{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:15:21Z","timestamp":1760188521949,"version":"build-2065373602"},"reference-count":53,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2019,5,3]],"date-time":"2019-05-03T00:00:00Z","timestamp":1556841600000},"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>Many methods have been proposed in the literature for estimating the number of materials\/endmembers in a hyperspectral image. This is sometimes called the \u201cintrinsic\u201d dimension (ID) of the image. A number of recent papers have proposed ID estimation methods based on various aspects of random matrix theory (RMT), under the assumption that the errors are uncorrelated, but with possibly unequal variances. A recent paper, which reviewed a number of the better known methods (including one RMT-based method), has shown that they are all biased, especially when the true ID is greater than about 20 or 30, even when the error structure is known. I introduce two RMT-based estimators (    R M  T G     , which is new, and     R M  T  K N      , which is a modification of an existing estimator), which are approximately unbiased when the error variances are known. However, they are biased when the error variance is unknown and needs to be estimated. This bias increases as ID increases. I show how this bias can be reduced. The results use semi-realistic simulations based on three real hyperspectral scenes. Despite this, when applied to the real scenes,     R M  T G      and     R M  T  K N       are larger than expected. Possible reasons for this are discussed, including the presence of errors which are either deterministic, spectrally and\/or spatially correlated, or signal-dependent. Possible future research into ID estimation in the presence of such errors is outlined.<\/jats:p>","DOI":"10.3390\/rs11091049","type":"journal-article","created":{"date-parts":[[2019,5,7]],"date-time":"2019-05-07T03:15:46Z","timestamp":1557198946000},"page":"1049","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Improved Estimation of the Intrinsic Dimension of a Hyperspectral Image Using Random Matrix Theory"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3124-1934","authenticated-orcid":false,"given":"Mark","family":"Berman","sequence":"first","affiliation":[{"name":"CSIRO Data61, Marsfield, NSW 2122, Australia"},{"name":"School of Computing, Engineering and Mathematics, Western Sydney University, Parramatta, NSW 2150, Australia"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,3]]},"reference":[{"key":"ref_1","first-page":"395","article-title":"Determining the number and identity of spectral endmembers: an integrated approach using Neyman-Pearson eigen-thresholding and iterative constrained RMS error minimization","volume":"Volume 1","author":"Harsanyi","year":"1993","journal-title":"Proceedings of the Thematic Conference on Geologic Remote Sensing"},{"key":"ref_2","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_3","doi-asserted-by":"crossref","first-page":"5579","DOI":"10.1109\/TSP.2007.901645","article-title":"Rank estimation and redundancy reduction of high-dimensional noisy signals with preservation of rare vectors","volume":"55","author":"Kuybeda","year":"2007","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_4","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_5","doi-asserted-by":"crossref","first-page":"3844","DOI":"10.1109\/TGRS.2009.2021764","article-title":"A new algorithm for robust estimation of the signal subspace in hyperspectral images in the presence of rare signal components","volume":"47","author":"Acito","year":"2009","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_6","first-page":"582","article-title":"Estimating the number of endmembers in hyperspectral images using the normal compositional model and a hierarchical Bayesian algorithm","volume":"4","author":"Eches","year":"2010","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. 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","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":"2013","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1109\/LGRS.2012.2189934","article-title":"Empirical automatic estimation of the number of endmembers in hyperspectral images","volume":"10","author":"Luo","year":"2013","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_10","first-page":"191","article-title":"Estimation of signal subspace on hyperspectral data","volume":"5982","author":"Nascimento","year":"2005","journal-title":"SPIE Proc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1301","DOI":"10.1109\/TIP.2012.2227765","article-title":"Determining the intrinsic dimension of a hyperspectral image using random matrix theory","volume":"22","author":"Damelin","year":"2013","journal-title":"IEEE Trans. Image Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3930","DOI":"10.1109\/TSP.2009.2022897","article-title":"Non-parametric detection of the number of signals: Hypothesis testing and random matrix theory","volume":"57","author":"Kritchman","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1150002","DOI":"10.1142\/S201032631150002X","article-title":"On determining the number of spikes in a high-dimensional spiked population model","volume":"1","author":"Passemier","year":"2012","journal-title":"Random Matrices Theory Appl."},{"key":"ref_14","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_15","doi-asserted-by":"crossref","first-page":"3588","DOI":"10.1109\/TGRS.2017.2676816","article-title":"A comparison between three sparse unmixing algorithms using a large library of shortwave infrared mineral spectra","volume":"55","author":"Berman","year":"2017","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"542","DOI":"10.1109\/36.297973","article-title":"Minimum-Volume Transforms for Remotely Sensed Data","volume":"32","author":"Craig","year":"1994","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Winter, M.E. (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.","DOI":"10.1117\/12.366289"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2085","DOI":"10.1109\/TGRS.2004.835299","article-title":"ICE: A statistical approach to identifying endmembers","volume":"42","author":"Berman","year":"2004","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_19","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_20","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_21","doi-asserted-by":"crossref","first-page":"4407","DOI":"10.1109\/JSTARS.2016.2580178","article-title":"Semi-realistic simulations of natural hyperspectral scenes","volume":"9","author":"Hao","year":"2016","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3279","DOI":"10.1109\/JSTARS.2018.2850047","article-title":"An investigation into the impact of band error variance estimation on intrinsic dimension estimation in hyperspectral images","volume":"11","author":"Berman","year":"2018","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/S0034-4257(98)00064-9","article-title":"Imaging spectroscopy and the Airborne Visible\/Infrared Imaging Spectrometer (AVIRIS)","volume":"65","author":"Green","year":"1998","journal-title":"Remote Sens. Environ."},{"key":"ref_24","unstructured":"Schaepman, M., Schlapfer, D., and Itten, K. (1998). The HyMap airborne hyperspectral sensor: The system, calibration and performance. Proceedings of the 1st EARSeL Workshop on Imaging Spectroscopy, Zurich, Switzerland, 6\u20138 October 1998, EARSeL."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2719","DOI":"10.1080\/01431169608949102","article-title":"Principal Components transform with simple automatic noise adjustment","volume":"17","author":"Roger","year":"1996","journal-title":"Int. J. Remote Sens."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1451","DOI":"10.1109\/TGRS.2016.2624505","article-title":"Modified residual method for the estimation of noise in hyperspectral images","volume":"55","author":"Mahmood","year":"2017","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"488","DOI":"10.1109\/JSTARS.2012.2227245","article-title":"A comparative study on linear regression-based noise estimation for hyperspectral imagery","volume":"6","author":"Gao","year":"2013","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2854","DOI":"10.1109\/JSTARS.2015.2432460","article-title":"Estimation of the intrinsic dimension of hyperspectral images: Comparison of current methods","volume":"8","author":"Robin","year":"2015","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Meyer, T.R., Drumetz, L., Chanussot, J., Bertozzi, A.L., and Jutten, C. (2016, January 25\u201328). Hyperspectral unmixing with material variability using social sparsity. Proceedings of the 2016 IEEE International Conference on Image Processing (ICIP), Phoenix, AZ, USA.","DOI":"10.1109\/ICIP.2016.7532746"},{"key":"ref_30","unstructured":"Zhou, Y., Wetherley, E.B., and Gader, P.D. (2018). Unmixing urban hyperspectral imagery with a Gaussian mixture model on endmember variability. arXiv."},{"key":"ref_31","first-page":"507","article-title":"Distribution of eigenvalues for some sets of random matrices","volume":"114","author":"Marchenko","year":"1967","journal-title":"Mat. Sb."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1382","DOI":"10.1016\/j.jmva.2005.08.003","article-title":"Eigenvalues of large sample covariance matrices of spiked population models","volume":"97","author":"Baik","year":"2006","journal-title":"J. Multivar. Anal."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1214\/aop\/1176994775","article-title":"A limit theorem for the norm of random matrices","volume":"8","author":"Geman","year":"1980","journal-title":"Ann. Probab."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1214\/aos\/1009210544","article-title":"On the distribution of the largest eigenvalue in principal components analysis","volume":"29","author":"Johnstone","year":"2001","journal-title":"Ann. Stat."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1214\/aoms\/1177729029","article-title":"On a heuristic method of test construction and its use in multivariate analysis","volume":"24","author":"Roy","year":"1953","journal-title":"Ann. Math. Stat."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1109\/TASSP.1985.1164557","article-title":"Detection of signals by information theoretic criteria","volume":"33","author":"Wax","year":"1985","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1109\/34.44408","article-title":"A fast parallel algorithm for blind estimation of noise variance","volume":"12","author":"Meer","year":"1990","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_38","first-page":"5131","article-title":"Earth and planetary remote sensing with the USGS Tetracorder and expert systems","volume":"83","author":"Clark","year":"2003","journal-title":"J. Geophys. Res."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1083","DOI":"10.1109\/36.841987","article-title":"Endmember bundles: A new approach to incorporating endmember variability into spectral mixture analysis","volume":"38","author":"Bateson","year":"2000","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1603","DOI":"10.1016\/j.rse.2011.03.003","article-title":"Endmember variability in spectral mixture analysis: A review","volume":"115","author":"Somers","year":"2011","journal-title":"Remote Sens. Environ."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1109\/MSP.2013.2279177","article-title":"Endmember variability in hyperspectral analysis: Addressing spectral variability during spectral unmixing","volume":"31","author":"Zare","year":"2014","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1109\/36.3001","article-title":"A transformation for ordering multispectral data in terms of image quality with implications for noise removal","volume":"26","author":"Green","year":"1988","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_43","first-page":"753208","article-title":"A method for blind estimation of spatially correlated noise characteristics","volume":"Volume 7532","author":"Ponomarenko","year":"2010","journal-title":"Image Processing: Algorithms and Systems VIII"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1615\/TelecomRadEng.v73.i6.40","article-title":"Automatic estimation of spatially correlated noise variance in spectral domain for images","volume":"73","author":"Abramova","year":"2014","journal-title":"Telecommun. Radio Eng."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2957","DOI":"10.1109\/TGRS.2011.2110657","article-title":"Signal-dependent noise modeling and model parameter estimation in hyperspectral images","volume":"49","author":"Acito","year":"2011","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3829","DOI":"10.1364\/AO.50.003829","article-title":"Modeling and estimation of signal-dependent noise in hyperspectral imagery","volume":"50","author":"Meola","year":"2011","journal-title":"Appl. Opt."},{"key":"ref_47","first-page":"469","article-title":"Local signal-dependent noise variance estimation from hyperspectral textural images","volume":"5","author":"Uss","year":"2011","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1016\/j.rse.2009.02.003","article-title":"Nonlinear hyperspectral mixture analysis for tree cover estimates in orchards","volume":"113","author":"Somers","year":"2009","journal-title":"Remote Sens. Environ."},{"key":"ref_49","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_50","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_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":"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_53","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."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/11\/9\/1049\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:48:55Z","timestamp":1760186935000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/11\/9\/1049"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,3]]},"references-count":53,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["rs11091049"],"URL":"https:\/\/doi.org\/10.3390\/rs11091049","relation":{},"ISSN":["2072-4292"],"issn-type":[{"type":"electronic","value":"2072-4292"}],"subject":[],"published":{"date-parts":[[2019,5,3]]}}}