Abstract

In this paper, we develop a robust and effective algorithm for texture segmentation and feature selection. The approach is to incorporate a patch-based subspace learning technique into the subspace Mumford–Shah (SMS) model to make the minimization of the SMS model robust and accurate. The proposed method is fully unsupervised in that it removes the need to specify training data, which is required by existing methods for the same model. We further propose a novel (to our knowledge) pairwise dissimilarity measure for pixels. Its novelty lies in the use of the relevance scores of the features of each pixel to improve its discriminating power. Some superior results are obtained compared to existing unsupervised algorithms, which do not use a subspace approach. This confirms the usefulness of the subspace approach and the proposed unsupervised algorithm.

© 2011 Optical Society of America

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  1. P. Petrou and P. G. Sevilla, Dealing with Texture (Wiley, 2006).
    [CrossRef]
  2. Y. Law, H. Lee, and A. Yip, “Supervised texture segmentation using the subspace Mumford-Shah model,” in Proceedings of the 2009 International Conference on Image Processing, Computer Vision and Pattern Recognition (IPCV ’09) (CSREA Press, 2009), pp. 554–560.
  3. B. Sandberg, T. Chan, and L. Vese, “A level-set and Gabor-based active contour algorithm for segmenting textured images,” CAM report (UCLA, 2002), ftp://ftp.math.ucla.edu/pub/camreport/cam02-39.ps.gz.
  4. L. Jing, M. Ng, and J. Huang, “An entropy weighting k-means algorithm for subspace clustering of high-dimensional sparse data,” IEEE Trans. Knowl. Data Eng. 19, 1026–1041(2007).
    [CrossRef]
  5. Y. Law, H. Lee, and A. Yip, “Semi-supervised subspace learning for Mumford-Shah model based texture segmentation,” Opt. Express 18, 4434–4448 (2010).
    [CrossRef] [PubMed]
  6. D. Gabor, “Theory of communication,” J. Inst. Electr. Eng., Part 3 93, 429–459 (1946).
    [CrossRef]
  7. B. Manjunath and W. Ma, “Texture features for browsing and retrieval of image data,” IEEE Trans. Pattern Anal. Machine Intell. 18, 837–842 (1996).
    [CrossRef]
  8. N. Paragios and R. Deriche, “Geodesic active regions and level set methods for supervised texture segmentation,” Int. J. Comput. Vis. 46, 223–247 (2002).
    [CrossRef]
  9. C. Sagiv, N. Sochen, and Y. Zeevi, “Texture segmentation via a diffusion-segmentation scheme in the Gabor feature space,” in Proceedings of the 2nd International Workshop on Texture Analysis and Synthesis (Texture02) (Heriot-Watt University, 2002), pp. 123–128.
  10. B. Sharif, A. Ahmadian, M. Oghabian, and N. Izadi, “Texture segmentation of endometrial images for aiding diagnosis of hyperplasia,” in Proceedings of the International Conference on Computer as a Tool, 2005, Vol.  2 (IEEE, 2005), pp. 983–986.
    [CrossRef]
  11. B.-W. Hong, S. Soatto, K. Ni, and T. Chan, “The scale of a texture and its application to segmentation,” in IEEE Conference on Computer Vision and Pattern Recognition, 2008 (IEEE, 2008), pp. 1–8.
    [CrossRef]
  12. L. Szumilas, B. Mičušík, and A. Hanbury, “Texture segmentation through salient texture patches,” in Proceedings of the Computer Vision Winter Workshop 2006 (Czech Pattern Recognition Society, 2006), pp. 111–116.
  13. J. Ward, “Hierarchical grouping to optimize an objective function,” J. Am. Stat. Assoc. 58, 236–244 (1963).
    [CrossRef]
  14. P. Brodatz, Textures: A Photographic Album for Artists and Designers (Dover, 1996).
  15. L. Roberts, J. Redan, and H. Reich, “Extraperitoneal endometriosis with catamenial pneumothoraces: a review of the literature,” JSLS 7, 371–375 (2003).
    [PubMed]
  16. E. Gelasca, B. Obara, D. Fedorov, K. Kvilekval, and B. Manjunath, “A biosegmentation benchmark for evaluation of bioimage analysis methods,” BMC Bioinf. 10, 368(2009).
    [CrossRef]
  17. K. Laws, “Texture energy measures,” in Proceedings of the Image Understanding Workshop (Defense Advanced Research Projects Agency, 1979), pp. 47–51.
  18. The Brodatz texture similarity measures are defined as follows. Let u be a given image and {vk} be the set of all images in the Brodatz database. The kth feature value fk(x,y) at pixel (x,y)∈Ω is defined by fk(x,y)=d(u(·,·|x,y),vk), where d(·,·) is defined by Eq.  and u(·,·|x,y) is a local patch in u around (x,y).
  19. T. Brox, M. Rousson, R. Deriche, and J. Weickert, “Unsupervised segmentation incorporating color, texture, and motion,” in Proceedings of the International Conference on Computer Analysis of Images and Patterns (Springer, 2003), pp. 353–360.
    [CrossRef]
  20. Y. Rubner, J. Puzicha, C. Tomasi, and J. Buhmann, “Empirical evalutation of dissimilarity measures for color and texture,” Comput. Vis. Image Understand. 84, 25–43 (2001).
    [CrossRef]
  21. R. Dobrushin, “Prescribing a system of random variables by conditional distributions,” Theory Probab. Appl. 15, 458–486 (1970).
    [CrossRef]
  22. M. Ng, G. Qiu, and A. Yip, “Numerical methods for interactive multiple-class image segmentation problems,” Int. J. Imag. Syst. Technol. 20, 191–201 (2010).
    [CrossRef]
  23. 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]
  24. A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imag. Vis. 20, 89–97(2004).
    [CrossRef]
  25. 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]
  26. X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thiran, and S. Osher, “Fast global minimization of the active contour/snake model,” J. Math. Imaging Vision 28, 151–167 (2007).
    [CrossRef]
  27. Y. Law, H. Lee, and A. Yip, “A multiresolution stochastic level set method for Mumford-Shah image segmentation,” IEEE Trans. Image Process. 17, 2289–2300 (2008).
    [CrossRef] [PubMed]

2010 (2)

Y. Law, H. Lee, and A. Yip, “Semi-supervised subspace learning for Mumford-Shah model based texture segmentation,” Opt. Express 18, 4434–4448 (2010).
[CrossRef] [PubMed]

M. Ng, G. Qiu, and A. Yip, “Numerical methods for interactive multiple-class image segmentation problems,” Int. J. Imag. Syst. Technol. 20, 191–201 (2010).
[CrossRef]

2009 (2)

Y. Law, H. Lee, and A. Yip, “Supervised texture segmentation using the subspace Mumford-Shah model,” in Proceedings of the 2009 International Conference on Image Processing, Computer Vision and Pattern Recognition (IPCV ’09) (CSREA Press, 2009), pp. 554–560.

E. Gelasca, B. Obara, D. Fedorov, K. Kvilekval, and B. Manjunath, “A biosegmentation benchmark for evaluation of bioimage analysis methods,” BMC Bioinf. 10, 368(2009).
[CrossRef]

2008 (2)

B.-W. Hong, S. Soatto, K. Ni, and T. Chan, “The scale of a texture and its application to segmentation,” in IEEE Conference on Computer Vision and Pattern Recognition, 2008 (IEEE, 2008), pp. 1–8.
[CrossRef]

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

2007 (2)

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

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

2006 (4)

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

L. Szumilas, B. Mičušík, and A. Hanbury, “Texture segmentation through salient texture patches,” in Proceedings of the Computer Vision Winter Workshop 2006 (Czech Pattern Recognition Society, 2006), pp. 111–116.

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]

2005 (1)

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

2004 (1)

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

2003 (2)

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

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

2002 (3)

B. Sandberg, T. Chan, and L. Vese, “A level-set and Gabor-based active contour algorithm for segmenting textured images,” CAM report (UCLA, 2002), ftp://ftp.math.ucla.edu/pub/camreport/cam02-39.ps.gz.

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

C. Sagiv, N. Sochen, and Y. Zeevi, “Texture segmentation via a diffusion-segmentation scheme in the Gabor feature space,” in Proceedings of the 2nd International Workshop on Texture Analysis and Synthesis (Texture02) (Heriot-Watt University, 2002), pp. 123–128.

2001 (1)

Y. Rubner, J. Puzicha, C. Tomasi, and J. Buhmann, “Empirical evalutation of dissimilarity measures for color and texture,” Comput. Vis. Image Understand. 84, 25–43 (2001).
[CrossRef]

1996 (2)

P. Brodatz, Textures: A Photographic Album for Artists and Designers (Dover, 1996).

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

1979 (1)

K. Laws, “Texture energy measures,” in Proceedings of the Image Understanding Workshop (Defense Advanced Research Projects Agency, 1979), pp. 47–51.

1970 (1)

R. Dobrushin, “Prescribing a system of random variables by conditional distributions,” Theory Probab. Appl. 15, 458–486 (1970).
[CrossRef]

1963 (1)

J. Ward, “Hierarchical grouping to optimize an objective function,” J. Am. Stat. Assoc. 58, 236–244 (1963).
[CrossRef]

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng., Part 3 93, 429–459 (1946).
[CrossRef]

Ahmadian, A.

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

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]

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 Vision 28, 151–167 (2007).
[CrossRef]

Brodatz, P.

P. Brodatz, Textures: A Photographic Album for Artists and Designers (Dover, 1996).

Brox, T.

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

Buhmann, J.

Y. Rubner, J. Puzicha, C. Tomasi, and J. Buhmann, “Empirical evalutation of dissimilarity measures for color and texture,” Comput. Vis. Image Understand. 84, 25–43 (2001).
[CrossRef]

Chambolle, A.

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

Chan, T.

B.-W. Hong, S. Soatto, K. Ni, and T. Chan, “The scale of a texture and its application to segmentation,” in IEEE Conference on Computer Vision and Pattern Recognition, 2008 (IEEE, 2008), pp. 1–8.
[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]

B. Sandberg, T. Chan, and L. Vese, “A level-set and Gabor-based active contour algorithm for segmenting textured images,” CAM report (UCLA, 2002), ftp://ftp.math.ucla.edu/pub/camreport/cam02-39.ps.gz.

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]

Deriche, R.

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

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

Dobrushin, R.

R. Dobrushin, “Prescribing a system of random variables by conditional distributions,” Theory Probab. Appl. 15, 458–486 (1970).
[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 Vision 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]

Fedorov, D.

E. Gelasca, B. Obara, D. Fedorov, K. Kvilekval, and B. Manjunath, “A biosegmentation benchmark for evaluation of bioimage analysis methods,” BMC Bioinf. 10, 368(2009).
[CrossRef]

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng., Part 3 93, 429–459 (1946).
[CrossRef]

Gelasca, E.

E. Gelasca, B. Obara, D. Fedorov, K. Kvilekval, and B. Manjunath, “A biosegmentation benchmark for evaluation of bioimage analysis methods,” BMC Bioinf. 10, 368(2009).
[CrossRef]

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]

Hanbury, A.

L. Szumilas, B. Mičušík, and A. Hanbury, “Texture segmentation through salient texture patches,” in Proceedings of the Computer Vision Winter Workshop 2006 (Czech Pattern Recognition Society, 2006), pp. 111–116.

Hong, B.-W.

B.-W. Hong, S. Soatto, K. Ni, and T. Chan, “The scale of a texture and its application to segmentation,” in IEEE Conference on Computer Vision and Pattern Recognition, 2008 (IEEE, 2008), pp. 1–8.
[CrossRef]

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. Knowl. Data Eng. 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,” in Proceedings of the International Conference on Computer as a Tool, 2005, Vol.  2 (IEEE, 2005), pp. 983–986.
[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. Knowl. Data Eng. 19, 1026–1041(2007).
[CrossRef]

Kvilekval, K.

E. Gelasca, B. Obara, D. Fedorov, K. Kvilekval, and B. Manjunath, “A biosegmentation benchmark for evaluation of bioimage analysis methods,” BMC Bioinf. 10, 368(2009).
[CrossRef]

Law, Y.

Y. Law, H. Lee, and A. Yip, “Semi-supervised subspace learning for Mumford-Shah model based texture segmentation,” Opt. Express 18, 4434–4448 (2010).
[CrossRef] [PubMed]

Y. Law, H. Lee, and A. Yip, “Supervised texture segmentation using the subspace Mumford-Shah model,” in Proceedings of the 2009 International Conference on Image Processing, Computer Vision and Pattern Recognition (IPCV ’09) (CSREA Press, 2009), pp. 554–560.

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

Laws, K.

K. Laws, “Texture energy measures,” in Proceedings of the Image Understanding Workshop (Defense Advanced Research Projects Agency, 1979), pp. 47–51.

Lee, H.

Y. Law, H. Lee, and A. Yip, “Semi-supervised subspace learning for Mumford-Shah model based texture segmentation,” Opt. Express 18, 4434–4448 (2010).
[CrossRef] [PubMed]

Y. Law, H. Lee, and A. Yip, “Supervised texture segmentation using the subspace Mumford-Shah model,” in Proceedings of the 2009 International Conference on Image Processing, Computer Vision and Pattern Recognition (IPCV ’09) (CSREA Press, 2009), pp. 554–560.

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

Ma, W.

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

Manjunath, B.

E. Gelasca, B. Obara, D. Fedorov, K. Kvilekval, and B. Manjunath, “A biosegmentation benchmark for evaluation of bioimage analysis methods,” BMC Bioinf. 10, 368(2009).
[CrossRef]

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

Micušík, B.

L. Szumilas, B. Mičušík, and A. Hanbury, “Texture segmentation through salient texture patches,” in Proceedings of the Computer Vision Winter Workshop 2006 (Czech Pattern Recognition Society, 2006), pp. 111–116.

Ng, M.

M. Ng, G. Qiu, and A. Yip, “Numerical methods for interactive multiple-class image segmentation problems,” Int. J. Imag. Syst. Technol. 20, 191–201 (2010).
[CrossRef]

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

Ni, K.

B.-W. Hong, S. Soatto, K. Ni, and T. Chan, “The scale of a texture and its application to segmentation,” in IEEE Conference on Computer Vision and Pattern Recognition, 2008 (IEEE, 2008), pp. 1–8.
[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]

Obara, B.

E. Gelasca, B. Obara, D. Fedorov, K. Kvilekval, and B. Manjunath, “A biosegmentation benchmark for evaluation of bioimage analysis methods,” BMC Bioinf. 10, 368(2009).
[CrossRef]

Oghabian, M.

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

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 Vision 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]

Paragios, N.

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

Petrou, P.

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

Puzicha, J.

Y. Rubner, J. Puzicha, C. Tomasi, and J. Buhmann, “Empirical evalutation of dissimilarity measures for color and texture,” Comput. Vis. Image Understand. 84, 25–43 (2001).
[CrossRef]

Qiu, G.

M. Ng, G. Qiu, and A. Yip, “Numerical methods for interactive multiple-class image segmentation problems,” Int. J. Imag. Syst. Technol. 20, 191–201 (2010).
[CrossRef]

Redan, J.

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

Reich, H.

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

Roberts, L.

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

Rousson, M.

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

Rubner, Y.

Y. Rubner, J. Puzicha, C. Tomasi, and J. Buhmann, “Empirical evalutation of dissimilarity measures for color and texture,” Comput. Vis. Image Understand. 84, 25–43 (2001).
[CrossRef]

Sagiv, C.

C. Sagiv, N. Sochen, and Y. Zeevi, “Texture segmentation via a diffusion-segmentation scheme in the Gabor feature space,” in Proceedings of the 2nd International Workshop on Texture Analysis and Synthesis (Texture02) (Heriot-Watt University, 2002), pp. 123–128.

Sandberg, B.

B. Sandberg, T. Chan, and L. Vese, “A level-set and Gabor-based active contour algorithm for segmenting textured images,” CAM report (UCLA, 2002), ftp://ftp.math.ucla.edu/pub/camreport/cam02-39.ps.gz.

Sevilla, P. G.

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

Sharif, B.

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

Soatto, S.

B.-W. Hong, S. Soatto, K. Ni, and T. Chan, “The scale of a texture and its application to segmentation,” in IEEE Conference on Computer Vision and Pattern Recognition, 2008 (IEEE, 2008), pp. 1–8.
[CrossRef]

Sochen, N.

C. Sagiv, N. Sochen, and Y. Zeevi, “Texture segmentation via a diffusion-segmentation scheme in the Gabor feature space,” in Proceedings of the 2nd International Workshop on Texture Analysis and Synthesis (Texture02) (Heriot-Watt University, 2002), pp. 123–128.

Szumilas, L.

L. Szumilas, B. Mičušík, and A. Hanbury, “Texture segmentation through salient texture patches,” in Proceedings of the Computer Vision Winter Workshop 2006 (Czech Pattern Recognition Society, 2006), pp. 111–116.

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 Vision 28, 151–167 (2007).
[CrossRef]

Tomasi, C.

Y. Rubner, J. Puzicha, C. Tomasi, and J. Buhmann, “Empirical evalutation of dissimilarity measures for color and texture,” Comput. Vis. Image Understand. 84, 25–43 (2001).
[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 Vision 28, 151–167 (2007).
[CrossRef]

Vese, L.

B. Sandberg, T. Chan, and L. Vese, “A level-set and Gabor-based active contour algorithm for segmenting textured images,” CAM report (UCLA, 2002), ftp://ftp.math.ucla.edu/pub/camreport/cam02-39.ps.gz.

Ward, J.

J. Ward, “Hierarchical grouping to optimize an objective function,” J. Am. Stat. Assoc. 58, 236–244 (1963).
[CrossRef]

Weickert, J.

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

Yip, A.

M. Ng, G. Qiu, and A. Yip, “Numerical methods for interactive multiple-class image segmentation problems,” Int. J. Imag. Syst. Technol. 20, 191–201 (2010).
[CrossRef]

Y. Law, H. Lee, and A. Yip, “Semi-supervised subspace learning for Mumford-Shah model based texture segmentation,” Opt. Express 18, 4434–4448 (2010).
[CrossRef] [PubMed]

Y. Law, H. Lee, and A. Yip, “Supervised texture segmentation using the subspace Mumford-Shah model,” in Proceedings of the 2009 International Conference on Image Processing, Computer Vision and Pattern Recognition (IPCV ’09) (CSREA Press, 2009), pp. 554–560.

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

Zeevi, Y.

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Y. Law, H. Lee, and A. Yip, “Supervised texture segmentation using the subspace Mumford-Shah model,” in Proceedings of the 2009 International Conference on Image Processing, Computer Vision and Pattern Recognition (IPCV ’09) (CSREA Press, 2009), pp. 554–560.

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Figures (8)

Fig. 1
Fig. 1

Two incorrect retrievals (i.e., ideal retrieval is not ranked first) using ER and WMV distances. The ideal retrieval is enclosed in a box. The ER distance gives patterns that are visually closer to the query than the WMV distance.

Fig. 2
Fig. 2

The figure demonstrates how to compare the true segmentation and the computed segmentation based on distribution of feature values. In each segment, the marginal pdf and the associated cdf of the jth feature of manual segmentation and computed segmentation are computed. Then, four kinds of distance be tween the pdf or cdf of the manual and computed segmentations are computed.

Fig. 3
Fig. 3

Results on four images: (a) RGB, (b) endometrial tissue, (c) stem cells, (d) cuttlefish. First row: original image, detected patches in background and in foreground, manual segmentation; second row: segmentation using k-means, MS NLST , MS, and auto-SMS.

Fig. 4
Fig. 4

Ratio of distance between segments from MS model and manual segments and distance between segments from auto-SMS model and manual segments in log scale. Auto-SMS outperforms MS in all cases.

Fig. 5
Fig. 5

Percentage error with respect to manual segmentation in a large range of γ values [0.05,10].

Fig. 6
Fig. 6

Percentage error with respect to manual segmentation in a large range of μ values.

Fig. 7
Fig. 7

Detected patches in the retinal layer image. From left to right: GCL, INL, ONL, and OS.

Fig. 8
Fig. 8

Results on retinal layer image. From left to right: Manual segmentation, k-means, MS, and auto-SMS. The respective percentage error is depicted in the bracket.

Tables (5)

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Table 1 Frequency of the Rank of the Ideal Retrievals Obtained by ER and WMV Distances

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Table 2 Percentage Error (Percentage Symmetric Difference) with Respect to Manual Segmentation

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Table 3 Algorithm 1: Auto Subspace Mumford–Shah

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Table 4 Algorithm 2: Alternating Minimization of F Auto

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Table 5 Algorithm 3: Minimization of F SMS with Respect to { Ω i }

Equations (37)

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F SMS ( { Ω i } , { c i j } , { λ i j } ) μ · Length ( C ) + i = 1 n x Ω i j = 1 m λ i j | f j ( x ) c i j | 2 + γ i = 1 n j = 1 m λ i j log λ i j ,
j = 1 m λ i j = 1 , for     i = 1 , 2 , , n ,
0 λ i j 1 , for     i = 1 , 2 , , n ; j = 1 , 2 , , m .
G ( x , y ) 1 2 π σ x σ y exp [ 1 2 ( x 2 σ x 2 + y 2 σ y 2 ) + 2 π i W x ] ,
G p q ( 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 ^ | .
R x , r { z Ω x z r }
N x , r R x , 3 r R x , r = { z Ω | r < x z 3 r } .
P X ( y ) 1 | X | z X δ y , u ( z ) ,
F X ( y ) 0 y P X ( x ) d x .
D W ( P 1 , P 2 ) 0 1 | F 1 ( y ) F 2 ( y ) | d y ,
H ( P ) 0 1 P ( y ) log P ( y ) d y .
r * argmin r { D W ( P R x , r , P N x , r ) + ν H ( P R x , r ) + η r } ,
F ( { Ω i } , { c i j } , { λ i j } ) i = 1 n x Ω i j = 1 m λ i j | f j ( x ) c i j | 2 + γ H ( { λ i j } ) ,
H ( { λ i j } ) i = 1 n j = 1 m λ i j log λ i j
c i j = f ¯ j ( Ω i ) 1 | Ω i | x Ω i f j ( x ) ,
λ i j = exp ( D i j γ ) k = 1 m exp ( D i k γ ) ,
D i j x Ω i | f j ( x ) f ¯ j ( Ω i ) | 2 .
1 | Ω i | x Ω i | f j ( x ) f ¯ j ( Ω i ) | 2 1 | R z | x R z | f j ( x ) f ¯ j ( R z ) | 2
λ i j λ ^ z , j exp ( D ^ z , j γ ) k = 1 m exp ( D ^ z , k γ ) ,
D ^ z , j x R z | f j ( x ) f ¯ j ( R z ) | 2 .
F ( { S i } , { λ i j } ) = i = 1 n 1 2 | S i | x S i x S i j = 1 m λ i j | f j ( x ) f j ( x ) | 2 + γ H ( { λ i j } ) .
F ( { S i } ) = i = 1 n 1 2 | S i | x S i x S i j = 1 m λ ^ x , j | f j ( x ) f j ( x ) | 2 .
d x ( x ) j = 1 m λ ^ x , j | f j ( x ) f j ( x ) | 2
F ( { S i } ) = i = 1 n 1 2 | S i | x S i x S i d ( x , x ) ,
d ( x , x ) j = 1 m ( λ ^ x , j + λ ^ x , j 2 ) | f j ( x ) f j ( x ) | 2 .
( S 1 * , S 2 * ) = argmin S 1 S 2 Δ ( S 1 , S 2 ) .
Δ ( S 1 , S 2 ) = 1 2 ( | S 1 | + | S 2 | ) x S 1 S 2 x S 1 S 2 d ( x , x ) 1 2 | S 1 | x S 1 x S 1 d ( x , x ) 1 2 | S 2 | x S 2 x S 2 d ( x , x ) .
F Patch ( { λ i j } ) | Ω | | x S R x | i = 1 n x R i j = 1 m λ i j | f j ( x ) f ¯ j ( R i ) | 2 + γ H ( { λ i j } ) ,
F Auto ( { Ω i } , { c i j } , { λ i j } ) β F Patch ( { λ i j } ) + ( 1 β ) F SMS ( { Ω i } , { c i j } , { λ i j } ) .
j = 1 24 | f ¯ j f ¯ j std ( f ¯ j ) | + | σ j σ j std ( σ j ) | ,
λ i j = exp ( D ˜ i j γ ) k = 1 m exp ( D ˜ i k γ ) ,
D ˜ i j ( 1 β ) x Ω i | f j ( x ) c i j | 2 + β | Ω | | x S R x | · x R i | f j ( x ) f ¯ j ( R i ) | 2 .
χ k + 1 ( x ) = { 1 if     u k ( x ) > 0.5 , 0 otherwise
v k + 1 ( x ) = min { max { u k ( x ) θ r ( x ) , 0 } , 1 }
p k + 1 ( x ) = p k ( x ) + δ [ ( div p k v k + 1 μ θ ) ] ( x ) 1 + δ | [ ( div p k v k + 1 μ θ ) ] ( x ) |
u k + 1 ( x ) = v k + 1 ( x ) μ θ ( div p k + 1 ) ( x ) .

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