Journal of Signal Processing Systems

, Volume 65, Issue 3, pp 479–496 | Cite as

Unmixing of Hyperspectral Images using Bayesian Non-negative Matrix Factorization with Volume Prior

  • Morten Arngren
  • Mikkel N. Schmidt
  • Jan Larsen


Hyperspectral imaging can be used in assessing the quality of foods by decomposing the image into constituents such as protein, starch, and water. Observed data can be considered a mixture of underlying characteristic spectra (endmembers), and estimating the constituents and their abundances requires efficient algorithms for spectral unmixing. We present a Bayesian spectral unmixing algorithm employing a volume constraint and propose an inference procedure based on Gibbs sampling. We evaluate the method on synthetic and real hyperspectral data of wheat kernels. Results show that our method perform as good or better than existing volume constrained methods. Further, our method gives credible intervals for the endmembers and abundances, which allows us to asses the confidence of the results.


Bayesian source separation Hyperspectral image analysis Volume regularization Gibbs sampling 


  1. 1.
    Arngren, M., Schmidt, M., & Larsen, J. (2009). Bayesian nonnegative matrix factorization with volume prior for unmixing of hyperspectral images. In Machine learning for signal processing, IEEE workshop on (MLSP). doi: 10.1109/MLSP.2009.5306262.
  2. 2.
    Arngren, M., Schmidt, M. N., & Larsen, J. (2010). Unmixing of hyperspectral images using Bayesian nonnegative matrix factorization with volume prior. Technical note.
  3. 3.
    Arngren, M., Schmidt, M. N., & Larsen, J. (2010). Unmixing of hyperspectral images using bayesian nonnegative matrix factorization with volume prior, Matlab toolbox.
  4. 4.
    Berman, M., Kiiveri, H., Lagerstrom, R., Ernst, A., Dunne, R., & Huntington, J. (2004). ICE: A statistical approach to identifying endmembers in hyperspectral images. IEEE Transactions on Geoscience and Remote Sensing, 42(10), 2085–2095.CrossRefGoogle Scholar
  5. 5.
    Bioucas-Dias, J., & Nascimento, J. (2008). Hyperspectral subspace identification. IEEE Transactions on Geoscience and Remote Sensing, 46(8), 2435–2445.CrossRefGoogle Scholar
  6. 6.
    Boardman, J. W., Kruse, F. A., & Green, R. O. (1995). Mapping target signatures via partial unmixing of aviris data. In Summaries of JPL airborne earth science workshop.Google Scholar
  7. 7.
    Chan, T. H., Chi, C. Y., Huang, Y. M., & Ma, W. K. (2009). A convex analysis-based minimum-volume enclosing simplex algorithm for hyperspectral unmixing. IEEE Transactions on Signal Processing, 57(11), 4418–4432.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dobigeon, N., Moussaoui, S., Coulon, M., Tourneret, J. Y., & Hero, A. O. (2009). Joint Bayesian endmember extraction and linear unmixing for hyperspectral imagery. IEEE Transactions on Signal Processings, 57(11), 4355–4368.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dobigeon, N., Moussaoui, S., Tourneret, J. Y. Y., & Carteret, C. (2009). Bayesian separation of spectral sources under non-negativity and full additivity constraints. Signal Processing, 89(12), 2657–2669.zbMATHCrossRefGoogle Scholar
  10. 10.
    Dobigeon, N., & Tourneret, J. Y. (2007). Efficient sampling according to a multivariate Gaussian distribution truncated on a simplex. Tech. Rep., IRIT/ENSEEIHT/TeSA.Google Scholar
  11. 11.
    Eches, O., Dobigeon, N., & Tourneret, J. Y. (2010). Estimating the number of endmembers in hyperspectral images using the normal compositional model and a hierarchical Bayesian algorithm. IEEE Journal of Selected Topics in Signal Processing, 4(3), 582–591.CrossRefGoogle Scholar
  12. 12.
    Gelfand, A. E., & Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association, 85(410), 398–401.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Geman, S., & Geman, D. (1984). Stochastic relaxation, gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741. doi: 10.1109/TPAMI.1984.4767596.zbMATHCrossRefGoogle Scholar
  14. 14.
    Geweke, J. (1991). Efficient simulation from the multivariate normal and student-t distributions subject to linear constraints and the evaluation of constraint probabilities. In Proceedings the 23rd symposium on the interface between computer sciences and statistics (pp. 571–578). doi:
  15. 15.
    Hoyer, P. O. (2002). Non-negative sparse coding. Neural Networks for Signal Processing, 7, 557–565.Google Scholar
  16. 16.
    Masalmah, Y. M. (2007). Unsupervised unmixing of hyperspectral imagery using the constrained positive matrix factorization. Ph.D. thesis, Computing And Information Science And Engineering, University Of Puerto Rico, Mayagez Campus.Google Scholar
  17. 17.
    Mazet, V., Brie, D., & Idier, J. (2005). Simulation of postive normal variables using several proposal distributions. In IEEE workshop on statistical signal processing (pp. 37–42). doi: 10.1109/SSP.2005.1628560.
  18. 18.
    Miao, L., & Qi, H. (2007). Endmember extraction from highly mixed data using minimum volume constrained nonnegative matrix factorization. IEEE Transactions on Geoscience and Remote Sensing, 45(3), 765–777.CrossRefGoogle Scholar
  19. 19.
    Moussaoui, S., Brie, D., Mohammad-Djafari, A., & Carteret, C. (2006). Separation of non-negative mixture of non-negative sources using a Bayesian approach and mcmc sampling. IEEE Transactions on Signal Processing, 54(11), 4133–4145.CrossRefGoogle Scholar
  20. 20.
    Nascimento, J. M. P., & Dias, J. M. B. (2005). Vertex component analysis: A fast algorithm to unmix hyperspectral data. IEEE Transactions on Geoscience and Remote Sensing, 43(4), 898–910.CrossRefGoogle Scholar
  21. 21.
    Ochs, M. F., Stoyanova, R. S., Arias-Mendoza, F., & Brown, T. R. (1999). A new method for spectral decomposition using a bilinear bayesian approach. Journal of Magnetic Resonance, 137(1), 161–176.CrossRefGoogle Scholar
  22. 22.
    Paatero, P., & Tapper, U. (1994). Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values. Environmetricsm 5(2), 111–126.CrossRefGoogle Scholar
  23. 23.
    Parra, L., Spence, C., Sajda, P., Ziehe, A., & Mller, K. R. (1999). Unmixing hyperspectral data. Neural Information Processing Systems, 12, 942–948.Google Scholar
  24. 24.
    Pauca, V. P., Piper, J., & Plemmons, R. J. (2006). Nonnegative matrix factorization for spectral data analysis. Linear Algebra and Its Applications, 416(1), 29–47.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Plaza, A., Martnez, P., Prez, R., & Plaza, J. (2004). A quantitative and comparative analysis of endmember extraction algorithms from hyperspectral data. IEEE Transactions on Geoscience and Remote Sensing, 42(3), 650–663.CrossRefGoogle Scholar
  26. 26.
    Sajda, P., Du, S., & Parra, L. (2003). Recovery of constituent spectra using non-negative matrix factorization. Proceedings of the SPIE - The International Society for Optical Engineering, 5207(1), 321–331.Google Scholar
  27. 27.
    Schachtner, R., Pppel, G., Tom, A. M., & Lang, E. W. (2009). Minimum determinant constraint for non-negative matrix factorization. Lecture Notes in Computer Science, 5441/2009, 106–113.CrossRefGoogle Scholar
  28. 28.
    Schmidt, M. N. (2009). Linearly constrained bayesian matrix factorization for blind source separation. In Y. Bengio, D. Schuurmans, J. Lafferty, C. K. I. Williams, & A. Culotta (Eds.), Advances in neural information processing systems (Vol. 22, pp. 1624–1632).Google Scholar
  29. 29.
    Schmidt, M. N., & Laurberg, H. (2008). Nonnegative matrix factorization with Gaussian process priors. Computational Intelligence and Neuroscience p. Article ID 361705.Google Scholar
  30. 30.
    Schmidt, M. N., Winther, O., & Hansen, L. K. (2009). Bayesian non-negative matrix factorization. In Independent component analysis and signal separation, international conference on lecture notes in computer science (LNCS) (Vol. 5441, pp. 540–547).Google Scholar
  31. 31.
    Winter, M. E. (1999). N-findr: An algorithm for fast autonomous spectral end-member determination in hyperspectral data. Proceedings of SPIE - The International Society for Optical Engineering, 3753, 266–275.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Morten Arngren
    • 1
    • 2
  • Mikkel N. Schmidt
    • 1
  • Jan Larsen
    • 1
  1. 1.DTU InformaticsTechnical University of DenmarkLyngbyDenmark
  2. 2.FOSS Analytical A/SHillerødDenmark

Personalised recommendations