{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T14:38:29Z","timestamp":1773931109758,"version":"3.50.1"},"reference-count":52,"publisher":"MDPI AG","issue":"20","license":[{"start":{"date-parts":[[2022,10,15]],"date-time":"2022-10-15T00:00:00Z","timestamp":1665792000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["61877066"],"award-info":[{"award-number":["61877066"]}]},{"name":"National Natural Science Foundation of China","award":["20175181013"],"award-info":[{"award-number":["20175181013"]}]},{"name":"National Natural Science Foundation of China","award":["21RGZN0010"],"award-info":[{"award-number":["21RGZN0010"]}]},{"name":"Aero-Science Fund","award":["61877066"],"award-info":[{"award-number":["61877066"]}]},{"name":"Aero-Science Fund","award":["20175181013"],"award-info":[{"award-number":["20175181013"]}]},{"name":"Aero-Science Fund","award":["21RGZN0010"],"award-info":[{"award-number":["21RGZN0010"]}]},{"name":"Science and technology plan project of Xi\u2019an","award":["61877066"],"award-info":[{"award-number":["61877066"]}]},{"name":"Science and technology plan project of Xi\u2019an","award":["20175181013"],"award-info":[{"award-number":["20175181013"]}]},{"name":"Science and technology plan project of Xi\u2019an","award":["21RGZN0010"],"award-info":[{"award-number":["21RGZN0010"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>Hyperspectral unmixing decomposes the observed mixed spectra into a collection of constituent pure material signatures and the associated fractional abundances. Because of the universal modeling ability of neural networks, deep learning (DL) techniques are gaining prominence in solving hyperspectral analysis tasks. The autoencoder (AE) network has been extensively investigated in linear blind source unmixing. However, the linear mixing model (LMM) may fail to provide good unmixing performance when the nonlinear mixing effects are nonnegligible in complex scenarios. Considering the limitations of LMM, we propose an unsupervised nonlinear spectral unmixing method, based on autoencoder architecture. Firstly, a deep neural network is employed as the encoder to extract the low-dimension feature of the mixed pixel. Then, the generalized bilinear model (GBM) is used to design the decoder, which has a linear mixing part and a nonlinear mixing one. The coefficient of the bilinear mixing part can be adjusted by a set of learnable parameters, which makes the method perform well on both nonlinear and linear data. Finally, some regular terms are imposed on the loss function and an alternating update strategy is utilized to train the network. Experimental results on synthetic and real datasets verify the effectiveness of the proposed model and show very competitive performance compared with several existing algorithms.<\/jats:p>","DOI":"10.3390\/rs14205167","type":"journal-article","created":{"date-parts":[[2022,10,17]],"date-time":"2022-10-17T03:43:58Z","timestamp":1665978238000},"page":"5167","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Nonlinear Unmixing via Deep Autoencoder Networks for Generalized Bilinear Model"],"prefix":"10.3390","volume":"14","author":[{"given":"Jinhua","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Artificial Intelligence, Xidian University, Xi\u2019an 710071, China"}]},{"given":"Xiaohua","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Xidian University, Xi\u2019an 710071, China"}]},{"given":"Hongyun","family":"Meng","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Xidian University, Xi\u2019an 710071, China"}]},{"given":"Caihao","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Xidian University, Xi\u2019an 710071, China"}]},{"given":"Li","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Xidian University, Xi\u2019an 710071, China"}]},{"given":"Xianghai","family":"Cao","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Xidian University, Xi\u2019an 710071, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1109\/MGRS.2013.2244672","article-title":"Hyperspectral Remote Sensing Data Analysis and Future Challenges","volume":"1","author":"Plaza","year":"2013","journal-title":"IEEE Geosci. Remote Sens. Mag."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Bioucas-Dias, J.M., and Plaza, A. (2011, January 24\u201329). An overview on hyperspectral unmixing: Geometrical, statistical, and sparse regression based approaches. Proceedings of the 2011 IEEE International Geoscience and Remote Sensing Symposium, Vancouver, BC, Canada.","DOI":"10.1109\/IGARSS.2011.6049397"},{"key":"ref_3","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_4","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_5","doi-asserted-by":"crossref","unstructured":"D\u00f3pido, I., Gamba, P., and Plaza, A. (2013, January 26\u201328). Spectral unmixing-based post-processing for hyperspectral image classification. Proceedings of the 2013 5th Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), Gainesville, FL, USA.","DOI":"10.1109\/WHISPERS.2013.8080675"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Andrejchenko, V., Heylen, R., Scheunders, P., Philips, W., and Liao, W. (2016, January 10\u201315). Classification of hyperspectral images with very small training size using sparse unmixing. Proceedings of the 2016 IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Beijing, China.","DOI":"10.1109\/IGARSS.2016.7730333"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Ibarrola-Ulzurrun, E., Drumetz, L., Chanussot, J., Marcello, J., and Gonzalo-Martin, C. (2018, January 23\u201326). Classification Using Unmixing Models in Areas With Substantial Endmember Variability. Proceedings of the 2018 9th Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), Amsterdam, The Netherlands.","DOI":"10.1109\/WHISPERS.2018.8747096"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TGRS.2022.3215431","article-title":"Hyperspectral Imagery Classification Based on Contrastive Learning","volume":"60","author":"Hou","year":"2022","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Yokoya, N., and Iwasaki, A. (2014, January 24\u201327). Effect of unmixing-based hyperspectral super-resolution on target detection. Proceedings of the 2014 6th Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), Lausanne, Switzerland.","DOI":"10.1109\/WHISPERS.2014.8077602"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ziemann, A.K. (2016, January 6\u20138). Local spectral unmixing for target detection. Proceedings of the 2016 IEEE Southwest Symposium on Image Analysis and Interpretation (SSIAI), Santa Fe, NM, USA.","DOI":"10.1109\/SSIAI.2016.7459179"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Glenn, T., Dranishnikov, D., Gader, P., and Zare, A. (2013, January 21\u201326). Subpixel target detection in hyperspectral imagery using piece-wise convex spatial-spectral unmixing, possibilistic and fuzzy clustering, and co-registered LiDAR. Proceedings of the 2013 IEEE International Geoscience and Remote Sensing Symposium-IGARSS, Melbourne, Australia.","DOI":"10.1109\/IGARSS.2013.6721347"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"6469","DOI":"10.1109\/TGRS.2015.2441954","article-title":"Robust Feature Matching for Remote Sensing Image Registration via Locally Linear Transforming","volume":"53","author":"Ma","year":"2015","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Cui, S., Zhong, Y., Ma, A., and Zhang, L. (August, January 28). A novel robust feature descriptor for multi-source remote sensing image registration. Proceedings of the IGARSS 2019\u20142019 IEEE International Geoscience and Remote Sensing Symposium, Yokohama, Japan.","DOI":"10.1109\/IGARSS.2019.8900521"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2881","DOI":"10.1109\/JSTARS.2018.2844295","article-title":"High-Resolution Remote-Sensing Image Registration Based on Angle Matching of Edge Point Features","volume":"11","author":"Guo","year":"2018","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_15","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_16","doi-asserted-by":"crossref","unstructured":"Veganzones, M.A., Drumetz, L., Tochon, G., Dalla Mura, M., Plaza, A., Bioucas-Dias, J., and Chanussot, J. (2014, January 24\u201327). A new extended linear mixing model to address spectral variability. Proceedings of the 2014 6th Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), Lausanne, Switzerland.","DOI":"10.1109\/WHISPERS.2014.8077595"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3890","DOI":"10.1109\/TIP.2016.2579259","article-title":"Blind Hyperspectral Unmixing Using an Extended Linear Mixing Model to Address Spectral Variability","volume":"25","author":"Drumetz","year":"2016","journal-title":"IEEE Trans. Image Process."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Xu, M., Zhang, L., Du, B., and Zhang, L. (2016, January 21\u201324). The linear mixed model constrained particle swarm optimization for hyperspectral endmember extraction from highly mixed data. Proceedings of the 2016 8th Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), Los Angeles, CA, USA.","DOI":"10.1109\/WHISPERS.2016.8071763"},{"key":"ref_19","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_20","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_21","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1109\/TIP.2010.2076294","article-title":"An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems","volume":"20","author":"Afonso","year":"2011","journal-title":"IEEE Trans. Image Process."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Bioucas-Dias, J.M., and Figueiredo, M.A. (2010, January 14\u201316). Alternating direction algorithms for constrained sparse regression: Application to hyperspectral unmixing. Proceedings of the 2010 2nd Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing, Reykjavik, Iceland.","DOI":"10.1109\/WHISPERS.2010.5594963"},{"key":"ref_23","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_24","doi-asserted-by":"crossref","first-page":"5567","DOI":"10.1109\/TGRS.2013.2290372","article-title":"Semisupervised Kernel Feature Extraction for Remote Sensing Image Analysis","volume":"52","author":"Bruzzone","year":"2014","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Mateo-Garc\u00eda, G., Laparra, V., and G\u00f3mez-Chova, L. (2018, January 22\u201327). Optimizing Kernel Ridge Regression for Remote Sensing Problems. Proceedings of the IGARSS 2018\u20142018 IEEE International Geoscience and Remote Sensing Symposium, Valencia, Spain.","DOI":"10.1109\/IGARSS.2018.8518016"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3039","DOI":"10.1029\/JB086iB04p03039","article-title":"Bidirectional reflectance spectroscopy: 1. Theory","volume":"86","author":"Hapke","year":"1981","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3055","DOI":"10.1029\/JB086iB04p03055","article-title":"Bidirectional reflectance spectroscopy: 2. Experiments and observations","volume":"86","author":"Hapke","year":"1981","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Chen, W., Cao, C., Zhang, H., Jia, H., Ji, W., Xu, M., Gao, M., Ni, X., Zhao, J., and Zheng, S. (2011, January 24\u201329). Estimation of shrub canopy cover based on a geometric-optical model using HJ-1 data. Proceedings of the 2011 IEEE International Geoscience and Remote Sensing Symposium, Vancouver, BC, Canada.","DOI":"10.1109\/IGARSS.2011.6049501"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Altmann, Y., Dobigeon, N., and Tourneret, J.-Y. (2011, January 6\u20139). Bilinear models for nonlinear unmixing of hyperspectral images. Proceedings of the 2011 3rd Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), Lisbon, Portugal.","DOI":"10.1109\/WHISPERS.2011.6080928"},{"key":"ref_30","unstructured":"Nascimento, J.M., and Bioucas-Dias, J.M. (September, January 31). Nonlinear mixture model for hyperspectral unmixing. Proceedings of the Image and Signal Processing for Remote Sensing XV, Berlin, Germany."},{"key":"ref_31","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_32","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_33","doi-asserted-by":"crossref","unstructured":"Halimi, A., Altmann, Y., Dobigeon, N., and Tourneret, J.-Y. (2011, January 24\u201329). Unmixing hyperspectral images using the generalized bilinear model. Proceedings of the 2011 IEEE International Geoscience and Remote Sensing Symposium, Vancouver, BC, Canada.","DOI":"10.1109\/IGARSS.2011.6049492"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1430","DOI":"10.1109\/TGRS.2013.2251349","article-title":"Nonlinear Unmixing of Hyperspectral Data Using Semi-Nonnegative Matrix Factorization","volume":"52","author":"Yokoya","year":"2014","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Plaza, J., Plaza, A.J., Martinez, P., and Perez, R.M. (2003, January 9\u201312). Nonlinear mixture models for analyzing laboratory simulated-forest hyperspectral data. Proceedings of the Image and Signal Processing for Remote Sensing IX, Barcelona, Spain.","DOI":"10.1117\/12.511127"},{"key":"ref_36","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_37","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_38","doi-asserted-by":"crossref","first-page":"4309","DOI":"10.1109\/TGRS.2018.2890633","article-title":"DAEN: Deep Autoencoder Networks for Hyperspectral Unmixing","volume":"57","author":"Su","year":"2019","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Palsson, B., Ulfarsson, M.O., and Sveinsson, J.R. (August, January 28). Convolutional autoencoder for spatial-spectral hyperspectral unmixing. Proceedings of the IGARSS 2019\u20142019 IEEE International Geoscience and Remote Sensing Symposium, Yokohama, Japan.","DOI":"10.1109\/IGARSS.2019.8900297"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"5505412","DOI":"10.1109\/TGRS.2021.3069476","article-title":"JMnet: Joint Metric Neural Network for Hyperspectral Unmixing","volume":"60","author":"Min","year":"2022","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1467","DOI":"10.1109\/LGRS.2019.2900733","article-title":"Nonlinear Unmixing of Hyperspectral Data via Deep Autoencoder Networks","volume":"16","author":"Wang","year":"2019","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"8615","DOI":"10.1109\/TGRS.2020.3041157","article-title":"Deep Autoencoders With Multitask Learning for Bilinear Hyperspectral Unmixing","volume":"59","author":"Su","year":"2021","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_43","first-page":"5506105","article-title":"Model-Based Deep Autoencoder Networks for Nonlinear Hyperspectral Unmixing","volume":"19","author":"Li","year":"2022","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_44","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_45","unstructured":"Ioffe, S., and Szegedy, C. (2015, January 6\u201311). Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift. Proceedings of the 32nd International Conference on Machine Learning, Lille, France."},{"key":"ref_46","unstructured":"Keskar, N.S., Mudigere, D., Nocedal, J., Smelyanskiy, M., and Tang, P.T.P. (2016). On large-batch training for deep learning: Generalization gap and sharp minima. arXiv."},{"key":"ref_47","unstructured":"Masters, D., and Luschi, C. (2018). Revisiting small batch training for deep neural networks. arXiv."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"25646","DOI":"10.1109\/ACCESS.2018.2818280","article-title":"Hyperspectral Unmixing Using a Neural Network Autoencoder","volume":"6","author":"Palsson","year":"2018","journal-title":"IEEE Access"},{"key":"ref_49","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_50","doi-asserted-by":"crossref","first-page":"2783","DOI":"10.1016\/j.neucom.2017.11.052","article-title":"Robust GBM hyperspectral image unmixing with superpixel segmentation based low rank and sparse representation","volume":"275","author":"Mei","year":"2018","journal-title":"Neurocomputing"},{"key":"ref_51","unstructured":"Zhu, F. (2017). Hyperspectral unmixing: Ground truth labeling, datasets, benchmark performances and survey. arXiv."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"6633","DOI":"10.1109\/TGRS.2019.2907567","article-title":"Blind Hyperspectral Unmixing Considering the Adjacency Effect","volume":"57","author":"Wang","year":"2019","journal-title":"IEEE Trans. Geosci. Remote Sens."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/14\/20\/5167\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:55:00Z","timestamp":1760144100000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/14\/20\/5167"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,15]]},"references-count":52,"journal-issue":{"issue":"20","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["rs14205167"],"URL":"https:\/\/doi.org\/10.3390\/rs14205167","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,15]]}}}