{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:25:45Z","timestamp":1760228745305,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,5,19]],"date-time":"2022-05-19T00:00:00Z","timestamp":1652918400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Inverse scattering problems stand at the center of many important imaging applications, such as geophysical explorations, radar imaging, and synthetic-aperture radar (SAR). Several methods have been proposed to solve them when the full data are available, usually providing satisfactory reconstructions. However, it is impossible to acquire the full data in many practical circumstances, such as target detection and ground penetrating radar (GPR); consequently, only limited data are available. Thus, this paper focuses on the mathematical analysis and some numerical simulations to estimate the achievable resolution in reconstructing an object from the knowledge of the scattered far-field when only limited data are available, with multi-view excitations at a single frequency. We focus on 2D full rectangular geometry as the investigation domain (ID). We also examine the number of degrees of freedom (NDF) and evaluate the point spread function (PSF). In particular, the NDF of the considered geometry can be estimated analytically. An approximated closed-form evaluation of the PSF is recalled, discussed, and compared with the exact one. Moreover, receiving, transmission, and angle sensing modes are considered to apply the analysis to more realistic scenarios to highlight the difference between the corresponding NDF and the resulting resolution performances. Finally, interesting numerical applications of the resolution analysis for the localization of a collection of point-like scatterers are presented to illustrate how it matches the expectations.<\/jats:p>","DOI":"10.3390\/s22103868","type":"journal-article","created":{"date-parts":[[2022,5,20]],"date-time":"2022-05-20T00:18:11Z","timestamp":1653005891000},"page":"3868","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Performance of the Linear Model Scattering of 2D Full Object with Limited Data"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2670-739X","authenticated-orcid":false,"given":"Ehsan Akbari","family":"Sekehravani","sequence":"first","affiliation":[{"name":"Department of Engineering, University of Campania \u201cLuigi Vanvitelli\u201d, I-81031 Aversa, 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 \u201cLuigi Vanvitelli\u201d, I-81031 Aversa, 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 \u201cLuigi Vanvitelli\u201d, I-81031 Aversa, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/0895-7177(90)90175-M","article-title":"Associate memories for system identification: Inverse problems in remote sensing","volume":"14","author":"Kagiwada","year":"1990","journal-title":"Math. Comput. Model."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1177\/0309133311420320","article-title":"Regularization for inverse models in remote sensing","volume":"36","author":"Wang","year":"2012","journal-title":"Prog. Phys. Geogr. Earth Environ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.matpur.2012.02.009","article-title":"Linearized inverse scattering based on seismic reverse time migration","volume":"98","author":"Stolk","year":"2012","journal-title":"J. Mat. Pures Appl."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Stolt, R.H., and Weglein, A.B. (2012). Seismic Imaging and Inversion, Cambridge University Press. Application of Linear Inverse Theory.","DOI":"10.1017\/CBO9781139056250"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1137\/140982787","article-title":"Inverse obstacle scattering in two dimensions with multiple frequency data and multiple angles of incidence","volume":"8","author":"Borges","year":"2015","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Cheney, M., and Borden, B. (2009). Fundamentals of Radar Imaging, SIAM.","DOI":"10.1137\/1.9780898719291"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Colton, D.L., and Kress, R. (1998). Inverse Acoustic and Electromagnetic Scattering Theory, Springer. [2nd ed.].","DOI":"10.1007\/978-3-662-03537-5"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1085","DOI":"10.1109\/LGRS.2012.2230314","article-title":"A linear inverse scattering algorithm for radar imaging in multipath environments","volume":"10","author":"Gennarelli","year":"2013","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_9","first-page":"415","article-title":"Derivation and discussion of the SAR migration algorithm within inverse scattering problem: Theoretical analysis","volume":"48","author":"Li","year":"2009","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1109\/MSP.2014.2311271","article-title":"SAR imaging algorithms and some unconventional applications: A unified mathematical overview","volume":"31","author":"Solimene","year":"2014","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"7305","DOI":"10.1109\/TGRS.2020.2981884","article-title":"A comparison of linear inverse scattering models for contactless GPR imaging","volume":"58","author":"Ludeno","year":"2020","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Persico, R. (2014). Introduction to Ground Penetrating Radar: Inverse Scattering and Data Processing, Wiley-IEEE Press.","DOI":"10.1002\/9781118835647"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6573","DOI":"10.1109\/TGRS.2015.2444391","article-title":"GPR prospecting through an inverse scattering frequency-hopping multi-focusing approach","volume":"53","author":"Salucci","year":"2015","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1926","DOI":"10.1109\/TIM.2006.884346","article-title":"A microwave tomographic imaging approach for multibistatic configuration: The choice of the frequency step","volume":"55","author":"Persico","year":"2006","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2630","DOI":"10.1109\/TAP.2008.927506","article-title":"Localizing thin metallic cylinders by a 2.5-D linear distributional approach: Experimental results","volume":"56","author":"Solimene","year":"2008","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"259","DOI":"10.2528\/PIER09021609","article-title":"TWI experimental results by a linear inverse scattering approach","volume":"91","author":"Solimene","year":"2009","journal-title":"Prog. Electromagn. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"251497","DOI":"10.1155\/2012\/251497","article-title":"Ultrawideband impulse radar through-the-wall imaging with compressive sensing","volume":"2012","author":"Zhang","year":"2012","journal-title":"Int. J. Antennas Propag."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1498","DOI":"10.1364\/JOSA.64.001498","article-title":"Degrees of freedom, and eigenfunctions, for the noisy image","volume":"64","author":"Bendinelli","year":"1974","journal-title":"J. Opt. Soc. Am."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Sekehravani, E.A., Leone, G., and Pierri, R. (September, January 28). NDF of the far zone field radiated by square sources. Proceedings of the 2021 XXXIV General Assembly and Scientific Symposium of the International Union of Radio Science, URSI GASS, Rome, Italy.","DOI":"10.23919\/URSIGASS51995.2021.9560561"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Sekehravani, E.A., Leone, G., and Pierri, R. (2021). NDF and PSF analysis in inverse source and scattering problems for circumference geometries. Electronics, 10.","DOI":"10.3390\/electronics10172157"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3882","DOI":"10.1109\/TAP.2006.886517","article-title":"Electromagnetic degrees of freedom in 2-D scattering environments","volume":"54","author":"Xu","year":"2006","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Akbari Sekehravani, E., Leone, G., and Pierri, R. (2021). NDF of scattered fields for strip geometries. Electronics, 10.","DOI":"10.2139\/ssrn.3919490"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1002\/lpor.201400467","article-title":"Solving inverse scattering problems in biological samples by quantitative phase imaging","volume":"10","author":"Kim","year":"2016","journal-title":"Laser Photonics Rev."},{"key":"ref_24","unstructured":"Silverstein, S.D., and Zheng, Y. (2003, January 9\u201312). Near-field inverse coherent imaging problems: Solutions, simulations and applications. Proceedings of the Thrity-Seventh Asilomar Conference on Signals, Systems & Computers, Pacific Grove, CA, USA."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Tomlins, P.H., Woolliams, P., Tedaldi, M., Beaumont, A., and Hart, C. (2008, January 18). Measurement of the three-dimensional point-spread function in an optical coherence tomography imaging system. Proceedings of the Coherence Domain Optical Methods and Optical Coherence Tomography in Biomedicine XII, International Society for Optics and Photonics, San Jose, CA, USA.","DOI":"10.1117\/12.766576"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Sekehravani, E.A., Leone, G., and Pierri, R. (2021). PSF Analysis of the inverse source and scattering problems for strip geometries. Electronics, 10.","DOI":"10.3390\/electronics10060754"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3661","DOI":"10.1109\/TGRS.2007.903403","article-title":"Localization of Interfaces Embedded in a Half-Space by a Linear Inverse Scattering Algorithm","volume":"45","author":"Brancaccio","year":"2007","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1398","DOI":"10.1109\/TAP.2004.829847","article-title":"Study of Resolution and Super Resolution in Electromagnetic Imaging for Half-Space Problems","volume":"52","author":"Cui","year":"2004","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"5624","DOI":"10.1109\/TGRS.2015.2426502","article-title":"Forward-looking ground-penetrating radar via a linear inverse scattering approach","volume":"53","author":"Catapano","year":"2015","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Sekehravani, E.A., Leone, G., and Pierri, R. (2021). Resolution of born scattering in curve geometries: Aspect-Limited observations and excitations. Electronics, 10.","DOI":"10.3390\/electronics10243089"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1109\/TAP.2005.863121","article-title":"Beyond physical optics SVD shape reconstruction of metallic cylinders","volume":"54","author":"Pierri","year":"2006","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/0030-4018(73)90091-6","article-title":"Shannon number and degrees of freedom of an image","volume":"7","author":"Gori","year":"1973","journal-title":"Opt. Commun."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/10\/3868\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:15:00Z","timestamp":1760138100000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/10\/3868"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,19]]},"references-count":32,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["s22103868"],"URL":"https:\/\/doi.org\/10.3390\/s22103868","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2022,5,19]]}}}