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Kronecker least angle regression for unsupervised unmixing of hyperspectral imaging data

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Abstract

Spectral unmixing is an important data processing task that is commonly applied to hyperspectral imaging data. It uses a set of spectral pixels, i.e., multichannel data, to separate each spectral pixel, a multicomponent signal comprised of a linear mixture of pure spectral signatures, into its individual spectral signatures commonly known as endmembers. When no prior information about the required endmembers is available, the resulting unsupervised unmixing problem will be underdetermined, and additional constraints become necessary. A recent approach to solving this problem required that these endmembers be sparse in some dictionary. Sparse signal recovery is commonly solved using a basis pursuit optimization algorithm that requires specifying a data-dependent regularization parameter. Least angle regression (LARS) is a very efficient method to simultaneously solve the basis pursuit optimization problem for all relevant regularization parameter values. However, despite this efficiency of LARS, it has not been applied to the spectral unmixing problem before. This is likely because the application of LARS to large multichannel data could be very challenging in practice, due to the need for generation and storage of extremely large arrays (~ 1010 bytes in a relatively small area of spectral unmixing problem). In this paper, we extend the standard LARS algorithm, using Kronecker products, to make it suitable for practical and efficient recovery of sparse signals from large multichannel data, i.e., without the need to construct or process very large arrays, or the need for trial and error to determine the regularization parameter value. We then apply this new Kronecker LARS (K-LARS) algorithm to successfully achieve spectral unmixing of both synthetic and AVIRIS hyperspectral imaging data. We also compare our results to ones obtained using an earlier basis pursuit-based spectral unmixing algorithm, generalized morphological component analysis (GMCA). We show that these two results are similar, albeit our results were obtained without trial and error, or arbitrary choices, in specifying the regularization parameter. More important, our K-LARS algorithm could be a very valuable research tool to the signal processing community, where it could be used to solve sparse least squares problems involving large multichannel data.

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References

  1. Manolakis, D., Marden, D., Shaw, G.A.: Hyperspectral image processing for automatic target detection applications. Linc. Lab. J. 14(1), 79–116 (2003)

    Google Scholar 

  2. Manolakis, D., et al.: Detection algorithms in hyperspectral imaging systems: an overview of practical algorithms. IEEE Signal Process. Mag. 31(1), 24–33 (2014)

    Article  Google Scholar 

  3. Nasrabadi, N.M.: Hyperspectral target detection: an overview of current and future challenges. IEEE Signal Process. Mag. 31(1), 34–44 (2014)

    Article  Google Scholar 

  4. Bioucas-Dias, J.M., et al.: Hyperspectral unmixing overview: geometrical, statistical, and sparse regression-based approaches. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 5(2), 354–379 (2012)

    Article  Google Scholar 

  5. Rajan, S., Ghosh, J., Crawford, M.M.: An active learning approach to hyperspectral data classification. IEEE Trans. Geosci. Remote Sens. 46(4), 1231–1242 (2008)

    Article  Google Scholar 

  6. Keshava, N.: Distance metrics and band selection in hyperspectral processing with applications to material identification and spectral libraries. IEEE Trans. Geosci. Remote Sens. 42(7), 1552–1565 (2004)

    Article  Google Scholar 

  7. Parente, M., Plaza, A.: Survey of geometric and statistical unmixing algorithms for hyperspectral images. In: 2010 2nd Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), pp. 1–4 (2010)

  8. Dobigeon, N., et al.: Bayesian separation of spectral sources under non-negativity and full additivity constraints. Signal Process 89(12), 2657–2669 (2009)

    Article  Google Scholar 

  9. Amiri, F., Kahaei, M.: A sparsity-based Bayesian approach for hyperspectral unmixing using normal compositional model. SIViP 12(7), 1361–1367 (2018)

    Article  Google Scholar 

  10. Shi, C., Wang, L.: Incorporating spatial information in spectral unmixing: a review. Remote Sens. Environ. 149, 70–87 (2014)

    Article  Google Scholar 

  11. Iordache, M., Plaza, A., Bioucas-Dias, J.: On the use of spectral libraries to perform sparse unmixing of hyperspectral data. In: 2010 2nd Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), pp. 1–4

  12. Iordache, M., Bioucas-Dias, J.M., Plaza, A.: Sparse unmixing of hyperspectral data. IEEE Trans. Geosci. Remote Sens. 49(6), 2014–2039 (2011)

    Article  Google Scholar 

  13. Keshava, N., Mustard, J.F.: Spectral unmixing. IEEE Signal Process. Mag. 19(1), 44–57 (2002)

    Article  Google Scholar 

  14. Yang, Z., et al.: Blind spectral unmixing based on sparse nonnegative matrix factorization. IEEE Trans. Image Process. 20(4), 1112–1125 (2011)

    Article  MathSciNet  Google Scholar 

  15. Qian, Y., et al.: Hyperspectral unmixing via l1-sparsity-constrained nonnegative matrix factorization. IEEE Trans. Geosci. Remote Sens. 49(11), 4282–4297 (2011)

    Article  Google Scholar 

  16. Wu, Z., et al.: Sparse non-negative matrix factorization on GPUs for hyperspectral unmixing. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 7(8), 3640–3649 (2014)

    Article  Google Scholar 

  17. Tong, L., et al.: Dual graph regularized NMF for hyperspectral unmixing. In: 2014 International Conference on Digital Lmage Computing: Techniques and Applications (DlCTA), pp. 1–8 (2014)

  18. Wang, W., Qian, Y., Tang, Y.Y.: Hypergraph-regularized sparse NMF for hyperspectral unmixing. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 9(2), 681–694 (2016)

    Article  Google Scholar 

  19. Li, X., Cui, J., Zhao, L.: Blind nonlinear hyperspectral unmixing based on constrained kernel nonnegative matrix factorization. Signal Image Video Process. 8(8), 1555 (2014)

    Article  Google Scholar 

  20. Bobin, J., et al.: Sparsity and morphological diversity in blind source separation. IEEE Trans. Image Process. 16(11), 2662–2674 (2007)

    Article  MathSciNet  Google Scholar 

  21. Rapin, J., et al.: Sparse and non-negative BSS for noisy data. IEEE Trans. Signal Process. 61(22), 5620–5632 (2013)

    Article  MathSciNet  Google Scholar 

  22. Chenot, C., Bobin, J., Rapin, J.: Robust sparse blind source separation. IEEE Signal Process. Lett. 22(11), 2172–2176 (2015)

    Article  Google Scholar 

  23. Bioucas-Dias, J.M., Figueiredo, M.A.: Alternating direction algorithms for constrained sparse regression: application to hyperspectral unmixing. In: 2010 2nd workshop on hyperspectral image and signal processing: evolution in remote sensing (WHISPERS), pp. 1–4 (2010)

  24. Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM Rev. 43(1), 129–159 (2001)

    Article  MathSciNet  Google Scholar 

  25. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)

    Article  MathSciNet  Google Scholar 

  26. Rish, I., Grabarnik, G.: Sparse Modeling: Theory, Algorithms, and Applications. CRC Press, Boca Raton (2014)

    Book  Google Scholar 

  27. Caiafa, C.F., Cichocki, A.: Computing sparse representations of multidimensional signals using kronecker bases. Neural Comput. 25(1), 186–220 (2013)

    Article  MathSciNet  Google Scholar 

  28. Bernstein, D.S.: Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory. Princeton University Press, Princeton (2005)

    MATH  Google Scholar 

  29. Comon, P., Jutten, C., Herault, J.: Blind separation of sources, Part II: problems statement. Signal Process 24(1), 11–20 (1991)

    Article  Google Scholar 

  30. Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper I: basic analytical and algebraic properties. Fuzzy Sets Syst. 143(1), 5–26 (2004)

    Article  MathSciNet  Google Scholar 

  31. Mallat, S.: A Wavelet Tour of Signal Processing. Academic press, Cambridge (1999)

    MATH  Google Scholar 

  32. Li, Y., Osher, S.: Coordinate descent optimization for l1-minimization with application to compressed sensing; a greedy algorithm. Inverse Probl. Imaging 3(3), 487–503 (2009)

    Article  MathSciNet  Google Scholar 

  33. Lawson, C.L., Hanson, R.J.: Solving Least Squares Problems. SIAM, Philadelphia (1995)

    Book  Google Scholar 

  34. Efron, B., et al.: Least angle regression. Ann. Stat. 32(2), 407–499 (2004)

    Article  MathSciNet  Google Scholar 

  35. Asif, M.S., Romberg, J.: Dynamic updating for l1-minimization. IEEE J. Sel. Top. Signal Process. 4(2), 421–434 (2010)

    Article  Google Scholar 

  36. Donoho, D.L., Tsaig, Y.: Fast Solution of l1-norm minimization problems when the solution may be sparse. IEEE Trans. Inf. Theory 54(11), 4789–4812 (2008)

    Article  Google Scholar 

  37. ASTER Spectral Library, http://speclib.jpl.nasa.gov. Accessed 16 Sept 2018

  38. AVIRIS NASA Website, http://aviris.jpl.nasa.gov/alt_locator/. Accessed 16 Sept 2018

  39. Elad, M., Aharon, M.: Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans. Image Process. 15(12), 3736–3745 (2006)

    Article  MathSciNet  Google Scholar 

  40. Dewangan, N., Goswami, A.D.: Image denoising using wavelet thresholding methods. Int. J. Eng. Sci. Manag. 2(2), 271–275 (2012)

    Google Scholar 

  41. Winter, M.E.: N-FINDR: an algorithm for fast autonomous spectral end-member determination in hyperspectral data. In: SPIE’s International Symposium on Optical Science, Engineering, and Instrumentation, pp. 266–275 (1999)

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  1. Electrical and Computer Engineering Department, Faculty of Engineering, University of Manitoba, Winnipeg, MB, R3T 5V6, Canada

    Ahmed Elrewainy & Sherif S. Sherif

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  1. Ahmed Elrewainy
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  2. Sherif S. Sherif
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Correspondence to Sherif S. Sherif.

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Elrewainy, A., Sherif, S.S. Kronecker least angle regression for unsupervised unmixing of hyperspectral imaging data. SIViP 14, 359–367 (2020). https://doi.org/10.1007/s11760-019-01562-w

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  • DOI: https://doi.org/10.1007/s11760-019-01562-w

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