{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T10:23:43Z","timestamp":1778581423973,"version":"3.51.4"},"reference-count":27,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,3,27]],"date-time":"2019-03-27T00:00:00Z","timestamp":1553644800000},"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":["61871472"],"award-info":[{"award-number":["61871472"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Information geometry is the study of the intrinsic geometric properties of manifolds consisting of a probability distribution and provides a deeper understanding of statistical inference. Based on this discipline, this letter reports on the influence of the signal processing on the geometric structure of the statistical manifold in terms of estimation issues. This letter defines the intrinsic parameter submanifold, which reflects the essential geometric characteristics of the estimation issues. Moreover, the intrinsic parameter submanifold is proven to be a tighter one after signal processing. In addition, the necessary and sufficient condition of invariant signal processing of the geometric structure, i.e., isometric signal processing, is given. Specifically, considering the processing with the linear form, the construction method of linear isometric signal processing is proposed, and its properties are presented in this letter.<\/jats:p>","DOI":"10.3390\/e21040332","type":"journal-article","created":{"date-parts":[[2019,3,29]],"date-time":"2019-03-29T03:50:21Z","timestamp":1553831421000},"page":"332","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Isometric Signal Processing under Information Geometric Framework"],"prefix":"10.3390","volume":"21","author":[{"given":"Hao","family":"Wu","sequence":"first","affiliation":[{"name":"College of Electronic Science, National University of Defense Technology, Changsha 410073, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0127-384X","authenticated-orcid":false,"given":"Yongqiang","family":"Cheng","sequence":"additional","affiliation":[{"name":"College of Electronic Science, National University of Defense Technology, Changsha 410073, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongqiang","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Electronic Science, National University of Defense Technology, Changsha 410073, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,27]]},"reference":[{"key":"ref_1","unstructured":"Kotz, S., and Johnson, N.L. (1992). Information and the Accuracy Attainable in the Estimation of Statistical Parameters. Breakthroughs in Statistics: Foundations and Basic Theory, Springer."},{"key":"ref_2","unstructured":"Chentsov, N.N. (1982). Statistical Decision Rules and Optimal Inference, American Mathematical Society. Number v. 53 in Translations of Mathematical Monographs."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1189","DOI":"10.1214\/aos\/1176343282","article-title":"Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency)","volume":"3","author":"Efron","year":"1975","journal-title":"Ann. Stat."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1214\/aos\/1176344130","article-title":"The Geometry of Exponential Families","volume":"6","author":"Efron","year":"1978","journal-title":"Ann. Stat."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Amari, S.I. (2016). Information Geometry and Its Applications, Springer Publishing Company, Incorporated. [1st ed.].","DOI":"10.1007\/978-4-431-55978-8"},{"key":"ref_6","first-page":"329","article-title":"Lectures on Differential Geometry","volume":"40","author":"Chern","year":"2014","journal-title":"Ann. Inst. Henri Poincare-Phys. Theor."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Rong, Y., Tang, M., and Zhou, J. (2017). Intrinsic Losses Based on Information Geometry and Their Applications. Entropy, 19.","DOI":"10.3390\/e19080405"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3514","DOI":"10.1109\/TSP.2011.2152399","article-title":"Tracking and Localizing Moving Targets in the Presence of Phase Measurement Ambiguities","volume":"59","author":"Cheng","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5674","DOI":"10.1109\/TSP.2011.2166547","article-title":"Optimal Nonlinear Estimation for Localization of Wireless Sensor Networks","volume":"59","author":"Cheng","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Cheng, Y., Wang, X., and Moran, B. (2017). Optimal Nonlinear Estimation in Statistical Manifolds with Application to Sensor Network Localization. Entropy, 19.","DOI":"10.3390\/e19070308"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1610","DOI":"10.1109\/TSP.2005.845428","article-title":"Covariance, subspace, and intrinsic Cram\u00e9r-Rao bounds","volume":"53","author":"Smith","year":"2005","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_12","first-page":"84","article-title":"MIMO radar adaptive waveform design for extended target recognition","volume":"2015","author":"Wang","year":"2016","journal-title":"Int. J. Distrib. Sens. Netw."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1109\/78.984753","article-title":"Optimization algorithms exploiting unitary constraints","volume":"50","author":"Manton","year":"2002","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1134","DOI":"10.1109\/TSP.2007.908999","article-title":"Steepest Descent Algorithms for Optimization Under Unitary Matrix Constraint","volume":"56","author":"Abrudan","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1704","DOI":"10.1016\/j.sigpro.2009.03.015","article-title":"Conjugate gradient algorithm for optimization under unitary matrix constraint","volume":"89","author":"Abrudan","year":"2009","journal-title":"Signal Process."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Barbaresco, F. (2008, January 26\u201330). Innovative Tools for Radar Signal Processing Based on Cartan\u2019s Geometry of SPD Matrices and Information Geometry. Proceedings of the Radar Conference, Rome, Italy.","DOI":"10.1109\/RADAR.2008.4720937"},{"key":"ref_17","unstructured":"Barbaresco, F. (2011, January 7\u20139). Robust statistical Radar Processing in Fr\u00e9chet metric space: OS-HDR-CFAR and OS-STAP Processing in Siegel homogeneous bounded domains. Proceedings of the International Radar Symposium, Leipzig, Germany."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Wu, H., Cheng, Y., Hua, X., and Wang, H. (2018). Vector Bundle Model of Complex Electromagnetic Space and Change Detection. Entropy, 21.","DOI":"10.3390\/e21010010"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.dsp.2017.06.019","article-title":"Matrix CFAR detectors based on symmetrized Kullback-Leibler and total Kullback-Leibler divergences","volume":"69","author":"Hua","year":"2017","journal-title":"Digit. Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Hua, X., Fan, H., Cheng, Y., Wang, H., and Qin, Y. (2018). Information Geometry for Radar Target Detection with Total Jensen-Bregman Divergence. Entropy, 20.","DOI":"10.3390\/e20040256"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2892","DOI":"10.1109\/TIP.2017.2692524","article-title":"Classification via Sparse Representation of Steerable Wavelet Frames on Grassmann Manifold: Application to Target Recognition in SAR Image","volume":"26","author":"Dong","year":"2017","journal-title":"IEEE Trans. Image Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"4918","DOI":"10.3390\/e17074918","article-title":"Fisher Information Properties","volume":"17","author":"Zegers","year":"2015","journal-title":"Entropy"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1246","DOI":"10.1109\/18.669301","article-title":"A proof of the Fisher information inequality via a data processing argument","volume":"44","author":"Zamir","year":"1998","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1214\/aoms\/1177730497","article-title":"Conditional Expectation and Unbiased Sequential Estimation","volume":"18","author":"Blackwell","year":"1947","journal-title":"Ann. Math. Stat."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Rojo, J. (2012). Completeness, Similar Regions, and Unbiased Estimation-Part I. Selected Works of E. L. Lehmann, Springer US.","DOI":"10.1007\/978-1-4614-1412-4"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Rojo, J. (2012). Completeness, Similar Regions, and Unbiased Estimation\u2014Part II. Selected Works of E. L. Lehmann, Springer US.","DOI":"10.1007\/978-1-4614-1412-4"},{"key":"ref_27","first-page":"465","article-title":"Fundamentals of statistical signal processing: Estimation theory","volume":"37","author":"Kay","year":"1994","journal-title":"Control Eng. Pract."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/4\/332\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:41:01Z","timestamp":1760186461000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/4\/332"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,27]]},"references-count":27,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,4]]}},"alternative-id":["e21040332"],"URL":"https:\/\/doi.org\/10.3390\/e21040332","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,27]]}}}