Abstract

In this work, we propose a new algorithm for spectral color image segmentation based on the use of a kernel matrix. A cost function for spectral kernel clustering is introduced to measure the correlation between clusters. An efficient multiscale method is presented for accelerating spectral color image segmentation. The multiscale strategy uses the lattice geometry of images to construct an image pyramid whose hierarchy provides a framework for rapidly estimating eigenvectors of normalized kernel matrices. To prevent the boundaries from deteriorating, the image size on the top level of the pyramid is generally required to be around 75×75, where the eigenvectors of normalized kernel matrices would be approximately solved by the Nyström method. Within this hierarchical structure, the coarse solution is increasingly propagated to finer levels and is refined using subspace iteration. In addition, to make full use of the abundant color information contained in spectral color images, we propose using spectrum extension to incorporate the geometric features of spectra into similarity measures. Experimental results have shown that the proposed method can perform significantly well in spectral color image segmentation as well as speed up the approximation of the eigenvectors of normalized kernel matrices.

© 2008 Optical Society of America

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References

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  1. P. Paclík, R. P. W. Duin, G. M. P. van Kempen, and R. Kohlus, “Segmentation of multispectral images using the combined classifier approach,” Image Vis. Comput. 21, 473-482 (2003).
    [CrossRef]
  2. J. L. Crespo, R. J. Duro, and F. L. Pena, “Gaussian synapse ANNs in multi- and hyperspectral image data analysis,” IEEE Trans. Instrum. Meas. 52, 724-732 (2003).
    [CrossRef]
  3. S. K. Pal and P. Mitra, “Multispectral image segmentation using the rough-set-initialized EM algorithm,” IEEE Trans. Geosci. Remote Sens. 40, 2495-2501 (2002).
    [CrossRef]
  4. G. Mercier, S. Derrode, and M. Lennon, “Hyperspectral image segmentation with Markov chain model,” in IEEE International Proceedings of the Geoscience and Remote Sensing Symposium (IGARSS'03) (IEEE, 2003), Vol. 122, pp. 21-25.
  5. A. Mohammad-Djafari, N. Bali, and A. Mohammadpour, “Hierarchical Markovian models for hyperspectral image segmentation,” in International Workshop on Intelligent Computing in Pattern Analysis/Systems, IWICPAS (Springer, 2006), pp. 416-424.
  6. H. Kwon and N. M. Nasrabadi, “Kernel matched subspace detectors for hyperspectral target detection,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 178-194 (2006).
    [CrossRef] [PubMed]
  7. Y. Weiss, “Segmentation using eigenvectors: a unifying view,” in IEEE International Conference on Computer Vision (ICCV'99) (IEEE, 1999), pp. 975-982.
    [CrossRef]
  8. A. Y. Ng, M. I. Jordan, and Y. Weiss, “On spectral clustering: analysis and an algorithm,” in Proceedings of the Neural Information Processing Systems, NIPS'01 (MIT, 2001), Vol. 14, pp. 849-856.
  9. S. X. Yu and J. Shi, “Multiclass spectral clustering,” in IEEE International Conference on Computor Vision (ICCV'03) (IEEE, 2003), pp. 313-319.
    [CrossRef]
  10. N. Cristianini, J. Shawe-Taylor, and J. S. Kandola, “Spectral kernel methods for clustering,” in Proceedings of the Neural Information Processing Systems, NIPS'01 (MIT, 2001), pp. 649-655.
  11. C. Fowlkes, S. Belongie, F. R. K. Chung, and J. Malik, “Spectral grouping using the Nyström method,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 214-225 (2004).
    [CrossRef] [PubMed]
  12. J. Shawe-Taylor and N. Cristianini, Kernel Methods for Pattern Analysis (Cambridge U. Press, 2004).
    [CrossRef]
  13. University of Joensuu Color Group, “Spectral database,” http://spectral.joensuu.fi/.
  14. F. R. Bach and M. I. Jordan, “Learning spectral clustering,” in Proceedings of the Neural Information Processing Systems, NIPS'03 (MIT, 2003), http://books.nips.cc/papers/files/nips16/NIPS2003_AA39.pdf.
  15. D. Achlioptas, F. McSherry, and B. Schölkopf, “Sampling techniques for kernel methods,” in Proceedings of the Neural Information Processing Systems, NIPS'01, (MIT, 2001), pp. 335-342.
  16. C. K. I. Williams and M. Seeger, “Using the Nyström method to speed up kernel machines,” in Proceedings of the Neural Information Processing Systems, NIPS'00 (MIT, 2000), pp. 682-688.
  17. T. Cour, F. Bénézit, and J. Shi, “Spectral segmentation with multiscale graph decomposition,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CPVR 2005 (IEEE, 2005), pp. 1124-1131.
  18. E. Sharon, M. Galun, D. Sharon, R. Basri, and A. Brandt, “Hierarchy and adaptivity in segmenting visual scenes,” Nature 442, 810-813 (2006).
    [CrossRef] [PubMed]
  19. D. Tolliver, R. T. Collins, and S. Baker, “Multilevel spectral partitioning for efficient image segmentation and tracking,” in the Seventh IEEE Workshop on Application of Computer Vision (WACA/MOTION'OE) (IEEE, 2005), Vol. 1, pp. 414-420.
    [CrossRef]
  20. B. N. Parlett, The Symmetric Eigenvalue Problem (Prentice Hall, 1998).
    [CrossRef]
  21. R. Unnikrishnan, C. Pantofaru, and M. Hebert, “Toward objective evaluation of image segmentation algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 929-944 (2007).
    [CrossRef] [PubMed]
  22. R. O. Duda, P. E. Hart, and D. F. Stork, Pattern Classification (Wiley, 2001).
  23. T. Cour, F. Benezit, and J. Shi, “Multiscale NCut Image Segmentation Toolbar,” http://www.seas.upenn.edu/~timothee/software/ncut_multiscale/ncut_multiscale.html.

2007

R. Unnikrishnan, C. Pantofaru, and M. Hebert, “Toward objective evaluation of image segmentation algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 929-944 (2007).
[CrossRef] [PubMed]

2006

E. Sharon, M. Galun, D. Sharon, R. Basri, and A. Brandt, “Hierarchy and adaptivity in segmenting visual scenes,” Nature 442, 810-813 (2006).
[CrossRef] [PubMed]

H. Kwon and N. M. Nasrabadi, “Kernel matched subspace detectors for hyperspectral target detection,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 178-194 (2006).
[CrossRef] [PubMed]

2004

C. Fowlkes, S. Belongie, F. R. K. Chung, and J. Malik, “Spectral grouping using the Nyström method,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 214-225 (2004).
[CrossRef] [PubMed]

2003

P. Paclík, R. P. W. Duin, G. M. P. van Kempen, and R. Kohlus, “Segmentation of multispectral images using the combined classifier approach,” Image Vis. Comput. 21, 473-482 (2003).
[CrossRef]

J. L. Crespo, R. J. Duro, and F. L. Pena, “Gaussian synapse ANNs in multi- and hyperspectral image data analysis,” IEEE Trans. Instrum. Meas. 52, 724-732 (2003).
[CrossRef]

2002

S. K. Pal and P. Mitra, “Multispectral image segmentation using the rough-set-initialized EM algorithm,” IEEE Trans. Geosci. Remote Sens. 40, 2495-2501 (2002).
[CrossRef]

Achlioptas, D.

D. Achlioptas, F. McSherry, and B. Schölkopf, “Sampling techniques for kernel methods,” in Proceedings of the Neural Information Processing Systems, NIPS'01, (MIT, 2001), pp. 335-342.

Bach, F. R.

F. R. Bach and M. I. Jordan, “Learning spectral clustering,” in Proceedings of the Neural Information Processing Systems, NIPS'03 (MIT, 2003), http://books.nips.cc/papers/files/nips16/NIPS2003_AA39.pdf.

Baker, S.

D. Tolliver, R. T. Collins, and S. Baker, “Multilevel spectral partitioning for efficient image segmentation and tracking,” in the Seventh IEEE Workshop on Application of Computer Vision (WACA/MOTION'OE) (IEEE, 2005), Vol. 1, pp. 414-420.
[CrossRef]

Bali, N.

A. Mohammad-Djafari, N. Bali, and A. Mohammadpour, “Hierarchical Markovian models for hyperspectral image segmentation,” in International Workshop on Intelligent Computing in Pattern Analysis/Systems, IWICPAS (Springer, 2006), pp. 416-424.

Basri, R.

E. Sharon, M. Galun, D. Sharon, R. Basri, and A. Brandt, “Hierarchy and adaptivity in segmenting visual scenes,” Nature 442, 810-813 (2006).
[CrossRef] [PubMed]

Belongie, S.

C. Fowlkes, S. Belongie, F. R. K. Chung, and J. Malik, “Spectral grouping using the Nyström method,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 214-225 (2004).
[CrossRef] [PubMed]

Benezit, F.

T. Cour, F. Benezit, and J. Shi, “Multiscale NCut Image Segmentation Toolbar,” http://www.seas.upenn.edu/~timothee/software/ncut_multiscale/ncut_multiscale.html.

Bénézit, F.

T. Cour, F. Bénézit, and J. Shi, “Spectral segmentation with multiscale graph decomposition,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CPVR 2005 (IEEE, 2005), pp. 1124-1131.

Brandt, A.

E. Sharon, M. Galun, D. Sharon, R. Basri, and A. Brandt, “Hierarchy and adaptivity in segmenting visual scenes,” Nature 442, 810-813 (2006).
[CrossRef] [PubMed]

Chung, F. R. K.

C. Fowlkes, S. Belongie, F. R. K. Chung, and J. Malik, “Spectral grouping using the Nyström method,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 214-225 (2004).
[CrossRef] [PubMed]

Collins, R. T.

D. Tolliver, R. T. Collins, and S. Baker, “Multilevel spectral partitioning for efficient image segmentation and tracking,” in the Seventh IEEE Workshop on Application of Computer Vision (WACA/MOTION'OE) (IEEE, 2005), Vol. 1, pp. 414-420.
[CrossRef]

Cour, T.

T. Cour, F. Bénézit, and J. Shi, “Spectral segmentation with multiscale graph decomposition,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CPVR 2005 (IEEE, 2005), pp. 1124-1131.

T. Cour, F. Benezit, and J. Shi, “Multiscale NCut Image Segmentation Toolbar,” http://www.seas.upenn.edu/~timothee/software/ncut_multiscale/ncut_multiscale.html.

Crespo, J. L.

J. L. Crespo, R. J. Duro, and F. L. Pena, “Gaussian synapse ANNs in multi- and hyperspectral image data analysis,” IEEE Trans. Instrum. Meas. 52, 724-732 (2003).
[CrossRef]

Cristianini, N.

N. Cristianini, J. Shawe-Taylor, and J. S. Kandola, “Spectral kernel methods for clustering,” in Proceedings of the Neural Information Processing Systems, NIPS'01 (MIT, 2001), pp. 649-655.

J. Shawe-Taylor and N. Cristianini, Kernel Methods for Pattern Analysis (Cambridge U. Press, 2004).
[CrossRef]

Derrode, S.

G. Mercier, S. Derrode, and M. Lennon, “Hyperspectral image segmentation with Markov chain model,” in IEEE International Proceedings of the Geoscience and Remote Sensing Symposium (IGARSS'03) (IEEE, 2003), Vol. 122, pp. 21-25.

Duda, R. O.

R. O. Duda, P. E. Hart, and D. F. Stork, Pattern Classification (Wiley, 2001).

Duin, R. P. W.

P. Paclík, R. P. W. Duin, G. M. P. van Kempen, and R. Kohlus, “Segmentation of multispectral images using the combined classifier approach,” Image Vis. Comput. 21, 473-482 (2003).
[CrossRef]

Duro, R. J.

J. L. Crespo, R. J. Duro, and F. L. Pena, “Gaussian synapse ANNs in multi- and hyperspectral image data analysis,” IEEE Trans. Instrum. Meas. 52, 724-732 (2003).
[CrossRef]

Fowlkes, C.

C. Fowlkes, S. Belongie, F. R. K. Chung, and J. Malik, “Spectral grouping using the Nyström method,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 214-225 (2004).
[CrossRef] [PubMed]

Galun, M.

E. Sharon, M. Galun, D. Sharon, R. Basri, and A. Brandt, “Hierarchy and adaptivity in segmenting visual scenes,” Nature 442, 810-813 (2006).
[CrossRef] [PubMed]

Hart, P. E.

R. O. Duda, P. E. Hart, and D. F. Stork, Pattern Classification (Wiley, 2001).

Hebert, M.

R. Unnikrishnan, C. Pantofaru, and M. Hebert, “Toward objective evaluation of image segmentation algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 929-944 (2007).
[CrossRef] [PubMed]

Jordan, M. I.

F. R. Bach and M. I. Jordan, “Learning spectral clustering,” in Proceedings of the Neural Information Processing Systems, NIPS'03 (MIT, 2003), http://books.nips.cc/papers/files/nips16/NIPS2003_AA39.pdf.

A. Y. Ng, M. I. Jordan, and Y. Weiss, “On spectral clustering: analysis and an algorithm,” in Proceedings of the Neural Information Processing Systems, NIPS'01 (MIT, 2001), Vol. 14, pp. 849-856.

Kandola, J. S.

N. Cristianini, J. Shawe-Taylor, and J. S. Kandola, “Spectral kernel methods for clustering,” in Proceedings of the Neural Information Processing Systems, NIPS'01 (MIT, 2001), pp. 649-655.

Kohlus, R.

P. Paclík, R. P. W. Duin, G. M. P. van Kempen, and R. Kohlus, “Segmentation of multispectral images using the combined classifier approach,” Image Vis. Comput. 21, 473-482 (2003).
[CrossRef]

Kwon, H.

H. Kwon and N. M. Nasrabadi, “Kernel matched subspace detectors for hyperspectral target detection,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 178-194 (2006).
[CrossRef] [PubMed]

Lennon, M.

G. Mercier, S. Derrode, and M. Lennon, “Hyperspectral image segmentation with Markov chain model,” in IEEE International Proceedings of the Geoscience and Remote Sensing Symposium (IGARSS'03) (IEEE, 2003), Vol. 122, pp. 21-25.

Malik, J.

C. Fowlkes, S. Belongie, F. R. K. Chung, and J. Malik, “Spectral grouping using the Nyström method,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 214-225 (2004).
[CrossRef] [PubMed]

McSherry, F.

D. Achlioptas, F. McSherry, and B. Schölkopf, “Sampling techniques for kernel methods,” in Proceedings of the Neural Information Processing Systems, NIPS'01, (MIT, 2001), pp. 335-342.

Mercier, G.

G. Mercier, S. Derrode, and M. Lennon, “Hyperspectral image segmentation with Markov chain model,” in IEEE International Proceedings of the Geoscience and Remote Sensing Symposium (IGARSS'03) (IEEE, 2003), Vol. 122, pp. 21-25.

Mitra, P.

S. K. Pal and P. Mitra, “Multispectral image segmentation using the rough-set-initialized EM algorithm,” IEEE Trans. Geosci. Remote Sens. 40, 2495-2501 (2002).
[CrossRef]

Mohammad-Djafari, A.

A. Mohammad-Djafari, N. Bali, and A. Mohammadpour, “Hierarchical Markovian models for hyperspectral image segmentation,” in International Workshop on Intelligent Computing in Pattern Analysis/Systems, IWICPAS (Springer, 2006), pp. 416-424.

Mohammadpour, A.

A. Mohammad-Djafari, N. Bali, and A. Mohammadpour, “Hierarchical Markovian models for hyperspectral image segmentation,” in International Workshop on Intelligent Computing in Pattern Analysis/Systems, IWICPAS (Springer, 2006), pp. 416-424.

Nasrabadi, N. M.

H. Kwon and N. M. Nasrabadi, “Kernel matched subspace detectors for hyperspectral target detection,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 178-194 (2006).
[CrossRef] [PubMed]

Ng, A. Y.

A. Y. Ng, M. I. Jordan, and Y. Weiss, “On spectral clustering: analysis and an algorithm,” in Proceedings of the Neural Information Processing Systems, NIPS'01 (MIT, 2001), Vol. 14, pp. 849-856.

Paclík, P.

P. Paclík, R. P. W. Duin, G. M. P. van Kempen, and R. Kohlus, “Segmentation of multispectral images using the combined classifier approach,” Image Vis. Comput. 21, 473-482 (2003).
[CrossRef]

Pal, S. K.

S. K. Pal and P. Mitra, “Multispectral image segmentation using the rough-set-initialized EM algorithm,” IEEE Trans. Geosci. Remote Sens. 40, 2495-2501 (2002).
[CrossRef]

Pantofaru, C.

R. Unnikrishnan, C. Pantofaru, and M. Hebert, “Toward objective evaluation of image segmentation algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 929-944 (2007).
[CrossRef] [PubMed]

Parlett, B. N.

B. N. Parlett, The Symmetric Eigenvalue Problem (Prentice Hall, 1998).
[CrossRef]

Pena, F. L.

J. L. Crespo, R. J. Duro, and F. L. Pena, “Gaussian synapse ANNs in multi- and hyperspectral image data analysis,” IEEE Trans. Instrum. Meas. 52, 724-732 (2003).
[CrossRef]

Schölkopf, B.

D. Achlioptas, F. McSherry, and B. Schölkopf, “Sampling techniques for kernel methods,” in Proceedings of the Neural Information Processing Systems, NIPS'01, (MIT, 2001), pp. 335-342.

Seeger, M.

C. K. I. Williams and M. Seeger, “Using the Nyström method to speed up kernel machines,” in Proceedings of the Neural Information Processing Systems, NIPS'00 (MIT, 2000), pp. 682-688.

Sharon, D.

E. Sharon, M. Galun, D. Sharon, R. Basri, and A. Brandt, “Hierarchy and adaptivity in segmenting visual scenes,” Nature 442, 810-813 (2006).
[CrossRef] [PubMed]

Sharon, E.

E. Sharon, M. Galun, D. Sharon, R. Basri, and A. Brandt, “Hierarchy and adaptivity in segmenting visual scenes,” Nature 442, 810-813 (2006).
[CrossRef] [PubMed]

Shawe-Taylor, J.

J. Shawe-Taylor and N. Cristianini, Kernel Methods for Pattern Analysis (Cambridge U. Press, 2004).
[CrossRef]

N. Cristianini, J. Shawe-Taylor, and J. S. Kandola, “Spectral kernel methods for clustering,” in Proceedings of the Neural Information Processing Systems, NIPS'01 (MIT, 2001), pp. 649-655.

Shi, J.

S. X. Yu and J. Shi, “Multiclass spectral clustering,” in IEEE International Conference on Computor Vision (ICCV'03) (IEEE, 2003), pp. 313-319.
[CrossRef]

T. Cour, F. Bénézit, and J. Shi, “Spectral segmentation with multiscale graph decomposition,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CPVR 2005 (IEEE, 2005), pp. 1124-1131.

T. Cour, F. Benezit, and J. Shi, “Multiscale NCut Image Segmentation Toolbar,” http://www.seas.upenn.edu/~timothee/software/ncut_multiscale/ncut_multiscale.html.

Stork, D. F.

R. O. Duda, P. E. Hart, and D. F. Stork, Pattern Classification (Wiley, 2001).

Tolliver, D.

D. Tolliver, R. T. Collins, and S. Baker, “Multilevel spectral partitioning for efficient image segmentation and tracking,” in the Seventh IEEE Workshop on Application of Computer Vision (WACA/MOTION'OE) (IEEE, 2005), Vol. 1, pp. 414-420.
[CrossRef]

Unnikrishnan, R.

R. Unnikrishnan, C. Pantofaru, and M. Hebert, “Toward objective evaluation of image segmentation algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 929-944 (2007).
[CrossRef] [PubMed]

van Kempen, G. M. P.

P. Paclík, R. P. W. Duin, G. M. P. van Kempen, and R. Kohlus, “Segmentation of multispectral images using the combined classifier approach,” Image Vis. Comput. 21, 473-482 (2003).
[CrossRef]

Weiss, Y.

Y. Weiss, “Segmentation using eigenvectors: a unifying view,” in IEEE International Conference on Computer Vision (ICCV'99) (IEEE, 1999), pp. 975-982.
[CrossRef]

A. Y. Ng, M. I. Jordan, and Y. Weiss, “On spectral clustering: analysis and an algorithm,” in Proceedings of the Neural Information Processing Systems, NIPS'01 (MIT, 2001), Vol. 14, pp. 849-856.

Williams, C. K. I.

C. K. I. Williams and M. Seeger, “Using the Nyström method to speed up kernel machines,” in Proceedings of the Neural Information Processing Systems, NIPS'00 (MIT, 2000), pp. 682-688.

Yu, S. X.

S. X. Yu and J. Shi, “Multiclass spectral clustering,” in IEEE International Conference on Computor Vision (ICCV'03) (IEEE, 2003), pp. 313-319.
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

S. K. Pal and P. Mitra, “Multispectral image segmentation using the rough-set-initialized EM algorithm,” IEEE Trans. Geosci. Remote Sens. 40, 2495-2501 (2002).
[CrossRef]

IEEE Trans. Instrum. Meas.

J. L. Crespo, R. J. Duro, and F. L. Pena, “Gaussian synapse ANNs in multi- and hyperspectral image data analysis,” IEEE Trans. Instrum. Meas. 52, 724-732 (2003).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

H. Kwon and N. M. Nasrabadi, “Kernel matched subspace detectors for hyperspectral target detection,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 178-194 (2006).
[CrossRef] [PubMed]

C. Fowlkes, S. Belongie, F. R. K. Chung, and J. Malik, “Spectral grouping using the Nyström method,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 214-225 (2004).
[CrossRef] [PubMed]

R. Unnikrishnan, C. Pantofaru, and M. Hebert, “Toward objective evaluation of image segmentation algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 929-944 (2007).
[CrossRef] [PubMed]

Image Vis. Comput.

P. Paclík, R. P. W. Duin, G. M. P. van Kempen, and R. Kohlus, “Segmentation of multispectral images using the combined classifier approach,” Image Vis. Comput. 21, 473-482 (2003).
[CrossRef]

Nature

E. Sharon, M. Galun, D. Sharon, R. Basri, and A. Brandt, “Hierarchy and adaptivity in segmenting visual scenes,” Nature 442, 810-813 (2006).
[CrossRef] [PubMed]

Other

D. Tolliver, R. T. Collins, and S. Baker, “Multilevel spectral partitioning for efficient image segmentation and tracking,” in the Seventh IEEE Workshop on Application of Computer Vision (WACA/MOTION'OE) (IEEE, 2005), Vol. 1, pp. 414-420.
[CrossRef]

B. N. Parlett, The Symmetric Eigenvalue Problem (Prentice Hall, 1998).
[CrossRef]

R. O. Duda, P. E. Hart, and D. F. Stork, Pattern Classification (Wiley, 2001).

T. Cour, F. Benezit, and J. Shi, “Multiscale NCut Image Segmentation Toolbar,” http://www.seas.upenn.edu/~timothee/software/ncut_multiscale/ncut_multiscale.html.

J. Shawe-Taylor and N. Cristianini, Kernel Methods for Pattern Analysis (Cambridge U. Press, 2004).
[CrossRef]

University of Joensuu Color Group, “Spectral database,” http://spectral.joensuu.fi/.

F. R. Bach and M. I. Jordan, “Learning spectral clustering,” in Proceedings of the Neural Information Processing Systems, NIPS'03 (MIT, 2003), http://books.nips.cc/papers/files/nips16/NIPS2003_AA39.pdf.

D. Achlioptas, F. McSherry, and B. Schölkopf, “Sampling techniques for kernel methods,” in Proceedings of the Neural Information Processing Systems, NIPS'01, (MIT, 2001), pp. 335-342.

C. K. I. Williams and M. Seeger, “Using the Nyström method to speed up kernel machines,” in Proceedings of the Neural Information Processing Systems, NIPS'00 (MIT, 2000), pp. 682-688.

T. Cour, F. Bénézit, and J. Shi, “Spectral segmentation with multiscale graph decomposition,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CPVR 2005 (IEEE, 2005), pp. 1124-1131.

G. Mercier, S. Derrode, and M. Lennon, “Hyperspectral image segmentation with Markov chain model,” in IEEE International Proceedings of the Geoscience and Remote Sensing Symposium (IGARSS'03) (IEEE, 2003), Vol. 122, pp. 21-25.

A. Mohammad-Djafari, N. Bali, and A. Mohammadpour, “Hierarchical Markovian models for hyperspectral image segmentation,” in International Workshop on Intelligent Computing in Pattern Analysis/Systems, IWICPAS (Springer, 2006), pp. 416-424.

Y. Weiss, “Segmentation using eigenvectors: a unifying view,” in IEEE International Conference on Computer Vision (ICCV'99) (IEEE, 1999), pp. 975-982.
[CrossRef]

A. Y. Ng, M. I. Jordan, and Y. Weiss, “On spectral clustering: analysis and an algorithm,” in Proceedings of the Neural Information Processing Systems, NIPS'01 (MIT, 2001), Vol. 14, pp. 849-856.

S. X. Yu and J. Shi, “Multiclass spectral clustering,” in IEEE International Conference on Computor Vision (ICCV'03) (IEEE, 2003), pp. 313-319.
[CrossRef]

N. Cristianini, J. Shawe-Taylor, and J. S. Kandola, “Spectral kernel methods for clustering,” in Proceedings of the Neural Information Processing Systems, NIPS'01 (MIT, 2001), pp. 649-655.

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Figures (9)

Fig. 1
Fig. 1

Three-stage flowchart of spectral color image segmentation: preprocessing, multiscale strategy, and discretization.

Fig. 2
Fig. 2

Spectra contained in a spectral color image: (a) spectral color image, colorchecker, (b) corresponding color image in the RGB space, (c) spectra corresponding to marked points in panel (b).

Fig. 3
Fig. 3

Smoothing and normalizing of color spectra: (a) original spectra from the Munsell data set, (b) spectra after smoothing via cubic spline curve, (c) spectra after normalizing, (d) spectra after smoothing and normalizing.

Fig. 4
Fig. 4

Slope and curvature of the spectra of the colorchecker image at a wavelength of 625 nm .

Fig. 5
Fig. 5

Scale extension. The weight values are different for points in the 4-neighborhood and 8-neighborhood of the given point.

Fig. 6
Fig. 6

Multiscale segmentation of spectral color images. (Left) Downsampling input spectral color images. (Right) Upper three images represent the approximate eigenvectors, and the bottom one is the segmentation result.

Fig. 7
Fig. 7

Segmentation performance analysis. Transparent pseudocolors are laid over the color image to mark segmentation results. (a)–(d) Results using four kernels, linear, polynomial, Gaussian, and ANOVA; (e)–(h) segmentation results using the polynomial kernel when varying the number of segments; (i)–(l), top four largest eigenvectors of matrix P, which are used to generate the segmentation shown in (f), (m)–(p), results when varying weight coefficients, α and β.

Fig. 8
Fig. 8

Method comparison: (a) corresponding RGB images; (b) ground truth; (c) segmentation using our method based on appropriate scales; (d) still our method, but using too many scales; (e) directly applying k-means to spectral data; (f) applying k-means after reducing the dimensionality of spectral data with PCA; (g) applying GMM to spectral data; (h) applying the CBS method to the color images (b) in the RGB space.

Fig. 9
Fig. 9

Segmentation of spectral color images with a variety of properties. For the ease of visualization, we put the boundaries between different segments on the corresponding RGB images.

Tables (4)

Tables Icon

Table 1 Notation Used in This Paper

Tables Icon

Table 2 Confusion Matrix (Percentage) for the “Hand” Image

Tables Icon

Table 3 Confusion Matrix (Percentage) for the “Toy0” Image

Tables Icon

Table 4 Tested Spectral Color Images

Equations (28)

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

ϴ = ( θ 1 θ i θ ι ) = ( θ 1 1 θ 1 j θ 1 ρ θ i 1 θ i j θ i ρ θ ι 1 θ ι j θ ι ρ ) ,
θ i j = θ i j 1 ρ k = 1 ρ θ i k ,
θ i j = θ i j k = 1 ρ ( θ i k ) 2 .
ϕ i j = ( θ i j + ϵ θ i j ϵ ) 2 ϵ ,
ψ i j = ( θ i j + ϵ + θ i j ϵ 2 θ i j ) ϵ 2 ,
δ i = ( α θ i , β ϕ i , γ ψ i ) = ( α θ i 1 , , α θ i j , , α θ i ρ , β ϕ i 1 , , β ϕ i j , , β ϕ i ρ 2 , γ ψ i 1 , , γ ψ i j , , γ ψ i ρ 2 ) .
C ( S ) = t = 1 η i S t , j E \ S t k i j ,
C ( Y ) = C ( S ) = t = 1 η i S t , j E \ S t k i j = t = 1 η ( i , j E k i j i , j S t k i j ) = i j E k i j t = 1 η i , j S t k i j = tr Y T D Y tr Y T K Y ,
C ( Y ) = tr Y T D Y tr Y T K Y = tr Z T Z tr Z T D 1 2 K D 1 2 Z = tr I η tr Z T D 1 2 K D 1 2 Z = η tr Z T D 1 2 K D 1 2 Z .
P = D 1 2 K D 1 2 ,
h = max ( log 2 max ( μ , ν ) 75 , 1 ) ,
P = [ A B B T C ] ,
P ̂ = [ A B B T B T A 1 B ] .
V = [ A B T ] A 1 2 U G Λ G 1 2 ,
v ( p , p i ) = v p × w i × k ( p , p i ) k ( p , p ) ,
v p i = I Neib ( p i ) v ( I , p i ) .
v p 1 = I { p , r } v ( I , p 1 ) = 1 2 v p × k ( p , p 1 ) k ( p , p ) + 1 2 v r × k ( r , p 1 ) k ( r , r ) ,
v p 5 = I { p , q , r , s } v ( I , p 5 ) = 1 4 v p × k ( p , p 5 ) k ( p , p ) + 1 4 v r × k ( r , p 5 ) k ( r , r ) + 1 4 v q × k ( q , p 5 ) k ( q , q ) + 1 4 v s × k ( s , p 5 ) k ( s , s ) .
V 0 = v 1 0 , v 2 0 , , v η 0 ,
FOR ID i = 1 , 2 , , t ,
V i = P V i 1 ,
END ,
k ( δ i , δ j ) = δ i , δ j ,
k ( δ i , δ j ) = ( δ i , δ j + 1 ) d ,
k ( δ i , δ j ) = exp ( δ i δ j 2 σ 2 ) ,
k 0 m ( δ i , δ j ) = 1 , if m 0 ,
k t m ( δ i , δ j ) = 0 , if m < t ,
k t m ( δ i , δ j ) = ( δ i m δ j m ) k t 1 m 1 ( δ i , δ j ) + k t m 1 ( δ i , δ j ) ,

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