{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:04:53Z","timestamp":1760148293184,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,4,17]],"date-time":"2023-04-17T00:00:00Z","timestamp":1681689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Italian Ministry of University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The solution to an ill-posed linear inverse problem requires the use of regularization methods to achieve a stable approximation solution. One powerful approach is the truncated singular value decomposition (TSVD), but it requires an appropriate choice of the truncation level. One suitable option is to take into account the number of degrees of freedom (NDF) of the scattered field, which is defined by the step-like behavior of the singular values of the relevant operator. Then, the NDF can be estimated as the number of singular values preceding the knee or the exponential decay. Therefore, an analytical estimation of the NDF is significant for obtaining a stable, regularized solution. This paper addresses the analytical estimation of the NDF of the field scattered by the surface of a cube geometry for a single frequency and the multi-view case in the far-zone. In addition, a method is proposed to find the minimum numbers of plane waves and their directions to achieve the total estimated NDF. The main results are that the NDF is related to the measure of the surface of the cube and can be achieved by only considering a limited number of impinging plane waves. The efficiency of the theoretical discussion is demonstrated through a reconstruction application for microwave tomography of a dielectric object. Numerical examples are provided to confirm the theoretical results.<\/jats:p>","DOI":"10.3390\/s23084056","type":"journal-article","created":{"date-parts":[[2023,4,17]],"date-time":"2023-04-17T06:27:47Z","timestamp":1681712867000},"page":"4056","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Evaluation of the Number of Degrees of Freedom of the Field Scattered by a 3D Geometry"],"prefix":"10.3390","volume":"23","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":[[2023,4,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hadamard, J. (1953). Lectures on Cauchy\u2019s Problem in Linear Partial Differential Equations, Dover Publications.","DOI":"10.1063\/1.3061337"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1137\/S0895479897326432","article-title":"Tikhonov regularization and total least squares","volume":"21","author":"Golub","year":"1999","journal-title":"J. Matrix Anal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Bertero, M., and Boccacci, P. (1998). Introduction to Inverse Problems in Imaging, Institute of Physics.","DOI":"10.1887\/0750304359"},{"key":"ref_4","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 Propag."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2123","DOI":"10.1029\/97RS01826","article-title":"Electromagnetic inverse scattering: Retrievable information and measurement strategies","volume":"32","author":"Bucci","year":"1997","journal-title":"Radio Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1109\/8.29386","article-title":"On the degrees of freedom of scattered fields. IEEE Trans","volume":"37","author":"Bucci","year":"1989","journal-title":"Antennas Propag."},{"key":"ref_7","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 XXXIVth General Assembly and Scientific Symposium of the International Union of Radio Science, Rome, Italy."},{"key":"ref_8","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_9","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_10","doi-asserted-by":"crossref","unstructured":"Sekehravani, E.A., Leone, G., and Pierri, R. (2022). Performance of the Linear Model Scattering of 2D Full Object with Limited Data. Sensors, 22.","DOI":"10.3390\/s22103868"},{"key":"ref_11","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_12","first-page":"177","article-title":"Advanced sampling techniques in electromagnetics","volume":"1","author":"Bucci","year":"1993","journal-title":"Rev. Radio Sci."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Leone, G., Munno, F., and Pierri, R. (2020). Synthesis of Angle Arrays by the NDF of the Radiation Integral. IEEE Trans. Antennas Propag., 1.","DOI":"10.36227\/techrxiv.12682376.v1"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1109\/TAP.2005.863108","article-title":"On the role of the number of degrees of freedom of the field in MIMO channels","volume":"54","author":"Migliore","year":"2006","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1109\/TIT.2004.840892","article-title":"Degrees of freedom in multiple-antenna channels: A signal space approach","volume":"51","author":"Poon","year":"2005","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_16","unstructured":"Bliss, D.W., and Forsythe, K.W. (2003, January 7\u201310). Multiple-input multiple-output (MIMO) radar and imaging: Degrees of freedom and resolution. Proceedings of the Thirty-Seventh Asilomar Conference on Signals, Systems and Computers 2003, Pacific Grove, CA, USA."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1109\/TAP.2004.823999","article-title":"Reduction of truncation error in near-field measurements of antennas of base-station mobile communication systems","volume":"52","author":"Bolomey","year":"2004","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2805","DOI":"10.1109\/JSAC.2022.3196106","article-title":"Degrees of freedom in 3D linear large-scale antenna array communications\u2014a spatial bandwidth approach","volume":"40","author":"Ding","year":"2022","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3086","DOI":"10.1109\/TAP.2004.835277","article-title":"Plane-wave synthesis by an antenna-array and RCS determination: Theoretical approach and numerical simulations","volume":"52","author":"Stupfel","year":"2004","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_20","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\u2014Opt. Image Sci. Vis."},{"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":"Li, H. (2022, January 16\u201320). Degrees of Freedom in Scattered Fields for Trade-off in Joint Communications and Sensing. Proceedings of the ICC 2022-IEEE International Conference on Communications, Seoul, Korea.","DOI":"10.1109\/ICC45855.2022.9838816"},{"key":"ref_23","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_24","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\/23\/8\/4056\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:17:27Z","timestamp":1760123847000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/8\/4056"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,17]]},"references-count":24,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["s23084056"],"URL":"https:\/\/doi.org\/10.3390\/s23084056","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2023,4,17]]}}}