carson, H. Greenspan, and J. Malik, “Color- and texture-based image segmentation using EM and its application to content-based image retrieval,” Proc. IEEE Conf. on Computer Vision p. 675 (1998).

Benhamou, C. L.

M. Rachidi, A. Chappard, C. Marchadier, C. Gadois, E. Lespessailles, and C. L. Benhamou, “Application of Laws’ masks to bone texture analysis: An innovative image analysis tool in osteoporosis,” Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro pp. 1191–1194 (2008).
[CrossRef]

Beucher, S.

S. Beucher and C. Lantuéjoul, Use of watersheds in contour detection,” Proc. International Workshop on Image Processing: Real-time Edge and Motion Detection/Estimation (1979).
[PubMed]

Bresson, X.

X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thiran, and S. Osher, “Fast global minimization of the active contour/snake model,” J. Math. Imaging Vis. 28, 151–167 (2007).
[CrossRef]

Brodatz, P.

P. Brodatz, Textures: A photographic album for artists and designers (Dover, New York, 1996).

Brox, T.

T. Brox, M. Rousson, R. Deriche, and J. Weickert, “Unsupervised segmentation incorporating color, texture, and motion,” Proc. of the Intl. Conf. on Computer Analysis of Images and Patterns pp. 353–360 (2003).
[CrossRef]

M. Rousson, T. Brox, and R. Deriche, “Active unsupervised texture segmentation on a diffusion based feature space,” Proc. of the 2003 IEEE Computer Vision and Pattern Recognition (2003).

Bu, W.

W. Yu, H. Lee, S. Hariharan, W. Bu, and S. Ahmed, “Level set segmentation of cellular images based on topological dependence,” Proc. of ISVC 1, 540–551 (2008).

Carson, C.

S. Belongie, C. Carson, H. Greenspan, and J. Malik, “Color- and texture-based image segmentation using EM and its application to content-based image retrieval,” Proc. IEEE Conf. on Computer Vision p. 675 (1998).

Chambolle, A.

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89–97 (2004).
[CrossRef]

Chan, T.

J.-F. Aujol, G. Gilboa, T. Chan, and S. Osher, “Structure-texture image decomposition - modeling, algorithms, and parameter selection,” Int. J. Comput. Vis. 67, 111–136 (2006).
[CrossRef]

W. Zhu, T. Chan, and S. Esedog¯lu, “Segmentation with depth: A level set approach,” SIAM J. Sci. Comput. 28, 1957–1973 (2006).
[CrossRef]

B. Sandberg, T. Chan, and L. Vese, “A level-set and Gabor-based active contour algorithm for segmentation textured images,” Cam report, UCLA (2002).

T. Chan, B. Sandberg, and L. Vese, “Active contours without edges for vector-valued images,” Journal of Visual Communication and Image Representation 11, 130–141 (2000).
[CrossRef]

Chan, T. F.

T. F. Chan, S. Esedoglu, and M. Nikolova, “Algorithms for finding global minimizers of denoising and segmentation models,” SIAM J. Appl. Math. 66, 1632–1648 (2006).
[CrossRef]

L. A. Vese and T. F. Chan, “A multiphase level set framework for image segmentation using the Mumford and Shah model,” Int. J. Comput. Vision 50, 271–293 (2002).
[CrossRef]

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

Chappard, A.

M. Rachidi, A. Chappard, C. Marchadier, C. Gadois, E. Lespessailles, and C. L. Benhamou, “Application of Laws’ masks to bone texture analysis: An innovative image analysis tool in osteoporosis,” Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro pp. 1191–1194 (2008).
[CrossRef]

Cheng, C.-H.

C.-H. Cheng, A. W. Fu, and Y. Zhang, “Entropy-based subspace clustering for mining numerical data,” Proc. of the 5th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining pp. 84–93 (1999).
[CrossRef]

Chiao, C.

C. Chiao and R. Hanlon, “Cuttlefish camouflage: Visual perception of size, contrast and number of white squares on artificial checkerboard substrata initiates disruptive coloration,” The Journal of Experimental Biology 204, 2119–2125 (2001).
[PubMed]

Cline, H.

M. Datar, D. Padfield, and H. Cline, “Color and texture based segmentation of molecular pathology images using HSOMs,” Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro pp. 292–295 (2008).
[CrossRef]

Datar, M.

M. Datar, D. Padfield, and H. Cline, “Color and texture based segmentation of molecular pathology images using HSOMs,” Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro pp. 292–295 (2008).
[CrossRef]

Deriche, R.

T. Brox, M. Rousson, R. Deriche, and J. Weickert, “Unsupervised segmentation incorporating color, texture, and motion,” Proc. of the Intl. Conf. on Computer Analysis of Images and Patterns pp. 353–360 (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]

M. Rousson, T. Brox, and R. Deriche, “Active unsupervised texture segmentation on a diffusion based feature space,” Proc. of the 2003 IEEE Computer Vision and Pattern Recognition (2003).

Esedog¯lu, S.

W. Zhu, T. Chan, and S. Esedog¯lu, “Segmentation with depth: A level set approach,” SIAM J. Sci. Comput. 28, 1957–1973 (2006).
[CrossRef]

Esedoglu, S.

X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thiran, and S. Osher, “Fast global minimization of the active contour/snake model,” J. Math. Imaging Vis. 28, 151–167 (2007).
[CrossRef]

T. F. Chan, S. Esedoglu, and M. Nikolova, “Algorithms for finding global minimizers of denoising and segmentation models,” SIAM J. Appl. Math. 66, 1632–1648 (2006).
[CrossRef]

Farrokhnia, F.

A. Jain and F. Farrokhnia, “Unsupervised texture segmentation using Gabor filters,” Pattern. Recog. 24, 1167–1186 (1991).
[CrossRef]

Friedman, J.

J. Friedman and J. Meulman, “Clustering objects on subsets of attributes,” J. R. Statist. Soc. B 66, 815–849 (2004).
[CrossRef]

Fu, A. W.

C.-H. Cheng, A. W. Fu, and Y. Zhang, “Entropy-based subspace clustering for mining numerical data,” Proc. of the 5th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining pp. 84–93 (1999).
[CrossRef]

Gabor, D.

D. Gabor, “Theory of communication,” J. IEEE 93, 429–459 (1946).

Gadois, C.

M. Rachidi, A. Chappard, C. Marchadier, C. Gadois, E. Lespessailles, and C. L. Benhamou, “Application of Laws’ masks to bone texture analysis: An innovative image analysis tool in osteoporosis,” Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro pp. 1191–1194 (2008).
[CrossRef]

Gehrke, J.

R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan, “Automatic subspace clustering of high dimensional data for data mining applications,” Proc. of the 1998 ACM SIGMOD International Conference on Management of Data pp. 94–105 (1998).

Gilboa, G.

J.-F. Aujol, G. Gilboa, T. Chan, and S. Osher, “Structure-texture image decomposition - modeling, algorithms, and parameter selection,” Int. J. Comput. Vis. 67, 111–136 (2006).
[CrossRef]

Greenspan, H.

S. Belongie, C. Carson, H. Greenspan, and J. Malik, “Color- and texture-based image segmentation using EM and its application to content-based image retrieval,” Proc. IEEE Conf. on Computer Vision p. 675 (1998).

Gunopulos, D.

R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan, “Automatic subspace clustering of high dimensional data for data mining applications,” Proc. of the 1998 ACM SIGMOD International Conference on Management of Data pp. 94–105 (1998).

Haindl, M.

M. Haindl and S. Mikeš, “Unsupervised texture segmentation,” in “Patten Recognition Techniques, Technology and Applications,” , P.-Y. Yin, ed. (I-Tech, 2008), pp. 227–248.

Hanlon, R.

C. Chiao and R. Hanlon, “Cuttlefish camouflage: Visual perception of size, contrast and number of white squares on artificial checkerboard substrata initiates disruptive coloration,” The Journal of Experimental Biology 204, 2119–2125 (2001).
[PubMed]

Hariharan, S.

W. Yu, H. Lee, S. Hariharan, W. Bu, and S. Ahmed, “Level set segmentation of cellular images based on topological dependence,” Proc. of ISVC 1, 540–551 (2008).

Huang, J.

L. Jing, M. Ng, and J. Huang, “An entropy weighting k-means algorithm for subspace clustering of high-dimensional sparse data,” IEEE Trans. Knowledge and Data Engineering 19, 1026–1041 (2007).
[CrossRef]

Izadi, N.

B. Sharif, A. Ahmadian, M. Oghabian, and N. Izadi, “Texture segmentation of endometrial images for aiding diagnosis of hyperplasia,” Proceedings of the International Conference on Computer as a Tool 2, 983–986 (2005).

Jain, A.

A. Jain and F. Farrokhnia, “Unsupervised texture segmentation using Gabor filters,” Pattern. Recog. 24, 1167–1186 (1991).
[CrossRef]

Jing, L.

L. Jing, M. Ng, and J. Huang, “An entropy weighting k-means algorithm for subspace clustering of high-dimensional sparse data,” IEEE Trans. Knowledge and Data Engineering 19, 1026–1041 (2007).
[CrossRef]

Kass, M.

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contour models,” International Journal of Computer Vision 1, 321–331 (1988).
[CrossRef]

Kneller, A.

A. Kneller, “The new age of bioimaging,” in “Paradigm,” (2006), pp. 18–25.

Lantuéjoul, C.

S. Beucher and C. Lantuéjoul, Use of watersheds in contour detection,” Proc. International Workshop on Image Processing: Real-time Edge and Motion Detection/Estimation (1979).
[PubMed]

Law, Y.

Y. Law, H. Lee, and A. Yip, “A multiresolution stochastic level set method for Mumford-Shah image segmentation,” IEEE Transactions on Image Processing 17, 2289–2300 (2008).
[CrossRef] [PubMed]

Laws, K.

K. Laws, “Texture energy measures,” Proc. of Image Understanding Workshop pp. 47–51 (1979).

Lee, H.

C. Yap and H. Lee, “Identification of cell nucleus using a Mumford-Shah ellipse detector,” Proc. of ISVC 1, 582–593 (2008).

W. Yu, H. Lee, S. Hariharan, W. Bu, and S. Ahmed, “Level set segmentation of cellular images based on topological dependence,” Proc. of ISVC 1, 540–551 (2008).

Y. Law, H. Lee, and A. Yip, “A multiresolution stochastic level set method for Mumford-Shah image segmentation,” IEEE Transactions on Image Processing 17, 2289–2300 (2008).
[CrossRef] [PubMed]

Lespessailles, E.

M. Rachidi, A. Chappard, C. Marchadier, C. Gadois, E. Lespessailles, and C. L. Benhamou, “Application of Laws’ masks to bone texture analysis: An innovative image analysis tool in osteoporosis,” Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro pp. 1191–1194 (2008).
[CrossRef]

Ma, W.

B. Manjunath and W. Ma, “Texture features for browsing and retrieval of image data,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 837–842 (1996).
[CrossRef]

Malik, J.

S. Belongie, C. Carson, H. Greenspan, and J. Malik, “Color- and texture-based image segmentation using EM and its application to content-based image retrieval,” Proc. IEEE Conf. on Computer Vision p. 675 (1998).

Manjunath, B.

B. Manjunath and W. Ma, “Texture features for browsing and retrieval of image data,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 837–842 (1996).
[CrossRef]

Marchadier, C.

M. Rachidi, A. Chappard, C. Marchadier, C. Gadois, E. Lespessailles, and C. L. Benhamou, “Application of Laws’ masks to bone texture analysis: An innovative image analysis tool in osteoporosis,” Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro pp. 1191–1194 (2008).
[CrossRef]

Meulman, J.

J. Friedman and J. Meulman, “Clustering objects on subsets of attributes,” J. R. Statist. Soc. B 66, 815–849 (2004).
[CrossRef]

Mikeš, S.

M. Haindl and S. Mikeš, “Unsupervised texture segmentation,” in “Patten Recognition Techniques, Technology and Applications,” , P.-Y. Yin, ed. (I-Tech, 2008), pp. 227–248.

Mumford, D.

D. Mumford and J. Shah, “Optimal approximation by piecewise smooth functions and associated variational problems,” Commun. Pure Appl. Math. 42, 577–685 (1989).
[CrossRef]

Ng, M.

L. Jing, M. Ng, and J. Huang, “An entropy weighting k-means algorithm for subspace clustering of high-dimensional sparse data,” IEEE Trans. Knowledge and Data Engineering 19, 1026–1041 (2007).
[CrossRef]

Nikolova, M.

T. F. Chan, S. Esedoglu, and M. Nikolova, “Algorithms for finding global minimizers of denoising and segmentation models,” SIAM J. Appl. Math. 66, 1632–1648 (2006).
[CrossRef]

Oghabian, M.

B. Sharif, A. Ahmadian, M. Oghabian, and N. Izadi, “Texture segmentation of endometrial images for aiding diagnosis of hyperplasia,” Proceedings of the International Conference on Computer as a Tool 2, 983–986 (2005).

Osher, S.

X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thiran, and S. Osher, “Fast global minimization of the active contour/snake model,” J. Math. Imaging Vis. 28, 151–167 (2007).
[CrossRef]

J.-F. Aujol, G. Gilboa, T. Chan, and S. Osher, “Structure-texture image decomposition - modeling, algorithms, and parameter selection,” Int. J. Comput. Vis. 67, 111–136 (2006).
[CrossRef]

Padfield, D.

M. Datar, D. Padfield, and H. Cline, “Color and texture based segmentation of molecular pathology images using HSOMs,” Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro pp. 292–295 (2008).
[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]

Petrou, P.

P. Petrou and P. G. Sevilla, Dealing with Texture (Wiley, 2006).
[CrossRef]

Rachidi, M.

M. Rachidi, A. Chappard, C. Marchadier, C. Gadois, E. Lespessailles, and C. L. Benhamou, “Application of Laws’ masks to bone texture analysis: An innovative image analysis tool in osteoporosis,” Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro pp. 1191–1194 (2008).
[CrossRef]

Raghavan, P.

R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan, “Automatic subspace clustering of high dimensional data for data mining applications,” Proc. of the 1998 ACM SIGMOD International Conference on Management of Data pp. 94–105 (1998).

Redan, J.

L. Roberts, J. Redan, and H. Reich, “Extraperitoneal endometriosis with catamenial pneumothoraces: A review of the literature,” Journal of the Society of Laparoendoscopic Surgeons 7, 371–375 (2003).
[PubMed]

Reed, T.

T. Reed, “A review of recent texture segmentation and feature extraction techniques,” CVGIP: Image Understanding 57, 359–372 (1993).
[CrossRef]

Reich, H.

L. Roberts, J. Redan, and H. Reich, “Extraperitoneal endometriosis with catamenial pneumothoraces: A review of the literature,” Journal of the Society of Laparoendoscopic Surgeons 7, 371–375 (2003).
[PubMed]

Roberts, L.

L. Roberts, J. Redan, and H. Reich, “Extraperitoneal endometriosis with catamenial pneumothoraces: A review of the literature,” Journal of the Society of Laparoendoscopic Surgeons 7, 371–375 (2003).
[PubMed]

Rousson, M.

T. Brox, M. Rousson, R. Deriche, and J. Weickert, “Unsupervised segmentation incorporating color, texture, and motion,” Proc. of the Intl. Conf. on Computer Analysis of Images and Patterns pp. 353–360 (2003).
[CrossRef]

M. Rousson, T. Brox, and R. Deriche, “Active unsupervised texture segmentation on a diffusion based feature space,” Proc. of the 2003 IEEE Computer Vision and Pattern Recognition (2003).

Sagiv, C.

C. Sagiv, N. Sochen, and Y. Zeevi, “Texture segmentation via a diffusion-segmentation scheme in the Gabor feature space,” Proc. of the 2nd Intl. Workshop on Texture Analysis and Synthesis (2002).

Sandberg, B.

B. Sandberg, T. Chan, and L. Vese, “A level-set and Gabor-based active contour algorithm for segmentation textured images,” Cam report, UCLA (2002).

T. Chan, B. Sandberg, and L. Vese, “Active contours without edges for vector-valued images,” Journal of Visual Communication and Image Representation 11, 130–141 (2000).
[CrossRef]

Sevilla, P. G.

P. Petrou and P. G. Sevilla, Dealing with Texture (Wiley, 2006).
[CrossRef]

Shah, J.

D. Mumford and J. Shah, “Optimal approximation by piecewise smooth functions and associated variational problems,” Commun. Pure Appl. Math. 42, 577–685 (1989).
[CrossRef]

Sharif, B.

B. Sharif, A. Ahmadian, M. Oghabian, and N. Izadi, “Texture segmentation of endometrial images for aiding diagnosis of hyperplasia,” Proceedings of the International Conference on Computer as a Tool 2, 983–986 (2005).

Sochen, N.

C. Sagiv, N. Sochen, and Y. Zeevi, “Texture segmentation via a diffusion-segmentation scheme in the Gabor feature space,” Proc. of the 2nd Intl. Workshop on Texture Analysis and Synthesis (2002).

Terzopoulos, D.

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contour models,” International Journal of Computer Vision 1, 321–331 (1988).
[CrossRef]

Thiran, J.-P.

X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thiran, and S. Osher, “Fast global minimization of the active contour/snake model,” J. Math. Imaging Vis. 28, 151–167 (2007).
[CrossRef]

Vandergheynst, P.

X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thiran, and S. Osher, “Fast global minimization of the active contour/snake model,” J. Math. Imaging Vis. 28, 151–167 (2007).
[CrossRef]

Vese, L.

B. Sandberg, T. Chan, and L. Vese, “A level-set and Gabor-based active contour algorithm for segmentation textured images,” Cam report, UCLA (2002).

T. Chan, B. Sandberg, and L. Vese, “Active contours without edges for vector-valued images,” Journal of Visual Communication and Image Representation 11, 130–141 (2000).
[CrossRef]

Vese, L. A.

L. A. Vese and T. F. Chan, “A multiphase level set framework for image segmentation using the Mumford and Shah model,” Int. J. Comput. Vision 50, 271–293 (2002).
[CrossRef]

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

Wang, H.

J. Yang, W. Wang, H. Wang, and P. Yu, “δ-clusters: capturing subspace correlation in a large data set,” Proc. of 18th Interational Conference on Data Engineering pp. 517–528 (2002).
[CrossRef]

Wang, W.

J. Yang, W. Wang, H. Wang, and P. Yu, “δ-clusters: capturing subspace correlation in a large data set,” Proc. of 18th Interational Conference on Data Engineering pp. 517–528 (2002).
[CrossRef]

Weickert, J.

T. Brox, M. Rousson, R. Deriche, and J. Weickert, “Unsupervised segmentation incorporating color, texture, and motion,” Proc. of the Intl. Conf. on Computer Analysis of Images and Patterns pp. 353–360 (2003).
[CrossRef]

Witkin, A.

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contour models,” International Journal of Computer Vision 1, 321–331 (1988).
[CrossRef]

Yang, J.

J. Yang, W. Wang, H. Wang, and P. Yu, “δ-clusters: capturing subspace correlation in a large data set,” Proc. of 18th Interational Conference on Data Engineering pp. 517–528 (2002).
[CrossRef]

Yap, C.

C. Yap and H. Lee, “Identification of cell nucleus using a Mumford-Shah ellipse detector,” Proc. of ISVC 1, 582–593 (2008).

Yip, A.

Y. Law, H. Lee, and A. Yip, “A multiresolution stochastic level set method for Mumford-Shah image segmentation,” IEEE Transactions on Image Processing 17, 2289–2300 (2008).
[CrossRef] [PubMed]

Yu, P.

J. Yang, W. Wang, H. Wang, and P. Yu, “δ-clusters: capturing subspace correlation in a large data set,” Proc. of 18th Interational Conference on Data Engineering pp. 517–528 (2002).
[CrossRef]

C. Aggarwal and P. Yu, “Finding generalized projected clusters in high dimensional spaces,” Proc. of the 2000 ACM SIGMOD International Conference on Management of Data pp. 70–81 (2000).

Yu, W.

W. Yu, H. Lee, S. Hariharan, W. Bu, and S. Ahmed, “Level set segmentation of cellular images based on topological dependence,” Proc. of ISVC 1, 540–551 (2008).

Zeevi, Y.

C. Sagiv, N. Sochen, and Y. Zeevi, “Texture segmentation via a diffusion-segmentation scheme in the Gabor feature space,” Proc. of the 2nd Intl. Workshop on Texture Analysis and Synthesis (2002).

Zhang, Y.

C.-H. Cheng, A. W. Fu, and Y. Zhang, “Entropy-based subspace clustering for mining numerical data,” Proc. of the 5th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining pp. 84–93 (1999).
[CrossRef]

Zhu, S.

A. Barbu and S. Zhu, “Multigrid and multi-level Swendsen-Wang cuts for hierarchic graph partition,” Proc. IEEE Conf. on Computer Vision and Pattern Recognition 2, 731–738 (2004).

Zhu, W.

W. Zhu, T. Chan, and S. Esedog¯lu, “Segmentation with depth: A level set approach,” SIAM J. Sci. Comput. 28, 1957–1973 (2006).
[CrossRef]

Commun. Pure Appl. Math. (1)

D. Mumford and J. Shah, “Optimal approximation by piecewise smooth functions and associated variational problems,” Commun. Pure Appl. Math. 42, 577–685 (1989).
[CrossRef]

CVGIP: Image Understanding (1)

T. Reed, “A review of recent texture segmentation and feature extraction techniques,” CVGIP: Image Understanding 57, 359–372 (1993).
[CrossRef]

IEEE Tran. Image Process. (1)

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

IEEE Trans. Knowledge and Data Engineering (1)

L. Jing, M. Ng, and J. Huang, “An entropy weighting k-means algorithm for subspace clustering of high-dimensional sparse data,” IEEE Trans. Knowledge and Data Engineering 19, 1026–1041 (2007).
[CrossRef]

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

B. Manjunath and W. Ma, “Texture features for browsing and retrieval of image data,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 837–842 (1996).
[CrossRef]

IEEE Transactions on Image Processing (1)

Y. Law, H. Lee, and A. Yip, “A multiresolution stochastic level set method for Mumford-Shah image segmentation,” IEEE Transactions on Image Processing 17, 2289–2300 (2008).
[CrossRef] [PubMed]

Int. J. Comput. Vis. (1)

J.-F. Aujol, G. Gilboa, T. Chan, and S. Osher, “Structure-texture image decomposition - modeling, algorithms, and parameter selection,” Int. J. Comput. Vis. 67, 111–136 (2006).
[CrossRef]

Int. J. Comput. Vision (2)

L. A. Vese and T. F. Chan, “A multiphase level set framework for image segmentation using the Mumford and Shah model,” Int. J. Comput. Vision 50, 271–293 (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]

International Journal of Computer Vision (1)

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contour models,” International Journal of Computer Vision 1, 321–331 (1988).
[CrossRef]

J. IEEE (1)

D. Gabor, “Theory of communication,” J. IEEE 93, 429–459 (1946).

J. Math. Imaging Vis. (1)

X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thiran, and S. Osher, “Fast global minimization of the active contour/snake model,” J. Math. Imaging Vis. 28, 151–167 (2007).
[CrossRef]

J. Math. Imaging Vision (1)

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89–97 (2004).
[CrossRef]

J. R. Statist. Soc. B (1)

J. Friedman and J. Meulman, “Clustering objects on subsets of attributes,” J. R. Statist. Soc. B 66, 815–849 (2004).
[CrossRef]

Journal of the Society of Laparoendoscopic Surgeons (1)

L. Roberts, J. Redan, and H. Reich, “Extraperitoneal endometriosis with catamenial pneumothoraces: A review of the literature,” Journal of the Society of Laparoendoscopic Surgeons 7, 371–375 (2003).
[PubMed]

Journal of Visual Communication and Image Representation (1)

T. Chan, B. Sandberg, and L. Vese, “Active contours without edges for vector-valued images,” Journal of Visual Communication and Image Representation 11, 130–141 (2000).
[CrossRef]

Pattern. Recog. (1)

A. Jain and F. Farrokhnia, “Unsupervised texture segmentation using Gabor filters,” Pattern. Recog. 24, 1167–1186 (1991).
[CrossRef]

Proc. IEEE Conf. on Computer Vision (1)

S. Belongie, C. Carson, H. Greenspan, and J. Malik, “Color- and texture-based image segmentation using EM and its application to content-based image retrieval,” Proc. IEEE Conf. on Computer Vision p. 675 (1998).

Proc. IEEE Conf. on Computer Vision and Pattern Recognition (1)

A. Barbu and S. Zhu, “Multigrid and multi-level Swendsen-Wang cuts for hierarchic graph partition,” Proc. IEEE Conf. on Computer Vision and Pattern Recognition 2, 731–738 (2004).

Proc. of 18th Interational Conference on Data Engineering (1)

J. Yang, W. Wang, H. Wang, and P. Yu, “δ-clusters: capturing subspace correlation in a large data set,” Proc. of 18th Interational Conference on Data Engineering pp. 517–528 (2002).
[CrossRef]

Proc. of Image Understanding Workshop (1)

K. Laws, “Texture energy measures,” Proc. of Image Understanding Workshop pp. 47–51 (1979).

Proc. of ISVC (2)

C. Yap and H. Lee, “Identification of cell nucleus using a Mumford-Shah ellipse detector,” Proc. of ISVC 1, 582–593 (2008).

W. Yu, H. Lee, S. Hariharan, W. Bu, and S. Ahmed, “Level set segmentation of cellular images based on topological dependence,” Proc. of ISVC 1, 540–551 (2008).

Proc. of the 1998 ACM SIGMOD International Conference on Management of Data (1)

R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan, “Automatic subspace clustering of high dimensional data for data mining applications,” Proc. of the 1998 ACM SIGMOD International Conference on Management of Data pp. 94–105 (1998).

Proc. of the 2000 ACM SIGMOD International Conference on Management of Data (1)

C. Aggarwal and P. Yu, “Finding generalized projected clusters in high dimensional spaces,” Proc. of the 2000 ACM SIGMOD International Conference on Management of Data pp. 70–81 (2000).

Proc. of the 5th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (1)

C.-H. Cheng, A. W. Fu, and Y. Zhang, “Entropy-based subspace clustering for mining numerical data,” Proc. of the 5th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining pp. 84–93 (1999).
[CrossRef]

Proc. of the Intl. Conf. on Computer Analysis of Images and Patterns (1)

T. Brox, M. Rousson, R. Deriche, and J. Weickert, “Unsupervised segmentation incorporating color, texture, and motion,” Proc. of the Intl. Conf. on Computer Analysis of Images and Patterns pp. 353–360 (2003).
[CrossRef]

Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro (2)

M. Datar, D. Padfield, and H. Cline, “Color and texture based segmentation of molecular pathology images using HSOMs,” Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro pp. 292–295 (2008).
[CrossRef]

M. Rachidi, A. Chappard, C. Marchadier, C. Gadois, E. Lespessailles, and C. L. Benhamou, “Application of Laws’ masks to bone texture analysis: An innovative image analysis tool in osteoporosis,” Proceedings of the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro pp. 1191–1194 (2008).
[CrossRef]

Proceedings of the International Conference on Computer as a Tool (1)

B. Sharif, A. Ahmadian, M. Oghabian, and N. Izadi, “Texture segmentation of endometrial images for aiding diagnosis of hyperplasia,” Proceedings of the International Conference on Computer as a Tool 2, 983–986 (2005).

SIAM J. Appl. Math. (1)

T. F. Chan, S. Esedoglu, and M. Nikolova, “Algorithms for finding global minimizers of denoising and segmentation models,” SIAM J. Appl. Math. 66, 1632–1648 (2006).
[CrossRef]

SIAM J. Sci. Comput. (1)

W. Zhu, T. Chan, and S. Esedog¯lu, “Segmentation with depth: A level set approach,” SIAM J. Sci. Comput. 28, 1957–1973 (2006).
[CrossRef]

The Journal of Experimental Biology (1)

C. Chiao and R. Hanlon, “Cuttlefish camouflage: Visual perception of size, contrast and number of white squares on artificial checkerboard substrata initiates disruptive coloration,” The Journal of Experimental Biology 204, 2119–2125 (2001).
[PubMed]

Other (8)

P. Brodatz, Textures: A photographic album for artists and designers (Dover, New York, 1996).

S. Beucher and C. Lantuéjoul, Use of watersheds in contour detection,” Proc. International Workshop on Image Processing: Real-time Edge and Motion Detection/Estimation (1979).
[PubMed]

A. Kneller, “The new age of bioimaging,” in “Paradigm,” (2006), pp. 18–25.

P. Petrou and P. G. Sevilla, Dealing with Texture (Wiley, 2006).
[CrossRef]

M. Haindl and S. Mikeš, “Unsupervised texture segmentation,” in “Patten Recognition Techniques, Technology and Applications,” , P.-Y. Yin, ed. (I-Tech, 2008), pp. 227–248.

B. Sandberg, T. Chan, and L. Vese, “A level-set and Gabor-based active contour algorithm for segmentation textured images,” Cam report, UCLA (2002).

C. Sagiv, N. Sochen, and Y. Zeevi, “Texture segmentation via a diffusion-segmentation scheme in the Gabor feature space,” Proc. of the 2nd Intl. Workshop on Texture Analysis and Synthesis (2002).

M. Rousson, T. Brox, and R. Deriche, “Active unsupervised texture segmentation on a diffusion based feature space,” Proc. of the 2003 IEEE Computer Vision and Pattern Recognition (2003).

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Figures (6)

Fig. 1.
Fig. 1.

A work flow of the Semi-Supervised Subspace Mumford-Shah method. Features such as Gabor and RGB are first extracted from the image. Then a vector-valued Subspace Mumford-Shah model is used to segment the image, where the feature weights {λij } are learned with the help of the selected ROI’s.

Fig. 2.
Fig. 2.

Segmentation of endometrial tissue: SSSMS and MS models with three RGB channels and twelve Gabor features as input features, and MSNLST with nonlinear structure tensor features. μ = 0.04 (SSSMS) and 0.17 (MS) and 0.002 (MSNLST), γ = 0.5.

Fig. 3.
Fig. 3.

Segmentation of a rat myocardium image: SSSMS and MS models with 3 Gabor features and 4 Brodatz texture similarity measures as input features, and MSNLST with nonlinear structure tensor features. μ = 0.5 (SSSMS) and 1.5 (MS) and 0.0085 (MSNLST), γ = 0.5.

Fig. 4.
Fig. 4.

Segmentation of two cuttlefishes: SSSMS and MS models with 84 input features before feature selection process, and MSNLST with nonlinear structure tensor features. μ = 0.1 (both SSSMS and MS) and 0.006 (MSNLST), γ = 0.5.

Fig. 5.
Fig. 5.

Segmentation of two zebras: SSSMS and MS models with 6 Gabor transforms as input features, and MSNLST with nonlinear structure tensor features. μ = 0.02 (SSSMS) and 0.2 (MS) and 0.05 (MSNLST), γ = 50.

Fig. 6.
Fig. 6.

Segmentation of three stem cell images: SSSMS model with red channel and 6 Gabor features as input features. μ = 0.3, γ = 5.

Tables (2)

Tables Icon

Table 1. The weights of the top two channels for each region.

Tables Icon

Table 2. Percentage errors (percentage symmetric difference) w.r.t. manual segmentations.

Equations (28)

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

F U c Λ = i = 1 n j = 1 m λ ij k = 1 N u ki f kj c ij 2 + γ i = 1 n j = 1 m λ ij log λ ij .
j = 1 m λ ij = 1 , for i = 1 , , n ,
i = 1 n u ki = 1 , for k = 1 , , N ,
0 λ ij 1 , for i = 1 , , n , and j = 1 , , m ,
u ki { 0,1 } . for k = 1 , , N , and i = 1 , , n .
F MS C c = 1 m i = 1 n j = 1 m Ω i f j x y c ij 2 dxdy + μ Length ( C ) .
G x y = 1 2 π σ x σ y exp [ 1 2 ( x 2 σ x 2 + y 2 σ y 2 ) + 2 πiW x ] ,
G pq x y = a p G ( x′ , y ) ,
x = a p ( x cos θ + y sin θ ) ,
y = a p ( x sin θ + y cos θ ) ,
f j x y = Ω u x ̂ y ̂ G p j q j x x ̂ y y ̂ ¯ d x ̂ d y ̂ ,
μ j ( u ) = 1 Ω Ω f j x y dxdy , σ j ( u ) = 1 Ω Ω [ f i x y μ j ( u ) ] 2 dxdy .
d u v j μ j ( u ) μ j ( v ) std ( μ j ) + σ j ( u ) σ j ( v ) std ( σ j ) .
f k x y = d ( u ( , x , y ) , v k )
F USSMS C c Λ = i = 1 n j = 1 m λ ij Ω i f i x y c ij 2 dxdy + γ i = 1 n j = 1 m λ ij log λ ij + μ Length ( C ) .
F SSSMS C c Λ = ( 1 β ) F USSMS C c Λ + β [ Ω k ROI k i = 1 n j = 1 m λ ij F ij + γ i = 1 n j = 1 m λ ij log λ ij ] .
F ij = ROI i f i x y c ˜ ij 2 dxdy
c ˜ ij = ROI i f j x y dxdy ROI i .
c ij = Ω i f j x y dxdy Ω i = Ω χ i x y f j x y dxdy Ω χ i x y dxdy
λ ij = exp ( D ij γ ) k = 1 m exp ( D ik γ ) ,
D ij = ( 1 β ) Ω i f j x y c ij 2 dxdy + β Ω F ij k ROI k .
min 0 u 1 { Ω rudxdy + μ Ω u dxdy } ,
r x y = j = 1 m { λ 1 j [ c 1 j f j x y ] 2 λ 2 j [ c 2 j f j x y ] 2 } .
χ n + 1 x y = { 1 if u n x y > 0.5 , 0 otherwise
r n + 1 x y = j = 1 m { λ 1 j [ c 1 j f j x y ] 2 λ 2 j [ c 2 j f j x y ] 2 }
v n + 1 x y = min { max { u n + 1 x y θ r n + 1 x y , 0 } , 1 }
p n + 1 x y = p n x y + δ [ ( div p n v n + 1 μθ ) ] x y 1 + δ [ ( div p n v n + 1 μθ ) ] x y
u n + 1 x y = v n + 1 x y μθ ( div p n + 1 ) x y .

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