{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T11:39:17Z","timestamp":1780054757723,"version":"3.54.0"},"reference-count":43,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2016,10,25]],"date-time":"2016-10-25T00:00:00Z","timestamp":1477353600000},"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":["61302149"],"award-info":[{"award-number":["61302149"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The problem of hypothesis testing in the Neyman\u2013Pearson formulation is considered from a geometric viewpoint. In particular, a concise geometric interpretation of deterministic and random signal detection in the philosophy of information geometry is presented. In such a framework, both hypotheses and detectors can be treated as geometrical objects on the statistical manifold of a parameterized family of probability distributions. Both the detector and detection performance are geometrically elucidated in terms of the Kullback\u2013Leibler divergence. Compared to the likelihood ratio test, the geometric interpretation provides a consistent but more comprehensive means to understand and deal with signal detection problems in a rather convenient manner. Example of the geometry based detector in radar constant false alarm rate (CFAR) detection is presented, which shows its advantage over the classical processing method.<\/jats:p>","DOI":"10.3390\/e18110381","type":"journal-article","created":{"date-parts":[[2016,10,25]],"date-time":"2016-10-25T10:28:49Z","timestamp":1477391329000},"page":"381","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":52,"title":["The Geometry of Signal Detection with Applications to Radar Signal Processing"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0127-384X","authenticated-orcid":false,"given":"Yongqiang","family":"Cheng","sequence":"first","affiliation":[{"name":"School of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7687-7720","authenticated-orcid":false,"given":"Xiaoqiang","family":"Hua","sequence":"additional","affiliation":[{"name":"School of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2522-9552","authenticated-orcid":false,"given":"Hongqiang","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuliang","family":"Qin","sequence":"additional","affiliation":[{"name":"School of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiang","family":"Li","sequence":"additional","affiliation":[{"name":"School of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2016,10,25]]},"reference":[{"key":"ref_1","unstructured":"Kay, S.M. (1993). Fundamentals of Statistical Signal Processing: Detection Theory, Prentice Hall PTR. [1st ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1701","DOI":"10.1109\/18.930911","article-title":"Information geometry on hierarchy of probability distributions","volume":"47","author":"Amari","year":"2001","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_3","unstructured":"Amari, S. (2002, January 20\u201325). Information geometry of statistical inference\u2014An overview. Proceedings of the IEEE Information Theory Workshop, Bangalore, India."},{"key":"ref_4","first-page":"81","article-title":"Information and accuracy attainable in the estimation of statistical parameters","volume":"37","author":"Rao","year":"1945","journal-title":"Bull. Calcutta Math. Soc."},{"key":"ref_5","unstructured":"Kobayashi, S., and Takesaki, M. (2000). Methods of Information Geometry, American Mathematical Society. Translations of Mathematical Monographs vol. 191."},{"key":"ref_6","unstructured":"Leifman, L.J. (1982). Statistical Decision Rules and Optimal Inference, American Mathematical Society. Translations of Mathematical Monographs vol. 53."},{"key":"ref_7","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_8","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_9","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1214\/aos\/1176345779","article-title":"Differential geometry of curved exponential families-curvatures and information loss","volume":"10","author":"Amari","year":"1982","journal-title":"Ann. Stat."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Kass, R.E., and Vos, P.W. (1997). Geometrical Foundations of Asymptotic Inference, Wiley.","DOI":"10.1002\/9781118165980"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"29","DOI":"10.2307\/3318651","article-title":"Information geometry of estimating functions in semi parametric statistical models","volume":"3","author":"Amari","year":"1997","journal-title":"Bernoulli"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1109\/72.125867","article-title":"Information geometry of Boltzmann machines","volume":"3","author":"Amari","year":"1992","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1162\/089976698300017746","article-title":"Natural gradient works efficiently in learning","volume":"10","author":"Amari","year":"1998","journal-title":"Neural Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1379","DOI":"10.1016\/0893-6080(95)00003-8","article-title":"Information geometry of the EM and em algorithms for neural networks","volume":"8","author":"Amari","year":"1995","journal-title":"Neural Netw."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1007\/BF00057730","article-title":"Fisher information under restriction of Shannon information","volume":"41","author":"Amari","year":"1989","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/0020-0255(85)90050-7","article-title":"The relation between information theory and the differential geometry approach to statistics","volume":"35","author":"Campbell","year":"1985","journal-title":"Inf. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/BF01692059","article-title":"Differential geometry of a parametric family of invertible linear systems\u2014Riemannian metric, dual affine connections and divergence","volume":"20","author":"Amari","year":"1987","journal-title":"Math. Syst. Theory"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/0024-3795(94)00348-3","article-title":"Dualistic differential geometry of positive definite matrices and its applications to related problems","volume":"247","author":"Ohara","year":"1996","journal-title":"Linear Algebra Appl."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Barndorff-Nielsen, O.E., and Vedel Jensen, E.B. (1998, January 16\u201318). Information geometric analysis of an interior point method for semidefinite programming. Proceedings of the Geometry in Present Day Science, Aarhus, Denmark.","DOI":"10.1142\/9789814527910"},{"key":"ref_20","unstructured":"Opper, M., and Saad, D. (2000). Recent Developments of Mean Field Approximation, MIT Press."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1109\/18.817505","article-title":"The geometry of turbo-decoding dynamics","volume":"46","author":"Richardson","year":"2000","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2723","DOI":"10.1109\/18.887888","article-title":"On a geometric view of multiuser detection for synchronous DS\/CDMA channels","volume":"46","author":"Li","year":"2000","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1610","DOI":"10.1109\/TSP.2005.845428","article-title":"Covariance, subspace, and intrinsic Cram\u00e9r\u2013Rao bounds","volume":"53","author":"Smith","year":"2005","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1390","DOI":"10.1109\/78.839985","article-title":"A Bayesian approach to geometric subspace estimation","volume":"48","author":"Srivastava","year":"2000","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1239\/aap\/1077134463","article-title":"Bayesian and geometric subspace tracking","volume":"36","author":"Srivastava","year":"2004","journal-title":"Adv. Appl. Probab."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3327","DOI":"10.1109\/TIT.2008.924656","article-title":"Asymptotic geometry of multiple hypothesis testing","volume":"54","author":"Westover","year":"2008","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4470","DOI":"10.1109\/TSP.2011.2159500","article-title":"Bayes risk error is a Bregman divergence","volume":"59","author":"Varshney","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_28","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 & information geometry. Proceedings of the 2008 IEEE Radar Conference, Rome, Italy.","DOI":"10.1109\/RADAR.2008.4720937"},{"key":"ref_29","unstructured":"Barbaresco, F. (2009, January 12\u201316). New foundation of radar Doppler signal processing based on advanced differential geometry of symmetric spaces: Doppler matrix CFAR and radar application. Proceedings of the International Radar Conference, Bordeaux, France."},{"key":"ref_30","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_31","first-page":"119","article-title":"An elementary proof of the strong law of large numbers","volume":"55","author":"Etemadi","year":"1981","journal-title":"Probab. Theory Relat. Fields"},{"key":"ref_32","unstructured":"Kullback, S. (1968). Information Theory and Statistics, Dover Publications."},{"key":"ref_33","unstructured":"Cover, T.M., and Thomas, J.A. (1991). Elements of Information Theory, Wiley."},{"key":"ref_34","unstructured":"Pinsker, M.S. (1964). Information and Information Stability of Random Variables and Processes, Holden-Day."},{"key":"ref_35","unstructured":"Lehmann, F.L. (1959). Testing Statistical Hypotheses, Wiley."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1354","DOI":"10.1109\/TIT.2006.871599","article-title":"Neyman\u2013Pearson detection of Gauss-Markov signals in noise: Closed-form error exponent and properties","volume":"52","author":"Sung","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1109\/TSP.2002.806982","article-title":"Decentralized detection in sensor networks","volume":"51","author":"Chamberland","year":"2003","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_38","unstructured":"Sullivant, S. Statistical Models Are Algebraic Varieties. Available online: https:\/\/www.researchgate.net\/profile\/Seth_Sullivant\/publication\/254069909_Statistical_Models_are_Algebraic_Varieties\/links\/552bdb0d0cf2e089a3aa889a.pdf?origin=publication_detail."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1137\/S0895479803436937","article-title":"A differential geometric approach to the geometric mean of symmetric positive-definite matrices","volume":"26","author":"Moakher","year":"2005","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Balaji, B., Barbaresco, F., and Decurninge, A. (2014, January 13\u201317). Information geometry and estimation of Toeplitz covariance matrices. Proceedings of the 2014 International Radar Conference, Lille, France.","DOI":"10.1109\/RADAR.2014.7060458"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1109\/7.381910","article-title":"Asymptotically optimum radar detection in compound-Gaussian clutter","volume":"31","author":"Conte","year":"1995","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Roy, L.P., and Kumar, R.V.R. (2010, January 29\u201331). A GLRT detector in partially correlated texture based compound-Gaussian clutter. Proceedings of the 2010 National Conference on Communications (NCC), Chennai, India.","DOI":"10.1109\/NCC.2010.5430182"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"806","DOI":"10.1049\/el:19980582","article-title":"Probability of false alarm of CA-CFAR detector in Weibull clutter","volume":"34","author":"Vela","year":"1998","journal-title":"Electron. Lett."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/11\/381\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:33:53Z","timestamp":1760211233000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/11\/381"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,10,25]]},"references-count":43,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2016,11]]}},"alternative-id":["e18110381"],"URL":"https:\/\/doi.org\/10.3390\/e18110381","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,10,25]]}}}