{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:25:45Z","timestamp":1760243145512,"version":"build-2065373602"},"reference-count":55,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2015,7,24]],"date-time":"2015-07-24T00:00:00Z","timestamp":1437696000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Assuming sparsity or compressibility of the underlying signals, compressed sensing or compressive sampling (CS) exploits the informational efficiency of under-sampled measurements for increased efficiency yet acceptable accuracy in information gathering, transmission and processing, though it often incurs extra computational cost in signal reconstruction. Shannon information quantities and theorems, such as source rate-distortion, trans-information and rate distortion theorem concerning lossy data compression, provide a coherent framework, which is complementary to classic CS theory, for analyzing informational quantities and for determining the necessary number of measurements in CS. While there exists some information-theoretic research in the past on CS in general and compressive radar imaging in particular, systematic research is needed to handle issues related to scene description in cluttered environments and trans-information quantification in complex sparsity-clutter-sampling-noise settings. The novelty of this paper lies in furnishing a general strategy for information-theoretic analysis of scene compressibility, trans-information of radar echo data about the scene and the targets of interest, respectively, and limits to undersampling ratios necessary for scene reconstruction subject to distortion given sparsity-clutter-noise constraints. A computational experiment was performed to demonstrate informational analysis regarding the scene-sampling-reconstruction process and to generate phase transition diagrams showing relations between undersampling ratios and sparsity-clutter-noise-distortion constraints. The strategy proposed in this paper is valuable for information-theoretic analysis and undersampling theorem developments in compressive radar imaging and other computational imaging applications.<\/jats:p>","DOI":"10.3390\/e17085171","type":"journal-article","created":{"date-parts":[[2015,7,24]],"date-time":"2015-07-24T10:44:26Z","timestamp":1437734666000},"page":"5171-5198","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Information-Theoretic Characterization and Undersampling Ratio Determination for Compressive Radar Imaging in a Simulated Environment"],"prefix":"10.3390","volume":"17","author":[{"given":"Jingxiong","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Remote Sensing and Information Engineering, Wuhan University, 129 Luoyu Road, 430079 Wuhan, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ke","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Remote Sensing and Information Engineering, Wuhan University, 129 Luoyu Road, 430079 Wuhan, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fengzhu","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Remote Sensing and Information Engineering, Wuhan University, 129 Luoyu Road, 430079 Wuhan, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ying","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Remote Sensing and Information Engineering, Wuhan University, 129 Luoyu Road, 430079 Wuhan, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,7,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4203","DOI":"10.1109\/TIT.2005.858979","article-title":"Decoding by Linear Programming","volume":"51","author":"Tao","year":"2005","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2006.871582","article-title":"Compressed Sensing","volume":"52","author":"Donoho","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1109\/LSP.2012.2224518","article-title":"Measure What Should Be Measured: Progress and Challenges in Compressive Sensing","volume":"19","author":"Strohmer","year":"2012","journal-title":"IEEE Signal. Process. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2275","DOI":"10.1109\/TSP.2009.2014277","article-title":"High-Resolution Radar via Compressed Sensing","volume":"57","author":"Herman","year":"2009","journal-title":"IEEE Trans. Signal. Process."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4285","DOI":"10.1109\/TGRS.2010.2051231","article-title":"A Novel Strategy for Radar Imaging Based on Compressive Sensing","volume":"48","author":"Alonso","year":"2010","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1402","DOI":"10.1016\/j.sigpro.2009.11.009","article-title":"On Compressive Sensing Applied to Radar","volume":"90","author":"Ender","year":"2010","journal-title":"Signal. Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1137\/090757034","article-title":"Compressed Remote Sensing of Sparse Objects","volume":"3","author":"Fannjiang","year":"2010","journal-title":"SIAM J. Imag. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1006","DOI":"10.1109\/JPROC.2009.2037526","article-title":"Sparsity and Compressed Sensing in Radar Imaging","volume":"98","author":"Potter","year":"2010","journal-title":"IEEE Proc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1722","DOI":"10.1007\/s11432-012-4633-4","article-title":"Sparse Microwave Imaging: Principles and Applications","volume":"55","author":"Zhang","year":"2012","journal-title":"Sci. China Inf. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4214","DOI":"10.1109\/TGRS.2012.2227060","article-title":"Segmented Reconstruction for Compressed Sensing SAR Imaging","volume":"51","author":"Yang","year":"2013","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3889","DOI":"10.3390\/e16073889","article-title":"Identifying Chaotic FitzHugh\u2013Nagumo Neurons Using Compressive Sensing","volume":"16","author":"Su","year":"2014","journal-title":"Entropy"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"C1","DOI":"10.1364\/AO.54.0000C1","article-title":"Pitfalls and Possibilities of Radar Compressive Sensing","volume":"54","author":"Goodman","year":"2015","journal-title":"Applied Optics"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2123","DOI":"10.1109\/TGRS.2014.2355592","article-title":"Sparse Regularization of Interferometric Phase and Amplitude for InSAR Image Formation Based on Bayesian Representation","volume":"53","author":"Xu","year":"2015","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"922","DOI":"10.1109\/LSP.2012.2224655","article-title":"Sparse and Redundant Representation Modeling-What Next?","volume":"19","author":"Elad","year":"2012","journal-title":"IEEE Signal. Process. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"7136","DOI":"10.3390\/s150407136","article-title":"Informational Analysis for Compressive Sampling in Radar Imaging","volume":"15","author":"Zhang","year":"2015","journal-title":"Sensors"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1483","DOI":"10.1109\/83.791978","article-title":"Compression of Complex-Valued SAR Images","volume":"8","author":"Eichel","year":"1999","journal-title":"IEEE Trans. Image Process."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory, Wiley. [2nd, ed.].","DOI":"10.1002\/047174882X"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2954","DOI":"10.1109\/TIT.2006.876351","article-title":"Limit Results on Pattern Entropy","volume":"52","author":"Orlitsky","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1016\/j.neucom.2014.07.007","article-title":"SAR Complex Image Data Compression Based on Quadtree and Zerotree Coding in Discrete Wavelet Transform Domain: A Comparative Study","volume":"148","author":"Hou","year":"2015","journal-title":"Neurocomputing."},{"key":"ref_20","unstructured":"Fry, D.W., and Higinbotham, W. (1953). Probability and Information Theory, with Applications to Radar: International Series of Monographs on Electronics and Instrumentation, Pergamon Press."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"768","DOI":"10.1109\/TAES.1983.309378","article-title":"The Information Content of Synthetic Aperture Radar Images of Terrain","volume":"5","author":"Frost","year":"1983","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1578","DOI":"10.1109\/18.259642","article-title":"Information Theory and Radar Waveform Design","volume":"39","author":"Bell","year":"1993","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"708","DOI":"10.1109\/TIT.1976.1055644","article-title":"A Lower Bound on the Date Rate for Synthetic Aperture Radar","volume":"22","author":"Zeoli","year":"1976","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Chance, Z., Raj, R.G., and Love, D.J. (2011, January 23\u201327). Information-Theoretic Structure of Multistatic Radar Imaging. Proceedings of the 2011 IEEE Conference on Radar Conference (RADAR), Kansas City, MO, USA.","DOI":"10.1109\/RADAR.2011.5960658"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Aksoylar, C., and Saligrama, V. (2014, January 22\u201325). Information-Theoretic Characterization of Sparse Recovery. Proceedings of the 17th International Conference on Artificial Intelligence and Statistics, Reykavik, Iceland.","DOI":"10.1109\/ISIT.2014.6875045"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6241","DOI":"10.1109\/TIT.2012.2205894","article-title":"Optimal Phase Transitions in Compressed Sensing","volume":"58","author":"Wu","year":"2012","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1109\/JPROC.2010.2045630","article-title":"Precise Undersampling Theorems","volume":"98","author":"Donoho","year":"2010","journal-title":"Proc. IEEE"},{"key":"ref_28","unstructured":"Sarvotham, S., Baron, D., and Baraniuk, R. (2006, January 27\u201329). Measurements vs. Bits: Compressed Sensing Meets Informaiton Theory. Proceedings of the 44th Allerton Conference on Communication, Control, Computing, Monticello, IL, USA."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3065","DOI":"10.1109\/TIT.2012.2184848","article-title":"The Sampling Rate-Distortion Tradeoff for Sparsity Pattern Recovery in Compressed Sensing","volume":"58","author":"Reeves","year":"2012","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5111","DOI":"10.1109\/TIT.2010.2059891","article-title":"Information Theoretic Bounds for Compressed Sensing","volume":"56","author":"Aeron","year":"2010","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"4969","DOI":"10.1109\/TIT.2012.2201335","article-title":"Rate Distortion Behavior of Sparse Sources","volume":"58","author":"Weidmann","year":"2012","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_32","unstructured":"Oliver, C., and Quegan, S. (2004). Understanding Synthetic Aperture Radar Images, Scitech Publishing."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"912","DOI":"10.1109\/TAES.2011.5751234","article-title":"Theory and Application of SNR and Mutual Information Matched Illumination Waveforms","volume":"47","author":"Romero","year":"2011","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1109\/TAES.2014.120523","article-title":"Compressed Sensing Radar Amid Noise and Clutter Using Interference Covariance Information","volume":"50","author":"Tuuk","year":"2014","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"5758","DOI":"10.1109\/TIT.2009.2032726","article-title":"Necessary Sufficient Conditions for Sparisity Pattern Recovery","volume":"55","author":"Fletcher","year":"2009","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"5728","DOI":"10.1109\/TIT.2009.2032816","article-title":"Information-Theoretic Limits on Sparsity Recovery in the High-Dimensional and Noisy Setting","volume":"55","author":"Wainwright","year":"2009","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1109\/TIT.2009.2034796","article-title":"Shannon-Theoretic Limits on Noisy Compressive Sampling","volume":"56","author":"Tarokh","year":"2010","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"3396","DOI":"10.1109\/TIT.2013.2239356","article-title":"Accurate Prediction of Phase Transitions in Compressed Sensing via a Connection to Minimax Denoising","volume":"59","author":"Donoho","year":"2013","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Aeron, S., Zhao, M., and Saligrama, V. (2007, January 26\u201329). On Sensing Capacity of Sensor Networks for a Class of Linear Observation Models. Proceedings of IEEE\/SP 14th Workshop on Statistical Signal Processing (SSP\u201907), Madison, WI, USA.","DOI":"10.1109\/SSP.2007.4301286"},{"key":"ref_40","unstructured":"Cumming, I.G., and Wong, F.H. (2005). Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation, Artech House."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1109\/7.489523","article-title":"Man-made Target Backscattering Behavior: Applicability of Conventional Radar Resolution Theory","volume":"32","author":"Rihaczek","year":"1996","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1293","DOI":"10.1109\/18.243446","article-title":"Proper Complex Random Processes with Applications to Information Theory","volume":"39","author":"Neeser","year":"1993","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"4655","DOI":"10.1109\/TIT.2007.909108","article-title":"Signal Recovery from Random Measurements via Orthogonal Matching Pursuit","volume":"53","author":"Tropp","year":"2007","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/j.acha.2008.07.002","article-title":"CoSaMP: Iterative Signal Recovery from Incomplete and Inaccurate Samples","volume":"26","author":"Needell","year":"2009","journal-title":"Appl. Comput. Harmonic Anal."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"017003","DOI":"10.1117\/1.2150368","article-title":"Feature-Preserving Regularization Method for Complex-Valued Inverse Problems with Application to Coherent Imaging","volume":"45","author":"Karl","year":"2006","journal-title":"Opt. Eng."},{"key":"ref_46","unstructured":"Rilling, G., Davies, M., and Mulgrew, B. (2009, January 6\u20139). Compressed Sensing Based Compression of SAR Raw Data. Proceedings of the SPARS\u201909\u2014Signal Processing with Adaptive Sparse Structured Representations, Saint Malo, France."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2481","DOI":"10.1109\/TIP.2011.2176743","article-title":"Solving Inverse Problems With Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity","volume":"21","author":"Yu","year":"2012","journal-title":"IEEE Trans. Image Process."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2265","DOI":"10.1109\/TSP.2014.2309560","article-title":"Reconstruction of Signals Drawn From a Gaussian Mixture via Noisy Compressive Measurements","volume":"62","author":"Renna","year":"2014","journal-title":"IEEE Trans. Signal. Process."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1109\/TSP.2009.2027773","article-title":"Bayesian Compressive Sensing via Belief Propagation","volume":"58","author":"Baron","year":"2010","journal-title":"IEEE Trans. Signal. Process."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Barbier, J., Krzakala, F., Mezard, M., and Zdeborova, L. (2012, January 1\u20135). Compressed Sensing of Approximately-Sparse Signals: Phase Transitions and Optimal Reconstruction. Proceedings 50th Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, USA.","DOI":"10.1109\/Allerton.2012.6483300"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1129","DOI":"10.1109\/TGRS.2003.810710","article-title":"Efficient Transmission and Classification of Hyperspectral Image Data","volume":"41","author":"Jia","year":"2003","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"776","DOI":"10.1109\/18.986045","article-title":"Joint Source-Channel Coding of a Gaussian Mixture Source over the Gaussian Broadcast Channel","volume":"48","author":"Reznic","year":"2002","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_53","unstructured":"Davenport, M.A., Duarte, M.F., Eldar, Y.C., and Kutyniok, G. (2012). Compressed Sensing: Theory and Applications, Cambridge University Press."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1016\/j.crma.2008.03.014","article-title":"The Restricted Isometry Property and Its Implications for Compressed Sensing","volume":"346","year":"2008","journal-title":"Comptes Rendus Math."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1109\/MSP.2014.2312834","article-title":"Sparsity-Driven Synthetic Aperture Radar Imaging: Reconstruction, Autofocusing, Moving Targets, and Compressed Sensing","volume":"31","author":"Stojanovic","year":"2014","journal-title":"IEEE Signal. Process. 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