{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T15:40:57Z","timestamp":1772206857248,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"14","license":[{"start":{"date-parts":[[2021,7,10]],"date-time":"2021-07-10T00:00:00Z","timestamp":1625875200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100009448","name":"Universit\u00e0 degli Studi della Campania Luigi Vanvitelli","doi-asserted-by":"publisher","award":["B66J20000680005."],"award-info":[{"award-number":["B66J20000680005."]}],"id":[{"id":"10.13039\/501100009448","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper deals with an inverse scattering problem under a linearized scattering model for a multi-static\/multi-frequency configuration. The focus is on the determination of a sampling strategy that allows the reduction of the number of measurement points and frequencies and at the same time keeping the same achievable performance in the reconstructions as for full data acquisition. For the sake of simplicity, a 2D scalar geometry is addressed, and the scattered far-field data are collected. The relevant scattering operator exhibits a singular value spectrum that abruptly decays (i.e., a step-like behavior) beyond a certain index, which identifies the so-called number of degrees of freedom (NDF) of the problem. Accordingly, the sampling strategy is derived by looking for a discrete finite set of data points for which the arising semi-discrete scattering operator approximation can reproduce the most significant part of the singular spectrum, i.e., the singular values preceding the abrupt decay. To this end, the observation variables are suitably transformed so that Fourier-based arguments can be used. The arising sampling grid returns several data that is close to the NDF. Unfortunately, the resulting data points (in the angle-frequency domain) leading to a complicated measurement configuration which requires collecting the data at different spatial positions for each different frequency. To simplify the measurement configuration, a suboptimal sampling strategy is then proposed which, by an iterative procedure, enforces the sampling points to belong to a rectangular grid in the angle-frequency domain. As a result of this procedure, the overall data points (i.e., the couples angle-frequency) actually increase but the number of different angles and frequencies reduce and lead to a measurement configuration that is more practical to implement. A few numerical examples are included to check the proposed sampling scheme.<\/jats:p>","DOI":"10.3390\/s21144724","type":"journal-article","created":{"date-parts":[[2021,7,11]],"date-time":"2021-07-11T22:16:48Z","timestamp":1626041808000},"page":"4724","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Scattered Far-Field Sampling in Multi-Static Multi-Frequency Configuration"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9266-5880","authenticated-orcid":false,"given":"Maria Antonia","family":"Maisto","sequence":"first","affiliation":[{"name":"Department of Engineering, University of Campania, 81031 Aversa, CE, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2832-3736","authenticated-orcid":false,"given":"Mehdi","family":"Masoodi","sequence":"additional","affiliation":[{"name":"Department of Engineering, University of Campania, 81031 Aversa, CE, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2139-3398","authenticated-orcid":false,"given":"Giovanni","family":"Leone","sequence":"additional","affiliation":[{"name":"Department of Engineering, University of Campania, 81031 Aversa, CE, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1808-4671","authenticated-orcid":false,"given":"Raffaele","family":"Solimene","sequence":"additional","affiliation":[{"name":"Department of Engineering, University of Campania, 81031 Aversa, CE, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5861-5388","authenticated-orcid":false,"given":"Rocco","family":"Pierri","sequence":"additional","affiliation":[{"name":"Department of Engineering, University of Campania, 81031 Aversa, CE, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Devaney, A.J. (2012). Mathematical Foundations of Imaging, Tomography and Wavefield Inversion, Cambridge University Press.","DOI":"10.1017\/CBO9781139047838"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"538","DOI":"10.1109\/TAP.2007.891568","article-title":"Stochastic optimization methods applied to microwave imaging: A review","volume":"55","author":"Pastorino","year":"2007","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"283","DOI":"10.2528\/PIER04102902","article-title":"Microwave imaging of buried inhomogeneous objects using parallel genetic algorithm combined with FDTD method","volume":"53","author":"Chen","year":"2005","journal-title":"Prog. Electromagn. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1109\/LGRS.2017.2788699","article-title":"Frequency domain inverse profiling of buried dielectric elliptical-cylindrical objects using evolutionary programming","volume":"15","author":"Hajebi","year":"2018","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1305","DOI":"10.1109\/TGRS.2016.2622061","article-title":"Multifrequency particle swarm optimization for enhanced multiresolution GPR microwave imaging","volume":"55","author":"Sallucci","year":"2017","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1052","DOI":"10.1109\/LSP.2016.2579647","article-title":"A recursive born approach to nonlinear inverse scattering","volume":"23","author":"Kamilov","year":"2016","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1596","DOI":"10.1109\/36.934091","article-title":"On the local minima in a tomographic imaging technique","volume":"39","author":"Isernia","year":"2001","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"8837","DOI":"10.1364\/OE.14.008837","article-title":"A family of approximations spanning the Born and Rytov scattering series","volume":"14","author":"Marks","year":"2006","journal-title":"Opt. Exp."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2062","DOI":"10.1109\/TAP.2006.877170","article-title":"On the Role of Measurement Configuration in Contactless GPR Data Processing by Means of Linear Inverse Scattering","volume":"54","author":"Persico","year":"2006","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1875","DOI":"10.1109\/TAP.2005.848468","article-title":"The role of the measurement configuration in inverse scattering from buried objects under the born approximation","volume":"53","author":"Persico","year":"2005","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"123005","DOI":"10.1088\/0266-5611\/25\/12\/123005","article-title":"Problems in synthetic-aperture radar imaging","volume":"25","author":"Cheney","year":"2009","journal-title":"Inverse Probl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2266","DOI":"10.1364\/JOSAA.30.002266","article-title":"Role of diversity on the singular values of linear scattering operators: The case of strip objects","volume":"30","author":"Solimene","year":"2013","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Maisto, M.A., Pierri, R., and Solimene, R. (2020). Spatial Sampling in Monostatic Radar Imaging. IEEE Geosci. Remote Sens. Lett.","DOI":"10.36227\/techrxiv.12690155.v1"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1109\/8.662654","article-title":"Representation of Electromagnetic Fields over Arbitrary Surfaces by a Finite and Nonredundant Number of Samples","volume":"46","author":"Bucci","year":"1998","journal-title":"IEEE Trans. Antennas Propagat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1109\/OJAP.2020.3037678","article-title":"Sensor Arrangement in Monostatic Subsurface Radar Imaging","volume":"2","author":"Maisto","year":"2021","journal-title":"IEEE Open J. Antennas Propag."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1109\/78.370637","article-title":"Selection of observations in signal reconstruction","volume":"43","author":"Reeves","year":"1995","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1109\/TSP.2008.2007095","article-title":"Sensor selection via convex optimization","volume":"57","author":"Joshi","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1135","DOI":"10.1109\/TSP.2014.2299518","article-title":"Near-optimal sensor placement for linear inverse problems","volume":"62","author":"Ranieri","year":"2014","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"5595","DOI":"10.1109\/TSP.2016.2573767","article-title":"Sensor placement by maximal projection on minimum eigenspace for linear inverse problems","volume":"64","author":"Jiang","year":"2015","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"648","DOI":"10.1109\/TCI.2018.2875332","article-title":"Sampling design of synthetic volume arrays for three-dimensional microwave imaging","volume":"4","author":"Wang","year":"2018","journal-title":"IEEE Trans. Comput. Imaging"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1094","DOI":"10.1109\/LAWP.2016.2622713","article-title":"Singular value optimization in inverse electromagnetic scattering","volume":"16","author":"Capozzoli","year":"2017","journal-title":"IEEE Antennas Wirel. Propag. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2040","DOI":"10.1364\/JOSAA.2.002040","article-title":"Essential dimension as a well-defined number of degrees of freedom of finite convolution operators appearing in optics","volume":"2","author":"Newsam","year":"1985","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"892","DOI":"10.1364\/JOSAA.17.000892","article-title":"Electromagnetic degrees of freedom of an optical system","volume":"17","author":"Piestun","year":"2000","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02403043","article-title":"On the characteristic values of linear integral equations","volume":"57","author":"Hille","year":"1931","journal-title":"Acta Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"39","DOI":"10.2528\/PIERC17090903","article-title":"Information Content in Inverse Source with Symmetry and Support Priors","volume":"80","author":"Solimene","year":"2018","journal-title":"Prog. Electromagn. Res. C"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"46","DOI":"10.7716\/aem.v5i2.396","article-title":"Metric entropy in linear inverse scattering","volume":"5","author":"Maisto","year":"2016","journal-title":"Adv. Electromagn."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1364\/JOSAA.35.000755","article-title":"Inverse Source in Near Field: The Case of Strip Current","volume":"35","author":"Solimene","year":"2018","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"025603","DOI":"10.1088\/2040-8978\/18\/2\/025603","article-title":"Inverse scattering in the presence of a reflecting plane","volume":"18","author":"Solimene","year":"2015","journal-title":"J. Opt."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1109\/8.29386","article-title":"On the degrees of freedom of scattered fields","volume":"37","author":"Bucci","year":"1989","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Sekehravani, E.A., Leone, G., and Pierri, R. (2021). NDF of Scattered Fields for Strip Geometries. Electronics, 10.","DOI":"10.3390\/electronics10020202"},{"key":"ref_31","first-page":"539","article-title":"Linear inverse and ill-posed problems","volume":"75","author":"Bertero","year":"1989","journal-title":"Adv. Electron. Electron Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"826","DOI":"10.1364\/JOSAA.36.000826","article-title":"Resolution limits in inverse source problem for strip currents not in Fresnel zone","volume":"36","author":"Maisto","year":"2019","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1137\/0117035","article-title":"\u03f5-entropy and approximation of bandlimited functions","volume":"17","author":"Jagerman","year":"1969","journal-title":"SIAM J. Appl. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1364\/JOSAA.22.000434","article-title":"Sampling-theory approach to eigenwavefronts of imaging systems","volume":"22","author":"Khare","year":"2005","journal-title":"JOSA A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1565","DOI":"10.1109\/PROC.1977.10771","article-title":"The Shannon sampling theorem\u2014Its various extensions and applications: A tutorial review","volume":"65","author":"Jerri","year":"1977","journal-title":"Proc. IEEE"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1364\/JOSAA.36.000353","article-title":"Sampling approach for singular system computation of a radiation operator","volume":"36","author":"Maisto","year":"2019","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Maisto, M.A., Pierri, R., and Solimene, R. (2020). Near-Field Warping Sampling Scheme for Broad-Side Antenna Characterization. Electronics, 9.","DOI":"10.3390\/electronics9061047"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"62255","DOI":"10.1109\/ACCESS.2021.3074786","article-title":"Efficient Planar Near-Field Measurements for Radiation Pattern Evaluation by a Warping Strategy","volume":"9","author":"Maisto","year":"2021","journal-title":"IEEE Access"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1454","DOI":"10.1109\/TGRS.2006.870402","article-title":"Derivation and comparison of SAR and frequency-wavenumber migration within a common inverse scalar wave problem formulation","volume":"44","author":"Gilmore","year":"2006","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_40","first-page":"26922703","article-title":"Modified Kirchhoff mi- gration for UWB MIMO array-based radar imaging","volume":"48","author":"Zhuge","year":"2010","journal-title":"IEEE Trans. Geosci. Remote Sens."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/14\/4724\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:28:46Z","timestamp":1760164126000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/14\/4724"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,10]]},"references-count":40,"journal-issue":{"issue":"14","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["s21144724"],"URL":"https:\/\/doi.org\/10.3390\/s21144724","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,10]]}}}