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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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]

2007

L. Jing, M. Ng, and J. Huang, "An entropy weighting k-means algorithm for subspace clustering of highdimensional 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

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. Esedoglu, "Segmentation with depth: A level set approach," SIAM J. Sci. Comput. 28, 1957-1973 (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]

2004

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

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

2003

L. Roberts, J. Redan, and H. Reich, "Extraperitoneal endometriosis with catamenial pneumothoraces: A review of the literature," J. Soc. Laparoendoscopic Surgeons 7, 371-375 (2003).
[PubMed]

2002

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]

2001

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

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

2000

T. Chan, B. Sandberg, and L. Vese, "Active contours without edges for vector-valued images," J. Visual Commun. Image Representation 11, 130-141 (2000).
[CrossRef]

1996

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

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

1991

A. Jain and F. Farrokhnia, "Unsupervised texture segmentation using Gabor filters," Pattern. Recogn. 24, 1167-1186 (1991).
[CrossRef]

1989

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

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

1946

D. Gabor, "Theory of communication," Proc. of J. IEE, (London) 93, 429-459 (1946).

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).

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 Vis. 28, 151-167 (2007).
[CrossRef]

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. Esedoglu, "Segmentation with depth: A level set approach," SIAM J. Sci. Comput. 28, 1957-1973 (2006).
[CrossRef]

T. Chan, B. Sandberg, and L. Vese, "Active contours without edges for vector-valued images," J. Visual Commun. 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]

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," J. Experimental Biology 204, 2119-2125 (2001).
[PubMed]

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]

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]

W. Zhu, T. Chan, and S. Esedoglu, "Segmentation with depth: A level set approach," SIAM J. Sci. Comput. 28, 1957-1973 (2006).
[CrossRef]

Farrokhnia, F.

A. Jain and F. Farrokhnia, "Unsupervised texture segmentation using Gabor filters," Pattern. Recogn. 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]

Gabor, D.

D. Gabor, "Theory of communication," Proc. of J. IEE, (London) 93, 429-459 (1946).

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]

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," J. Experimental Biology 204, 2119-2125 (2001).
[PubMed]

Huang, J.

L. Jing, M. Ng, and J. Huang, "An entropy weighting k-means algorithm for subspace clustering of highdimensional 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. Recogn. 24, 1167-1186 (1991).
[CrossRef]

Jing, L.

L. Jing, M. Ng, and J. Huang, "An entropy weighting k-means algorithm for subspace clustering of highdimensional 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]

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]

Lee, H.

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]

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]

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]

Meulman, J.

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

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 highdimensional 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]

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]

Redan, J.

L. Roberts, J. Redan, and H. Reich, "Extraperitoneal endometriosis with catamenial pneumothoraces: A review of the literature," J. Soc. 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," J. Soc. 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," J. Soc. Laparoendoscopic Surgeons 7, 371-375 (2003).
[PubMed]

Sandberg, B.

T. Chan, B. Sandberg, and L. Vese, "Active contours without edges for vector-valued images," J. Visual Commun. Image Representation 11, 130-141 (2000).
[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).

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.

T. Chan, B. Sandberg, and L. Vese, "Active contours without edges for vector-valued images," J. Visual Commun. 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]

Witkin, A.

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

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]

Zhu, W.

W. Zhu, T. Chan, and S. Esedoglu, "Segmentation with depth: A level set approach," SIAM J. Sci. Comput. 28, 1957-1973 (2006).
[CrossRef]

Commun. Pure Appl. Math.

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

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

IEEE Tran. Image Process.

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

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

IEEE Trans. Pattern Anal. Mach. Intell.

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

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.

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

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

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

J. Experimental Biology

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

J. Math. Imaging Vis.

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

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

J. R. Statist. Soc. B

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

J. Soc. Laparoendoscopic Surgeons

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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)

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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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