{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T16:37:07Z","timestamp":1764175027172,"version":"build-2065373602"},"reference-count":69,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,5,28]],"date-time":"2021-05-28T00:00:00Z","timestamp":1622160000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>Unsupervised hyperspectral unmixing methods aim to extract endmember spectra and infer the proportion of each of these spectra in each observed pixel when considering linear mixtures. However, the interaction between sunlight and the Earth\u2019s surface is often very complex, so that observed spectra are then composed of nonlinear mixing terms. This nonlinearity is generally bilinear or linear quadratic. In this work, unsupervised hyperspectral unmixing methods, designed for the bilinear and linear-quadratic mixing models, are proposed. These methods are based on bilinear or linear-quadratic matrix factorization with non-negativity constraints. Two types of algorithms are considered. The first ones only use the projection of the gradient, and are therefore linked to an optimal manual choice of their learning rates, which remains the limitation of these algorithms. The second developed algorithms, which overcome the above drawback, employ multiplicative projective update rules with automatically chosen learning rates. In addition, the endmember proportions estimation, with three alternative approaches, constitutes another contribution of this work. Besides, the reduction of the number of manipulated variables in the optimization processes is also an originality of the proposed methods. Experiments based on realistic synthetic hyperspectral data, generated according to the two considered nonlinear mixing models, and also on two real hyperspectral images, are carried out to evaluate the performance of the proposed approaches. The obtained results show that the best proposed approaches yield a much better performance than various tested literature methods.<\/jats:p>","DOI":"10.3390\/rs13112132","type":"journal-article","created":{"date-parts":[[2021,5,31]],"date-time":"2021-05-31T03:45:29Z","timestamp":1622432729000},"page":"2132","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Hyperspectral Unmixing Based on Constrained Bilinear or Linear-Quadratic Matrix Factorization"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2606-7011","authenticated-orcid":false,"given":"Fatima Zohra","family":"Benhalouche","sequence":"first","affiliation":[{"name":"Agence Spatiale Alg\u00e9rienne, Centre des Techniques Spatiales, Arzew 31200, Algeria"},{"name":"Institut de Recherche en Astrophysique et Plan\u00e9tologie (IRAP), Universit\u00e9 de Toulouse, UPS, CNRS, OMP, CNES, 31400 Toulouse, France"},{"name":"Universit\u00e9 des Sciences et de la Technologie d\u2019Oran Mohamed Boudiaf, Oran 31000, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8769-2446","authenticated-orcid":false,"given":"Yannick","family":"Deville","sequence":"additional","affiliation":[{"name":"Institut de Recherche en Astrophysique et Plan\u00e9tologie (IRAP), Universit\u00e9 de Toulouse, UPS, CNRS, OMP, CNES, 31400 Toulouse, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6223-4863","authenticated-orcid":false,"given":"Moussa Sofiane","family":"Karoui","sequence":"additional","affiliation":[{"name":"Agence Spatiale Alg\u00e9rienne, Centre des Techniques Spatiales, Arzew 31200, Algeria"},{"name":"Institut de Recherche en Astrophysique et Plan\u00e9tologie (IRAP), Universit\u00e9 de Toulouse, UPS, CNRS, OMP, CNES, 31400 Toulouse, France"},{"name":"Universit\u00e9 des Sciences et de la Technologie d\u2019Oran Mohamed Boudiaf, Oran 31000, Algeria"}]},{"given":"Abdelaziz","family":"Ouamri","sequence":"additional","affiliation":[{"name":"Universit\u00e9 des Sciences et de la Technologie d\u2019Oran Mohamed Boudiaf, Oran 31000, Algeria"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Weber, C., Aguejdad, R., Briottet, X., Avala, J., Fabre, S., Demuynck, J., Zenou, E., Deville, Y., Karoui, M.S., and Benhalouche, F.Z. (2018, January 22\u201327). Hyperspectral imagery for environmental urban planning. Proceedings of the IGARSS 2018\u20142018 IEEE International Geoscience and Remote Sensing Symposium, Valencia, Spain.","DOI":"10.1109\/IGARSS.2018.8519085"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Brabant, C., Alvarez-Vanhard, E., Laribi, A., Morin, G., Thanh Nguyen, K., Thomas, A., and Houet, T. (2019). Comparison of hyperspectral techniques for urban tree diversity classification. Remote Sens., 11.","DOI":"10.3390\/rs11111269"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Karoui, M.S., Benhalouche, F.Z., Deville, Y., Djerriri, K., Briottet, X., Houet, T., Le Bris, A., and Weber, C. (2019). Partial linear nmf-based unmixing methods for detection and area estimation of photovoltaic panels in urban hyperspectral remote sensing data. Remote Sens., 11.","DOI":"10.3390\/rs11182164"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Chang, C.-I. (2013). Hyperspectral Data Processing: Algorithm Design and Analysis, John Wiley&Sons.","DOI":"10.1002\/9781118269787"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1109\/79.974727","article-title":"Spectral unmixing","volume":"19","author":"Keshava","year":"2002","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1109\/36.992799","article-title":"Linear spectral random mixture analysis for hyperspectral imagery","volume":"40","author":"Chang","year":"2002","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1109\/JSTARS.2012.2194696","article-title":"Hyperspectral unmixing overview: Geometrical, statistical, and sparse regression-based approaches","volume":"5","author":"Plaza","year":"2012","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_8","unstructured":"Comon, P., and Jutten, C. (2010). Handbook of Blind Source Separation: Independent Component Analysis and Applications, Academic Press. [1st ed.]."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Deville, Y. (2016). Blind source separation and blind mixture identification methods. Wiley Encyclopedia of Electrical and Electronics Engineering, John Wiley & Sons, Inc.","DOI":"10.1002\/047134608X.W8300"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2601","DOI":"10.1109\/TGRS.2006.874135","article-title":"Applications of independent component analysis in endmember extraction and abundance quantification for hyperspectral imagery","volume":"44","author":"Wang","year":"2006","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2194","DOI":"10.1016\/j.neucom.2007.07.034","article-title":"On the decomposition of mars hyperspectral data by ica and bayesian positive source separation","volume":"71","author":"Moussaoui","year":"2008","journal-title":"Neurocomputing"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2165","DOI":"10.1109\/TGRS.2010.2101609","article-title":"Independent component analysis for blind unmixing of hyperspectral imagery with additional constraints","volume":"49","author":"Xia","year":"2011","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1109\/LSP.2002.800502","article-title":"Conditions for nonnegative independent component analysis","volume":"9","author":"Plumbley","year":"2002","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1109\/TNN.2003.810616","article-title":"Algorithms for nonnegative independent component analysis","volume":"14","author":"Plumbley","year":"2003","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/978-3-540-30110-3_7","article-title":"Optimization using fourier expansion over a geodesic for non-negative ICA","volume":"Volume 3195","author":"Puntonet","year":"2004","journal-title":"Independent Component Analysis and Blind Signal Separation"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"522","DOI":"10.1007\/978-3-642-15995-4_65","article-title":"Non-negative independent component analysis algorithm based on 2D givens rotations and a newton optimization","volume":"Volume 6365","author":"Ouedraogo","year":"2010","journal-title":"Latent Variable Analysis and Signal Separation"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2014","DOI":"10.1109\/TGRS.2010.2098413","article-title":"Sparse unmixing of hyperspectral data","volume":"49","author":"Iordache","year":"2011","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4263","DOI":"10.1016\/j.patcog.2012.05.008","article-title":"Blind spatial unmixing of multispectral images: New methods combining sparse component analysis, clustering and non-negativity constraints","volume":"45","author":"Karoui","year":"2012","journal-title":"Pattern Recognit."},{"key":"ref_19","unstructured":"Naik, G.R. (2013). Blind unmixing of hyperspectral data with some pure pixels: Spatial variance-based methods exploiting sparsity and non-negativity properties. Signal Processing: New Research, Nova Science Publishers."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3265","DOI":"10.1109\/TGRS.2018.2797200","article-title":"Spectral\u2013spatial weighted sparse regression for hyperspectral image unmixing","volume":"56","author":"Zhang","year":"2018","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1038\/44565","article-title":"Learning the parts of objects by non-negative matrix factorization","volume":"401","author":"Lee","year":"1999","journal-title":"Nature"},{"key":"ref_22","first-page":"556","article-title":"Algorithms for non-negative matrix factorization","volume":"13","author":"Lee","year":"2001","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Cichocki, A., Zdunek, R., Phan, A.H., and Amari, S. (2009). Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-Way Data Analysis and Blind Source Separation, John Wiley and Sons.","DOI":"10.1002\/9780470747278"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2815","DOI":"10.1109\/TGRS.2012.2213825","article-title":"Manifold regularized sparse NMF for hyperspectral unmixing","volume":"51","author":"Lu","year":"2013","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1109\/TCI.2017.2693967","article-title":"Distributed blind hyperspectral unmixing via joint sparsity and low-rank constrained non-negative matrix factorization","volume":"3","author":"Tsinos","year":"2017","journal-title":"IEEE Trans. Comput. Imaging"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1232","DOI":"10.1109\/TAES.2003.1261124","article-title":"Automatic spectral target recognition in hyperspectral imagery","volume":"39","author":"Ren","year":"2003","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1109\/LGRS.2005.856701","article-title":"A fast iterative algorithm for implementation of pixel purity index","volume":"3","author":"Chang","year":"2006","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_28","unstructured":"Descour, M.R., and Shen, S.S. (1999, January 18\u201323). N-FINDR: An algorithm for fast autonomous spectral end-member determination in hyperspectral data. Proceedings of the SPIE\u2019s International Symposium on Optical Science, Engineering, and Instrumentation, Denver, CO, USA."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"898","DOI":"10.1109\/TGRS.2005.844293","article-title":"Vertex component analysis: A fast algorithm to unmix hyperspectral data","volume":"43","author":"Nascimento","year":"2005","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2804","DOI":"10.1109\/TGRS.2006.881803","article-title":"A New growing method for simplex-based endmember extraction algorithm","volume":"44","author":"Chang","year":"2006","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1834","DOI":"10.1109\/TGRS.2009.2034979","article-title":"Real-time simplex growing algorithms for hyperspectral endmember extraction","volume":"48","author":"Chang","year":"2010","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"4280","DOI":"10.1109\/JSTARS.2016.2555960","article-title":"Comparative study and analysis among ATGP, VCA, and SGA for finding endmembers in hyperspectral imagery","volume":"9","author":"Chang","year":"2016","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1109\/TGRS.2006.888466","article-title":"Endmember extraction from highly mixed data using minimum volume constrained nonnegative matrix factorization","volume":"45","author":"Miao","year":"2007","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Li, J., and Bioucas-Dias, J.M. (2008, January 8\u201311). Minimum volume simplex analysis: A fast algorithm to unmix hyperspectral data. Proceedings of the IGARSS 2008\u20142008 IEEE International Geoscience and Remote Sensing Symposium, Boston, MA, USA.","DOI":"10.1109\/IGARSS.2008.4779330"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Bioucas-Dias, J.M. (2009, January 26\u201328). A variable splitting augmented lagrangian approach to linear spectral unmixing. Proceedings of the 2009 First Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing, Grenoble, France.","DOI":"10.1109\/WHISPERS.2009.5289072"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1109\/MSP.2013.2279274","article-title":"Nonlinear unmixing of hyperspectral images: Models and algorithms","volume":"31","author":"Dobigeon","year":"2014","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1844","DOI":"10.1109\/JSTARS.2014.2320576","article-title":"A review of nonlinear hyperspectral unmixing methods","volume":"7","author":"Heylen","year":"2014","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1109\/TGRS.2015.2453915","article-title":"A multilinear mixing model for nonlinear spectral unmixing","volume":"54","author":"Heylen","year":"2016","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"544","DOI":"10.1109\/TGRS.2013.2242475","article-title":"Linear\u2013quadratic mixing model for reflectances in urban environments","volume":"52","author":"Meganem","year":"2014","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_40","unstructured":"Nascimento, J.M.P., and Bioucas-Dias, J.M. (September, January 31). Nonlinear mixture model for hyperspectral unmixing. Proceedings of the SPIE 7477, Image and Signal Processing for Remote Sensing, Berlin, Germany."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2951","DOI":"10.1080\/01431160802558659","article-title":"Comparative study between a new nonlinear model and common linear model for analysing laboratory simulated-forest hyperspectral data","volume":"30","author":"Fan","year":"2009","journal-title":"Int. J. Remote Sens."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"4153","DOI":"10.1109\/TGRS.2010.2098414","article-title":"Nonlinear unmixing of hyperspectral images using a generalized bilinear model","volume":"49","author":"Halimi","year":"2011","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"3017","DOI":"10.1109\/TIP.2012.2187668","article-title":"Supervised nonlinear spectral unmixing using a postnonlinear mixing model for hyperspectral imagery","volume":"21","author":"Altmann","year":"2012","journal-title":"IEEE Trans. Image Process."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2442","DOI":"10.1109\/TSP.2013.2245127","article-title":"Nonlinear spectral unmixing of hyperspectral images using gaussian processes","volume":"61","author":"Altmann","year":"2013","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2663","DOI":"10.1109\/TIP.2014.2314022","article-title":"Unsupervised post-nonlinear unmixing of hyperspectral images using a hamiltonian monte carlo algorithm","volume":"23","author":"Altmann","year":"2014","journal-title":"IEEE Trans. Image Process."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"778","DOI":"10.1109\/LGRS.2013.2278993","article-title":"A bilinear\u2013bilinear nonnegative matrix factorization method for hyperspectral unmixing","volume":"11","author":"Eches","year":"2014","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1822","DOI":"10.1109\/TSP.2014.2306181","article-title":"Linear-quadratic blind source separation using NMF to unmix urban hyperspectral images","volume":"62","author":"Meganem","year":"2014","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"4534","DOI":"10.1109\/TGRS.2017.2693366","article-title":"Unsupervised nonlinear spectral unmixing based on a multilinear mixing model","volume":"55","author":"Wei","year":"2017","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1287","DOI":"10.1109\/TGRS.2014.2336858","article-title":"Constrained least squares algorithms for nonlinear unmixing of hyperspectral imagery","volume":"53","author":"Pu","year":"2015","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Broadwater, J., Chellappa, R., Banerjee, A., and Burlina, P. (2007, January 23\u201327). Kernel fully constrained least squares abundance estimates. Proceedings of the IGARSS 2007\u20142007 IEEE International Geoscience and Remote Sensing Symposium, Barcelona, Spain.","DOI":"10.1109\/IGARSS.2007.4423736"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1109\/TSP.2012.2222390","article-title":"Nonlinear unmixing of hyperspectral data based on a linear-mixture\/nonlinear-fluctuation model","volume":"61","author":"Chen","year":"2013","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1555","DOI":"10.1007\/s11760-012-0392-3","article-title":"Blind nonlinear hyperspectral unmixing based on constrained kernel nonnegative matrix factorization","volume":"8","author":"Li","year":"2014","journal-title":"Signal Image Video Process."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1109\/TIP.2016.2627815","article-title":"Nonlinear unmixing of hyperspectral data with vector-valued kernel functions","volume":"26","author":"Ammanouil","year":"2017","journal-title":"IEEE Trans. Image Process."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Li, Z., Chen, J., and Rahardja, S. (2019). Kernel-based nonlinear spectral unmixing with dictionary pruning. Remote Sens., 11.","DOI":"10.3390\/rs11050529"},{"key":"ref_55","unstructured":"Plaza, J., Plaza, A., Perez, R., and Martinez, P. (2005, January 25\u201329). Automated generation of semi-labeled training samples for nonlinear neural network-based abundance estimation in hyperspectral data. Proceedings of the IGARSS 2005\u20142005 IEEE International Geoscience and Remote Sensing Symposium, Seoul, Korea."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"4163","DOI":"10.1109\/TGRS.2011.2160950","article-title":"Pixel unmixing in hyperspectral data by means of neural networks","volume":"49","author":"Licciardi","year":"2011","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"2644","DOI":"10.1109\/JSTARS.2015.2427517","article-title":"Nonlinear hyperspectral unmixing using nonlinearity order estimation and polytope decomposition","volume":"8","author":"Marinoni","year":"2015","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1002","DOI":"10.1109\/LGRS.2016.2560222","article-title":"Hopfield neural network approach for supervised nonlinear spectral unmixing","volume":"13","author":"Li","year":"2016","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1755","DOI":"10.1109\/LGRS.2018.2857804","article-title":"Hyperspectral unmixing via deep convolutional neural networks","volume":"15","author":"Zhang","year":"2018","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Benhalouche, F.Z., Deville, Y., Karoui, M.S., and Ouamri, A. (2016, January 10\u201315). Bilinear matrix factorization using a gradient method for hyperspectral endmember spectra extraction. Proceedings of the IGARSS 2016\u20142016 IEEE International Geoscience and Remote Sensing Symposium, Beijing, China.","DOI":"10.1109\/IGARSS.2016.7730715"},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Benhalouche, F.Z., Deville, Y., Karoui, M.S., and Ouamri, A. (2016, January 13\u201316). Hyperspectral endmember spectra extraction based on constrained linear-quadratic matrix factorization using a projected gradient method. Proceedings of the 2016 IEEE 26th International Workshop on Machine Learning for Signal Processing (MLSP), Vietri sul Mare, Salerno, Italy.","DOI":"10.1109\/MLSP.2016.7738868"},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Deville, Y. (September, January 31). Matrix factorization for bilinear blind source separation: Methods, separability and conditioning. Proceedings of the 2015 23rd European Signal Processing Conference (EUSIPCO), Nice, France.","DOI":"10.1109\/EUSIPCO.2015.7362714"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.dsp.2019.01.011","article-title":"From separability\/identifiability properties of bilinear and linear-quadratic mixture matrix factorization to factorization algorithms","volume":"87","author":"Deville","year":"2019","journal-title":"Digit. Signal Process."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1109\/36.911111","article-title":"Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery","volume":"39","author":"Heinz","year":"2001","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"5776","DOI":"10.1109\/JSTARS.2016.2602882","article-title":"A New algorithm for bilinear spectral unmixing of hyperspectral images using particle swarm optimization","volume":"9","author":"Luo","year":"2016","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Clark, R.N., Swayze, G.A., Wise, R.A., Livo, K.E., Hoefen, T.M., Kokaly, R.F., and Sutley, S.J. (2007). USGS Digital Spectral Library Splib06a.","DOI":"10.3133\/ds231"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1016\/j.rse.2008.11.007","article-title":"The ASTER spectral library version 2.0","volume":"113","author":"Baldridge","year":"2009","journal-title":"Remote Sens. Environ."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"5412","DOI":"10.1109\/TIP.2014.2363423","article-title":"Spectral unmixing via data-guided sparsity","volume":"23","author":"Zhu","year":"2014","journal-title":"IEEE Trans. Image Process."},{"key":"ref_69","unstructured":"Chang, C.-I. (July, January 28). Spectral information divergence for hyperspectral image analysis. Proceedings of the IGARSS 1999\u20141999 IEEE International Geoscience and Remote Sensing Symposium, Hamburg, Germany."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/13\/11\/2132\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:09:58Z","timestamp":1760162998000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/13\/11\/2132"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,28]]},"references-count":69,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["rs13112132"],"URL":"https:\/\/doi.org\/10.3390\/rs13112132","relation":{},"ISSN":["2072-4292"],"issn-type":[{"type":"electronic","value":"2072-4292"}],"subject":[],"published":{"date-parts":[[2021,5,28]]}}}