{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T05:29:21Z","timestamp":1761629361710,"version":"build-2065373602"},"reference-count":72,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2018,5,21]],"date-time":"2018-05-21T00:00:00Z","timestamp":1526860800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61572133"],"award-info":[{"award-number":["61572133"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Research Fund for the State Key Laboratory of Earth Surface Processes and Resource Ecology","award":["2017-KF-19"],"award-info":[{"award-number":["2017-KF-19"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>Bilinear mixture model-based methods have recently shown promising capability in nonlinear spectral unmixing. However, relying on the endmembers extracted in advance, their unmixing accuracies decrease, especially when the data is highly mixed. In this paper, a strategy of geometric projection has been provided and combined with constrained nonnegative matrix factorization for unsupervised nonlinear spectral unmixing. According to the characteristics of bilinear mixture models, a set of facets are determined, each of which represents the partial nonlinearity neglecting one endmember. Then, pixels\u2019 barycentric coordinates with respect to every endmember are calculated in several newly constructed simplices using a distance measure. In this way, pixels can be projected into their approximate linear mixture components, which reduces greatly the impact of collinearity. Different from relevant nonlinear unmixing methods in the literature, this procedure effectively facilitates a more accurate estimation of endmembers and abundances in constrained nonnegative matrix factorization. The updated endmembers are further used to reconstruct the facets and get pixels\u2019 new projections. Finally, endmembers, abundances, and pixels\u2019 projections are updated alternately until a satisfactory result is obtained. The superior performance of the proposed algorithm in nonlinear spectral unmixing has been validated through experiments with both synthetic and real hyperspectral data, where traditional and state-of-the-art algorithms are compared.<\/jats:p>","DOI":"10.3390\/rs10050801","type":"journal-article","created":{"date-parts":[[2018,5,22]],"date-time":"2018-05-22T04:34:03Z","timestamp":1526963643000},"page":"801","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Unsupervised Nonlinear Hyperspectral Unmixing Based on Bilinear Mixture Models via Geometric Projection and Constrained Nonnegative Matrix Factorization"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9762-0788","authenticated-orcid":false,"given":"Bin","family":"Yang","sequence":"first","affiliation":[{"name":"Key Laboratory for Information Science of Electromagnetic Waves (MoE), Fudan University, Shanghai 200433, China"},{"name":"State Key Laboratory of Earth Surface Processes and Resource Ecology, Beijing Normal University, Beijing 100875, China"},{"name":"Research Center of Smart Networks and Systems, School of Information Science and Technology, Fudan University, Shanghai 200433, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4748-6426","authenticated-orcid":false,"given":"Bin","family":"Wang","sequence":"additional","affiliation":[{"name":"Key Laboratory for Information Science of Electromagnetic Waves (MoE), Fudan University, Shanghai 200433, China"},{"name":"State Key Laboratory of Earth Surface Processes and Resource Ecology, Beijing Normal University, Beijing 100875, China"},{"name":"Research Center of Smart Networks and Systems, School of Information Science and Technology, Fudan University, Shanghai 200433, China"}]},{"given":"Zongmin","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Fudan University, Shanghai 200433, China"}]}],"member":"1968","published-online":{"date-parts":[[2018,5,21]]},"reference":[{"key":"ref_1","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. 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