Sparsity-Based Unsupervised Unmixing of Hyperspectral Imaging Data Using Basis Pursuit
Mixing in the hyperspectral imaging occurs due to the low spatial resolutions of the used cameras. The existing pure materials “endmembers” in the scene share the spectra pixels with different amounts called “abundances”. Unmixing of the data cube is an important task to know the present endmembers in the cube for the analysis of these images. Unsupervised unmixing is done with no information about the given data cube. Sparsity is one of the recent approaches used in the source recovery or unmixing techniques. The l1-norm optimization problem “basis pursuit” could be used as a sparsity-based approach to solve this unmixing problem where the endmembers is assumed to be sparse in an appropriate domain known as dictionary. This optimization problem is solved using proximal method “iterative thresholding”. The l1-norm basis pursuit optimization problem as a sparsity-based unmixing technique was used to unmix real and synthetic hyperspectral data cubes.
 D. Manolakis, D. Marden and G. A. Shaw, "Hyperspectral image processing for automatic target detection applications," Lincoln Laboratory Journal, vol. 14, (1), pp. 79-116, 2003.
 J. M. Bioucas-Dias et al, "Hyperspectral unmixing overview: Geometrical, statistical, and sparse regression-based approaches," IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 5, (2), pp. 354-379, 2012.
 P. Comon, C. Jutten and J. Herault, "Blind separation of sources, Part II: Problems statement," Signal Process, vol. 24, (1), pp. 11-20, 1991.
 M. Parente and A. Plaza, "Survey of geometric and statistical unmixing algorithms for hyperspectral images," in Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), 2010 2nd Workshop On, 2010, pp. 1-4.
 N. Dobigeon et al, "Bayesian separation of spectral sources under non-negativity and full additivity constraints," Signal Process, vol. 89, (12), pp. 2657-2669, 2009.
 C. Shi and L. Wang, "Incorporating spatial information in spectral unmixing: A review," Remote Sens. Environ., vol. 149, pp. 70-87, 2014.
 J. Starck, F. Murtagh and J. M. Fadili, Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity. Cambridge university press, 2010.
 D. L. Donoho and I. M. Johnstone, "Ideal denoising in an orthonormal basis chosen from a library of bases," Comptes Rendus De L'Académie Des Sciences.Série I, Mathématique, vol. 319, (12), pp. 1317-1322, 1994.
 M. Iordache, A. Plaza and J. Bioucas-Dias, "On the use of spectral libraries to perform sparse unmixing of hyperspectral data," in Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), 2010 2nd Workshop On, 2010, pp. 1-4.
 Z. Wu et al, "Sparse non-negative matrix factorization on GPUs for hyperspectral unmixing," IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 7, (8), pp. 3640-3649, 2014.
 J. A. Tropp and A. C. Gilbert, "Signal recovery from random measurements via orthogonal matching pursuit," IEEE Trans. Inf. Theory, vol. 53, (12), pp. 4655-4666, 2007.
 S. S. Chen, D. L. Donoho and M. A. Saunders, "Atomic decomposition by basis pursuit," SIAM Rev, vol. 43, (1), pp. 129-159, 2001.
 S. Mallat, A Wavelet Tour of Signal Processing. Academic press, 1999.
 E. Candes et al, "Fast discrete curvelet transforms," Multiscale Modeling & Simulation, vol. 5, (3), pp. 861-899, 2006.
 R. Rubinstein, A. M. Bruckstein and M. Elad, "Dictionaries for sparse representation modeling," Proc IEEE, vol. 98, (6), pp. 1045-1057, 2010.
 J. Bobin et al, "Sparsity and morphological diversity in blind source separation," IEEE Trans. Image Process., vol. 16, (11), pp. 2662-2674, 2007.
 AVIRIS NASA Website, http://aviris.jpl.nasa.gov/alt_locator/, last accessed on April 2016.
 ASTER Spectral Library, http://speclib.jpl.nasa.gov, last accessed on July 2016.
 M. E. Winter, "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, 1999, pp. 266-275.