{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T17:33:23Z","timestamp":1776101603625,"version":"3.50.1"},"reference-count":39,"publisher":"MDPI AG","issue":"20","license":[{"start":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T00:00:00Z","timestamp":1665446400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CNPq (Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico)","award":["303267\/2019-4"],"award-info":[{"award-number":["303267\/2019-4"]}]},{"name":"CNPq (Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico)","award":["309538\/2021-1"],"award-info":[{"award-number":["309538\/2021-1"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>Polarimetric synthetic aperture radar (PolSAR) systems are an important remote sensing tool. Such systems can provide high spacial resolution images, but they are contaminated by an interference pattern called multidimensional speckle. This fact requires that PolSAR images receive specialised treatment; particularly, tailored models which are close to PolSAR physical formation are sought. In this paper, we propose two new matrix models which arise from applying the stochastic summation approach to PolSAR, called compound truncated Poisson complex Wishart (CTPCW) and compound geometric complex Wishart (CGCW) distributions. These models offer the unique ability to express multimodal data. Some of their mathematical properties are derived and discussed\u2014characteristic function and Mellin-kind log-cumulants (MLCs). Moreover, maximum likelihood (ML) estimation procedures via expectation maximisation algorithm for CTPCW and CGCW parameters are furnished as well as MLC-based goodness-of-fit graphical tools. Monte Carlo experiment results indicate ML estimates perform at what is asymptotically expected (small bias and mean square error) even for small sample sizes. Finally, our proposals are employed to describe actual PolSAR images, presenting evidence that they can outperform other well-known distributions, such as WmC, Gm0, and Km.<\/jats:p>","DOI":"10.3390\/rs14205083","type":"journal-article","created":{"date-parts":[[2022,10,12]],"date-time":"2022-10-12T02:10:27Z","timestamp":1665540627000},"page":"5083","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["PolSAR Models with Multimodal Intensities"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2131-6464","authenticated-orcid":false,"given":"Jodavid A.","family":"Ferreira","sequence":"first","affiliation":[{"name":"Departamento de Estat\u00edstica, Universidade Federal da Para\u00edba, Jo\u00e3o Pessoa 58051-900, PB, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2673-219X","authenticated-orcid":false,"given":"Abra\u00e3o D. C.","family":"Nascimento","sequence":"additional","affiliation":[{"name":"Departamento de Estat\u00edstica, Universidade Federal de Pernambuco, Recife 50670-901, PE, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8002-5341","authenticated-orcid":false,"given":"Alejandro C.","family":"Frery","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Victoria University of Wellington, Wellington 6140, New Zealand"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2299","DOI":"10.1080\/01431169408954244","article-title":"Classification of Multi-Look Polarimetric SAR Imagery Based on Complex Wishart Distribution","volume":"15","author":"Lee","year":"1994","journal-title":"Int. J. Remote Sens."},{"key":"ref_2","first-page":"5600","article-title":"Unsupervised Classification of Multilook Polarimetric SAR Data Using Spatially Variant Wishart Mixture Model with Double Constraints","volume":"56","author":"Liu","year":"2018","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"6104","DOI":"10.1109\/TGRS.2019.2904401","article-title":"Unsupervised Segmentation of Multilook Polarimetric Synthetic Aperture Radar Images","volume":"57","author":"Bouhlel","year":"2019","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.patcog.2013.04.001","article-title":"Speckle Reduction in Polarimetric SAR Imagery with Stochastic Distances and Nonlocal Means","volume":"47","author":"Torres","year":"2014","journal-title":"Pattern Recognit."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1109\/TGRS.2016.2608501","article-title":"Application of Mixture Regression for Improved Polarimetric SAR Speckle Filtering","volume":"55","author":"Wang","year":"2017","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"648","DOI":"10.1109\/JSTARS.2013.2266319","article-title":"Comparing Edge Detection Methods based on Stochastic Entropies and Distances for PolSAR Imagery","volume":"7","author":"Nascimento","year":"2014","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1380","DOI":"10.1109\/TGRS.2018.2866367","article-title":"Detecting Changes in Fully Polarimetric SAR Imagery With Statistical Information Theory","volume":"57","author":"Nascimento","year":"2019","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"682","DOI":"10.1080\/2150704X.2014.960611","article-title":"Multilook polarimetric SAR data probability density function estimation using a generalized form of multivariate K-distribution","volume":"5","author":"Bian","year":"2014","journal-title":"Remote Sens. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"4932","DOI":"10.1109\/TGRS.2013.2285927","article-title":"Bias Correction and Modified Profile Likelihood Under the Wishart Complex Distribution","volume":"52","author":"Nascimento","year":"2014","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2281","DOI":"10.1109\/TGRS.2010.2103945","article-title":"Application of the Matrix-Variate Mellin Transform to Analysis of Polarimetric Radar Images","volume":"49","author":"Anfinsen","year":"2011","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Deng, X., L\u00f3pez-Mart\u00ednez, C., Chen, J., and Han, P. (2017). Statistical Modeling of Polarimetric SAR Data: A Survey and Challenges. Remote Sens., 9.","DOI":"10.3390\/rs9040348"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2947","DOI":"10.1109\/TGRS.2019.2958125","article-title":"A Generalized Gaussian Coherent Scatterer Model for Correlated SAR Texture","volume":"58","author":"Yue","year":"2020","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_13","unstructured":"Nicolas, J., and Tupin, F. (2002, January 24\u201328). Gamma mixture modeled with \u201csecond kind statistics\u201d: Application to SAR image processing. Proceedings of the IEEE International Geoscience and Remote Sensing Symposium, Toronto, ON, Canada."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1109\/JSTSP.2010.2103925","article-title":"Supervised High-Resolution Dual-Polarization SAR Image Classification by Finite Mixtures and Copulas","volume":"5","author":"Krylov","year":"2011","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Solarna, D., Moser, G., and Serpico, S.B. (2018). A Markovian Approach to Unsupervised Change Detection with Multiresolution and Multimodality SAR Data. Remote Sens., 10.","DOI":"10.3390\/rs10111671"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1109\/LGRS.2008.923262","article-title":"Fisher Distribution for Texture Modeling of Polarimetric SAR Data","volume":"5","author":"Bombrun","year":"2008","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1002\/env.658","article-title":"The Polarimetric G Distribution for SAR Data Analysis","volume":"16","author":"Freitas","year":"2005","journal-title":"Environmetrics"},{"key":"ref_18","unstructured":"Lee, J.S., Schuler, D.L., Lang, R.H., and Ranson, K.J. (1994, January 8\u201312). K-distribution for multi-look processed polarimetric SAR imagery. Proceedings of the International Geoscience and Remote Sensing Symposium (IGARSS\u20191994), Pasadena, CA, USA."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3795","DOI":"10.1109\/TGRS.2009.2019269","article-title":"Estimation of the Equivalent Number of Looks in Polarimetric Synthetic Aperture Radar Imagery","volume":"47","author":"Anfinsen","year":"2009","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1430","DOI":"10.1109\/TGRS.2002.800234","article-title":"Modeling Non-Rayleigh Speckle Distribution in SAR Images","volume":"40","author":"Delignon","year":"2002","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1109\/MGRS.2020.3027609","article-title":"SAR Image Statistical Modeling Part II: Spatial Correlation Models and Simulation","volume":"9","author":"Yue","year":"2021","journal-title":"IEEE Geosci. Remote Sens. Mag."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Ulaby, F.T., and Elachi, C. (1990). Radar Polarimetriy for Geoscience Applications, Artech House.","DOI":"10.1080\/10106049009354274"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1214\/aoms\/1177704250","article-title":"Statistical Analysis Based on a Certain Complex Gaussian Distribution (an Introduction)","volume":"34","author":"Goodman","year":"1963","journal-title":"Ann. Math. Stat."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1016\/j.jmva.2005.05.014","article-title":"A Trivariate Chi-Squared Distribution Derived from the Complex Wishart Distribution","volume":"97","author":"Hagedorn","year":"2006","journal-title":"J. Multivar. Anal."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1093\/biomet\/39.3-4.247","article-title":"The estimation of the Poisson parameter from a truncated distribution","volume":"39","author":"Moore","year":"1952","journal-title":"Biometrika"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"485","DOI":"10.2307\/3001439","article-title":"The truncated Poisson distribution","volume":"9","author":"Plackett","year":"1953","journal-title":"Biometrics"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.1081\/SAC-120023878","article-title":"Tests of fit for the geometric distribution","volume":"32","author":"Best","year":"2003","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"609","DOI":"10.2307\/3212709","article-title":"A characterization of the geometric distribution","volume":"11","author":"Dallas","year":"1974","journal-title":"J. Appl. Probab."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1163\/156939389X00412","article-title":"K-Distribution and Polarimetric Terrain Radar Clutter","volume":"3","author":"Yueh","year":"1989","journal-title":"J. Electromagn. Waves Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1049\/ip-rsn:20000493","article-title":"Calculation of Moments of Complex Wishart and Complex Inverse Wishart Distributed Matrices","volume":"147","author":"Maiwald","year":"2000","journal-title":"IEE Proc.-Radar, Sonar Navig."},{"key":"ref_31","unstructured":"Fine, T.L. (2006). Probability and Probabilistic Reasoning for Electrical Engineering, Prentice Hall."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"787","DOI":"10.3934\/ipi.2019036","article-title":"Compound truncated Poisson normal distribution: Mathematical properties and Moment estimation","volume":"13","author":"Nascimento","year":"2019","journal-title":"Inverse Probl. Imaging"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","article-title":"Maximum likelihood from incomplete data via the EM algorithm","volume":"39","author":"Dempster","year":"1977","journal-title":"J. R. Stat. Soc. Ser. B (Methodol.)"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2764","DOI":"10.1109\/TGRS.2010.2104158","article-title":"Goodness-of-Fit Tests for Multilook Polarimetric Radar Data Based on the Mellin Transform","volume":"49","author":"Anfinsen","year":"2011","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Whelen, T., and Siqueira, P. (2017, January 23\u201328). Time series analysis of L-Band SAR for agricultural landcover classification. Proceedings of the 2017 IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Fort Worth, TX, USA.","DOI":"10.1109\/IGARSS.2017.8128211"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3770","DOI":"10.3390\/rs6053770","article-title":"Land Cover Classification for Polarimetric SAR Images Based on Mixture Models","volume":"6","author":"Gao","year":"2014","journal-title":"Remote Sens."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"288","DOI":"10.1109\/JOE.2009.2025645","article-title":"Reliable Methods for Estimating the K-Distribution Shape Parameter","volume":"35","author":"Abraham","year":"2010","journal-title":"IEEE J. Ocean. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"648","DOI":"10.1109\/36.581981","article-title":"A Model for Extremely Heterogeneous Clutter","volume":"35","author":"Frery","year":"1997","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1007\/BF01025868","article-title":"On the histogram as a density estimator: L2 theory","volume":"57","author":"Freedman","year":"1981","journal-title":"Z. F\u00fcr Wahrscheinlichkeitstheorie Und Verwandte Geb."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/14\/20\/5083\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:50:15Z","timestamp":1760143815000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/14\/20\/5083"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,11]]},"references-count":39,"journal-issue":{"issue":"20","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["rs14205083"],"URL":"https:\/\/doi.org\/10.3390\/rs14205083","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,11]]}}}