{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T14:26:14Z","timestamp":1775744774838,"version":"3.50.1"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,24]],"date-time":"2020-12-24T00:00:00Z","timestamp":1608768000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key R&amp;D Program of China","award":["2017YFB0502700"],"award-info":[{"award-number":["2017YFB0502700"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Synthetic aperture radar (SAR) is a widely used remote sensing observation technique. However, SAR raw echo data may be lost during the process of data acquisition by radar platform. In this paper, the imaging problem of SAR echo signal with periodically missing data along the azimuth is analyzed and a novel imaging method is proposed. Firstly, the problem of artificial artifact targets caused by periodically missing data is explained in detail, and the corresponding mathematical model is established. Then, the recovery method based on the RELAX algorithm with periodic notches data is proposed. In addition, when the size of two-dimensional (2D) echo data are large, block restoration along the azimuth is proposed to reduce the amount of calculation. Finally, the advantages of the algorithm proposed in this paper is demonstrated by the points target simulated SAR echo data processing and the real raw SAR echo data processing. When the azimuth periodically missing data rate is 50%, the SAR echo data can be recovered and the well-focused image can be obtained. Comparing the image entropy value and structural similarity index (SSIM) of the focused image, it proves the superiority of the proposed algorithm in solving the imaging problem of SAR azimuth periodically missing data.<\/jats:p>","DOI":"10.3390\/s21010049","type":"journal-article","created":{"date-parts":[[2020,12,24]],"date-time":"2020-12-24T09:02:44Z","timestamp":1608800564000},"page":"49","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["SAR Image Formation Method with Azimuth Periodically Missing Data Based on RELAX Algorithm"],"prefix":"10.3390","volume":"21","author":[{"given":"Weixing","family":"Yang","sequence":"first","affiliation":[{"name":"Key Laboratory of Radar Imaging and Microwave Photonics &amp; Ministry of Education, College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daiyin","family":"Zhu","sequence":"additional","affiliation":[{"name":"Key Laboratory of Radar Imaging and Microwave Photonics &amp; Ministry of Education, College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1109\/MGRS.2013.2248301","article-title":"A tutorial on synthetic aperture radar","volume":"1","author":"Moreira","year":"2013","journal-title":"Geosci. Remote Sens. Mag. IEEE"},{"key":"ref_2","unstructured":"Cumming, I.G., and Wong, F.H. (2004). Digital Signal Processing of Synthetic Aperture Radar Data: Algorithms and Implementation, Artech House."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1109\/62.1001990","article-title":"Interrupted synthetic aperture radar (SAR)","volume":"17","author":"Salzman","year":"2002","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bruder, J.A., and Schneible, R. (2007, January 15\u201318). Interrupted SAR waveforms for high interrupt ratios. Proceedings of the 2007 IET International Conference on Radar Systems, Edinburgh, UK.","DOI":"10.1049\/cp:20070658"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Wang, Y., Li, J., and Stoica, P. (2005). Spectral Analysis of Signals: The Missing Data Case, Morgan & Claypool.","DOI":"10.1007\/978-3-031-02525-9"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1876","DOI":"10.1109\/TGRS.2014.2350255","article-title":"Reconstruction of Coherent Pairs of Synthetic Aperture Radar Data Acquired in Interrupted Mode","volume":"53","author":"Pinheiro","year":"2015","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"987","DOI":"10.1109\/TAES.2014.120519","article-title":"Interrupted SAR persistent surveillance via group sparse reconstruction of multipass data","volume":"50","author":"Stojanovic","year":"2014","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_8","unstructured":"Ahmed, N., and Underwood, C. (2010, January 7\u201310). Monostatic CW SAR Concept for Microsatellites. Proceedings of the 8th European Conference on Synthetic Aperture Radar, Aachen, Germany."},{"key":"ref_9","unstructured":"Broersen, P.M.T., Waele, S.D., and Bos, R. (2003, January 20\u201322). Estimation of autoregressive spectra with randomly missing data. Proceedings of the IEEE Instrumentation and Measurement Technology Conference, Vail, CO, USA."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1089","DOI":"10.1109\/TAES.2003.1238761","article-title":"Spectral analysis of periodically gapped data","volume":"39","author":"Larsson","year":"2003","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2163","DOI":"10.1086\/301572","article-title":"Adaptive filter bank approach to restoration and spectral analysis of gapped data","volume":"120","author":"Stoica","year":"2000","journal-title":"Astron. J."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/j.dsp.2004.10.004","article-title":"Nonparametric spectral analysis with missing data via the EM algorithm","volume":"15","author":"Wang","year":"2005","journal-title":"Digit. Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1109\/TAES.2010.5417172","article-title":"Source localization and sensing: A nonparametric iterative approach based on weighted least squares","volume":"46","author":"Yardibi","year":"2010","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1109\/LSP.2009.2014114","article-title":"Missing data recovery via a nonparametric iterative adaptive approach","volume":"16","author":"Stoica","year":"2009","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/JSTSP.2011.2168192","article-title":"Nonparametric missing sample spectral analysis and its applications to interrupted SAR","volume":"6","author":"Vu","year":"2012","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1109\/JSTSP.2009.2039181","article-title":"Compressed synthetic aperture radar","volume":"4","author":"Patel","year":"2010","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_17","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":"Mallorqui","year":"2010","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Baraniuk, R., and Steeghs, P. (2007, January 17\u201320). Compressive Radar Imaging. Proceedings of the IEEE Radar Conference, Boston, MA, USA.","DOI":"10.1109\/RADAR.2007.374203"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1755","DOI":"10.1007\/s11432-012-4632-5","article-title":"Sparse SAR imaging based on L1\/2 regularization","volume":"55","author":"Zeng","year":"2012","journal-title":"Sci. China Inf. Sci."},{"key":"ref_20","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_21","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1109\/JSTARS.2013.2263309","article-title":"Fast compressed sensing SAR imaging based on approximated observation","volume":"7","author":"Fang","year":"2014","journal-title":"IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/BF01827811","article-title":"An efficient algorithm for two-dimensional frequency estimation","volume":"7","author":"Li","year":"1996","journal-title":"Multidimens. Syst. Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Huan, S., Zhang, M., Dai, G., and Gan, H. (2020). Low Elevation Angle Estimation with Range Super-Resolution in Wideband Radar. Sensors, 20.","DOI":"10.3390\/s20113104"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1109\/7.745697","article-title":"Super resolution SAR imaging via parametric spectral estimation methods","volume":"35","author":"Bi","year":"1999","journal-title":"IEEE Tans. Aerosp. Electron. Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1109\/TAES.1982.309269","article-title":"Modeling and a correlation algorithm for spaceborne SAR signals","volume":"5","author":"Wu","year":"1982","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"786","DOI":"10.1109\/36.298008","article-title":"Precision SAR processing using chirp scaling","volume":"32","author":"Raney","year":"1994","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1109\/78.485924","article-title":"Efficient mixed-spectrum estimation with applications to target feature extraction","volume":"44","author":"Li","year":"1996","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1109\/7.670348","article-title":"Implementation of the RELAX algorithm","volume":"34","author":"Liu","year":"1998","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_29","unstructured":"Kragh, T. (2006, January 6\u20137). Monotonic iterative algorithm for minimum entropy autofocus. Proceedings of the Adaptive Sensor Array Processing (ASAP) Workshop, Lexington, MA, USA."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3425","DOI":"10.1109\/TSP.2015.2422686","article-title":"Fast Entropy Minimization Based Autofocusing Technique for ISAR Imaging","volume":"63","author":"Zhang","year":"2015","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1109\/TIP.2003.819861","article-title":"Image quality assessment: From error visibility to structural similarity","volume":"13","author":"Wang","year":"2004","journal-title":"Image Process. IEEE Trans."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/1\/49\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:45:30Z","timestamp":1760179530000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/1\/49"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,24]]},"references-count":31,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["s21010049"],"URL":"https:\/\/doi.org\/10.3390\/s21010049","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12,24]]}}}