{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T14:28:47Z","timestamp":1768832927504,"version":"3.49.0"},"reference-count":46,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,2,15]],"date-time":"2021-02-15T00:00:00Z","timestamp":1613347200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Spanish Ministry","award":["FPU15\/02090"],"award-info":[{"award-number":["FPU15\/02090"]}]},{"DOI":"10.13039\/501100014181","name":"Junta de Extremadura","doi-asserted-by":"publisher","award":["GR18060"],"award-info":[{"award-number":["GR18060"]}],"id":[{"id":"10.13039\/501100014181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000780","name":"European Union","doi-asserted-by":"publisher","award":["734541-EXPOSURE"],"award-info":[{"award-number":["734541-EXPOSURE"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>Endmember estimation plays a key role in hyperspectral image unmixing, often requiring an estimation of the number of endmembers and extracting endmembers. However, most of the existing extraction algorithms require prior knowledge regarding the number of endmembers, being a critical process during unmixing. To bridge this, a new maximum distance analysis (MDA) method is proposed that simultaneously estimates the number and spectral signatures of endmembers without any prior information on the experimental data containing pure pixel spectral signatures and no noise, being based on the assumption that endmembers form a simplex with the greatest volume over all pixel combinations. The simplex includes the farthest pixel point from the coordinate origin in the spectral space, which implies that: (1) the farthest pixel point from any other pixel point must be an endmember, (2) the farthest pixel point from any line must be an endmember, and (3) the farthest pixel point from any plane (or affine hull) must be an endmember. Under this scenario, the farthest pixel point from the coordinate origin is the first endmember, being used to create the aforementioned point, line, plane, and affine hull. The remaining endmembers are extracted by repetitively searching for the pixel points that satisfy the above three assumptions. In addition to behaving as an endmember estimation algorithm by itself, the MDA method can co-operate with existing endmember extraction techniques without the pure pixel assumption via generalizing them into more effective schemes. The conducted experiments validate the effectiveness and efficiency of our method on synthetic and real data.<\/jats:p>","DOI":"10.3390\/rs13040713","type":"journal-article","created":{"date-parts":[[2021,2,15]],"date-time":"2021-02-15T22:58:01Z","timestamp":1613429881000},"page":"713","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["Endmember Estimation with Maximum Distance Analysis"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1093-0079","authenticated-orcid":false,"given":"Xuanwen","family":"Tao","sequence":"first","affiliation":[{"name":"Hyperspectral Computing Laboratory (HyperComp), Department of Computer Technology and Communications, Escuela Politecnica de Caceres, University of Extremadura, Avenida de la Universidad sn, E-10002 Caceres, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1030-3729","authenticated-orcid":false,"given":"Mercedes E.","family":"Paoletti","sequence":"additional","affiliation":[{"name":"Hyperspectral Computing Laboratory (HyperComp), Department of Computer Technology and Communications, Escuela Politecnica de Caceres, University of Extremadura, Avenida de la Universidad sn, E-10002 Caceres, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6701-961X","authenticated-orcid":false,"given":"Juan M.","family":"Haut","sequence":"additional","affiliation":[{"name":"Department of Communication and Control Systems, Higher Technical School of Computer Engineering, National Distance Education University (UNED), E-28040 Madrid, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3949-985X","authenticated-orcid":false,"given":"Peng","family":"Ren","sequence":"additional","affiliation":[{"name":"College of Oceanography and Space Informatics, China University of Petroleum (East China), Qingdao 266580, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2384-9141","authenticated-orcid":false,"given":"Javier","family":"Plaza","sequence":"additional","affiliation":[{"name":"Hyperspectral Computing Laboratory (HyperComp), Department of Computer Technology and Communications, Escuela Politecnica de Caceres, University of Extremadura, Avenida de la Universidad sn, E-10002 Caceres, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9613-1659","authenticated-orcid":false,"given":"Antonio","family":"Plaza","sequence":"additional","affiliation":[{"name":"Hyperspectral Computing Laboratory (HyperComp), Department of Computer Technology and Communications, Escuela Politecnica de Caceres, University of Extremadura, Avenida de la Universidad sn, E-10002 Caceres, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2406","DOI":"10.1109\/TCYB.2018.2810806","article-title":"Feature learning using spatial-spectral hypergraph discriminant analysis for hyperspectral image","volume":"49","author":"Luo","year":"2018","journal-title":"IEEE Trans. Cybern."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/0034-4257(93)90012-M","article-title":"The airborne visible\/infrared imaging spectrometer (AVIRIS)","volume":"44","author":"Vane","year":"1993","journal-title":"Remote Sens. Environ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1147","DOI":"10.1126\/science.228.4704.1147","article-title":"Imaging spectrometry for earth remote sensing","volume":"228","author":"Goetz","year":"1985","journal-title":"Science"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.isprsjprs.2019.09.006","article-title":"Deep learning classifiers for hyperspectral imaging: A review","volume":"158","author":"Paoletti","year":"2019","journal-title":"ISPRS J. Photogramm. Remote Sens."},{"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":"220","DOI":"10.1016\/j.neucom.2017.06.004","article-title":"A novel target detection method for SAR images based on shadow proposal and saliency analysis","volume":"267","author":"Gao","year":"2017","journal-title":"Neurocomputing"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"8098","DOI":"10.1029\/JB091iB08p08098","article-title":"Spectral mixture modeling: A new analysis of rock and soil types at the Viking Lander 1 site","volume":"91","author":"Adams","year":"1986","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Chan, T.H., Ma, W.K., Ambikapathi, A., and Chi, C.Y. (2011, January 24\u201329). An optimization perspective onwinter\u2019s endmember extraction belief. Proceedings of the IEEE International Geoscience and Remote Sensing Symposium, Vancouver, BC, Canada.","DOI":"10.1109\/IGARSS.2011.6049399"},{"key":"ref_9","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_10","doi-asserted-by":"crossref","first-page":"6076","DOI":"10.1109\/TGRS.2016.2580702","article-title":"Robust Collaborative Nonnegative Matrix Factorization for Hyperspectral Unmixing","volume":"54","author":"Li","year":"2016","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"8952","DOI":"10.1109\/TGRS.2020.2992542","article-title":"Simultaneously Counting and Extracting Endmembers in a Hyperspectral Image Based on Divergent Subsets","volume":"58","author":"Tao","year":"2020","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Xu, X., Pan, B., Chen, Z., Shi, Z., and Li, T. (2020). Simultaneously Multiobjective Sparse Unmixing and Library Pruning for Hyperspectral Imagery. IEEE Trans. Geosci. Remote Sens.","DOI":"10.1109\/TGRS.2020.3016941"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"9612","DOI":"10.1109\/TGRS.2019.2928021","article-title":"A classification-based model for multi-objective hyperspectral sparse unmixing","volume":"57","author":"Xu","year":"2019","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_14","unstructured":"Geng, X. (2005). Study on Hyperspectral Target Detection and Classification. [Ph.D. Thesis, Chinese Academy of Sciences]."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1146","DOI":"10.1109\/LGRS.2017.2700542","article-title":"Hyperspectral unmixing using double reweighted sparse regression and total variation","volume":"14","author":"Wang","year":"2017","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"6344","DOI":"10.1109\/TGRS.2018.2837150","article-title":"Hyperspectral unmixing based on dual-depth sparse probabilistic latent semantic analysis","volume":"56","author":"Plaza","year":"2018","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"952","DOI":"10.1109\/JSTARS.2015.2495128","article-title":"Fast spatial preprocessing for spectral unmixing of hyperspectral data on graphics processing units","volume":"9","author":"Delgado","year":"2015","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2633","DOI":"10.1109\/TGRS.2010.2040284","article-title":"Hybrid detectors based on selective endmembers","volume":"48","author":"Zhang","year":"2010","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"782","DOI":"10.1109\/LGRS.2016.2544839","article-title":"A fast spatial\u2013spectral preprocessing module for hyperspectral endmember extraction","volume":"13","author":"Kowkabi","year":"2016","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/j.isprsjprs.2019.10.005","article-title":"Using spectral Geodesic and spatial Euclidean weights of neighbourhood pixels for hyperspectral Endmember Extraction preprocessing","volume":"158","author":"Kowkabi","year":"2019","journal-title":"ISPRS J. Photogramm. Remote Sens."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.isprsjprs.2017.02.005","article-title":"Endmember extraction from hyperspectral image based on discrete firefly algorithm (EE-DFA)","volume":"126","author":"Zhang","year":"2017","journal-title":"ISPRS J. Photogramm. Remote Sens."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.isprsjprs.2017.08.001","article-title":"Pure endmember extraction using robust kernel archetypoid analysis for hyperspectral imagery","volume":"131","author":"Sun","year":"2017","journal-title":"ISPRS J. Photogramm. Remote Sens."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"6003","DOI":"10.1109\/TGRS.2019.2903875","article-title":"An improved quantum-behaved particle swarm optimization for endmember extraction","volume":"57","author":"Du","year":"2019","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"7872","DOI":"10.1109\/TGRS.2019.2917001","article-title":"An improved multiobjective discrete particle swarm optimization for hyperspectral endmember extraction","volume":"57","author":"Tong","year":"2019","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4022","DOI":"10.1109\/TNNLS.2017.2749279","article-title":"Detection of sources in non-negative blind source separation by minimum description length criterion","volume":"29","author":"Lin","year":"2017","journal-title":"IEEE Trans. Neural Networks Learn. Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1016\/0005-1098(78)90005-5","article-title":"Modeling by shortest data description","volume":"14","author":"Rissanen","year":"1978","journal-title":"Automatica"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1214\/aos\/1176344136","article-title":"Estimating the dimension of a model","volume":"6","author":"Schwarz","year":"1978","journal-title":"Ann. Stat."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Akaike, H. (1974). Akaike, H. A new look at the statistical model identification. Selected Papers of Hirotugu Akaike, Springer.","DOI":"10.1007\/978-1-4612-1694-0_16"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TSP.2006.870586","article-title":"Unsupervised learning of parsimonious mixtures on large spaces with integrated feature and component selection","volume":"54","author":"Graham","year":"2006","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"974","DOI":"10.1016\/j.csda.2004.06.015","article-title":"How many principal components? Stopping rules for determining the number of non-trivial axes revisited","volume":"49","author":"Jackson","year":"2005","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2435","DOI":"10.1109\/TGRS.2008.918089","article-title":"Hyperspectral subspace identification","volume":"46","author":"Nascimento","year":"2008","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1109\/TGRS.2003.819189","article-title":"Estimation of number of spectrally distinct signal sources in hyperspectral imagery","volume":"42","author":"Chang","year":"2004","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3811","DOI":"10.1109\/TGRS.2016.2528298","article-title":"Estimating the intrinsic dimension of hyperspectral images using a noise-whitened eigengap approach","volume":"54","author":"Halimi","year":"2016","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2753","DOI":"10.1109\/TGRS.2012.2213261","article-title":"Hyperspectral data geometry-based estimation of number of endmembers using p-norm-based pure pixel identification algorithm","volume":"51","author":"Ambikapathi","year":"2012","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Parente, M., and Plaza, A. (2010, January 14\u201316). Survey of geometric and statistical unmixing algorithms for hyperspectral images. Proceedings of the 2010 2nd Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing, Reykjavik, Iceland.","DOI":"10.1109\/WHISPERS.2010.5594929"},{"key":"ref_36","unstructured":"Boardman, J.W., Kruse, F.A., and Green, R.O. (1995, January 23\u201326). Mapping target signatures via partial unmixing of AVIRIS data. Proceedings of the Summaries 5th JPL Airborne Earth Science Workshop, Pasadena, CA, USA."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Tao, X., Cui, T., Yu, Z., and Ren, P. (2018, January 28\u201331). Locality Preserving Endmember Extraction for Estimating Green Algae Area. Proceedings of the 2018 OCEANS-MTS\/IEEE Kobe Techno-Oceans (OTO), Kobe, Japan.","DOI":"10.1109\/OCEANSKOBE.2018.8558824"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1109\/LGRS.2018.2888574","article-title":"Cofactor-Based Efficient Endmember Extraction for Green Algae Area Estimation","volume":"16","author":"Tao","year":"2019","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_39","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_40","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_41","doi-asserted-by":"crossref","first-page":"4418","DOI":"10.1109\/TSP.2009.2025802","article-title":"A convex analysis-based minimum-volume enclosing simplex algorithm for hyperspectral unmixing","volume":"57","author":"Chan","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Li, J., and Bioucas-Dias, J.M. (2008, January 7\u201311). Minimum volume simplex analysis: A fast algorithm to unmix hyperspectral data. Proceedings of the IGARSS 2008-2008 IEEE International Geoscience and Remote Sensing Symposium, Boston, MA, USA.","DOI":"10.1109\/IGARSS.2008.4779330"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"5067","DOI":"10.1109\/TGRS.2015.2417162","article-title":"Minimum volume simplex analysis: A fast algorithm for linear hyperspectral unmixing","volume":"53","author":"Li","year":"2015","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1946","DOI":"10.1109\/TSP.2015.2508778","article-title":"A fast hyperplane-based minimum-volume enclosing simplex algorithm for blind hyperspectral unmixing","volume":"64","author":"Lin","year":"2015","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1109\/5254.708428","article-title":"Support vector machines","volume":"13","author":"Hearst","year":"1998","journal-title":"IEEE Intell. Syst. Their Appl."},{"key":"ref_46","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"}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/13\/4\/713\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:24:32Z","timestamp":1760160272000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/13\/4\/713"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,15]]},"references-count":46,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["rs13040713"],"URL":"https:\/\/doi.org\/10.3390\/rs13040713","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,2,15]]}}}