{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:27:47Z","timestamp":1787318867641,"version":"build-2736575974"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Imaging Sci."],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>This article is concerned with the problem of reconstructing the curl and divergence of two-dimensional (2D) vector fields from ultrasound time-of-flight measurements taking the refraction of the emitted ultrasound rays into account. Usually, in 2D vector field tomography the refraction of the ultrasound signal is neglected; i.e., one assumes that the signal propagates along straight lines. We address refractive effects, assuming that the ultrasound signal propagates along the geodesics of a Riemannian metric that is associated with the refractive index which is due to Fermat's principle. The investigated vector field, however, is still defined on a domain in $\\mathbb{R}^2$ that is equipped with the Euclidean metric. We show a connection between the ray transforms and the Radon transform along geodesic curves which generalizes a well-known result from the Euclidean case. Relying on this relation, we develop an asymptotic reconstruction formula for computing the curl and divergence of the vector field using Fourier integral operators. This article concludes with detailed numerical tests proving on the one hand a good performance of our method and showing on the other hand an improvement compared with computations that assume a propagation along straight lines.<\/jats:p>","DOI":"10.1137\/100786770","type":"journal-article","created":{"date-parts":[[2011,1,18]],"date-time":"2011-01-18T18:06:43Z","timestamp":1295374003000},"page":"40-56","source":"Crossref","is-referenced-by-count":10,"title":["Tomographic Reconstruction of the Curl and Divergence of 2D Vector Fields Taking Refractions into Account"],"prefix":"10.1137","volume":"4","author":[{"given":"Tim","family":"Pfitzenreiter","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Thomas","family":"Schuster","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2011,1,18]]},"reference":[{"key":"R1","unstructured":"Yu. E. Anikonov,\n                      Some Methods for the Study of Multidimensional Inverse Problems for Differential Equations\n                      , \u201cNauka\" Sibirsk. Otdel., Novosibirsk, Russia, 1978 (in Russian)."},{"key":"R2","doi-asserted-by":"crossref","unstructured":"Yu. E. Anikonov,\n                      Multidimensional Inverse and Ill-Posed Problems for Differential Equations\n                      , VSP, Utrecht, The Netherlands, 1995.","DOI":"10.1515\/9783110271478"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1515\/jiip.1997.5.6.487"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160370503"},{"key":"R5","unstructured":"M. Born and E. Wolf,\n                      Principles of Optics\n                      , Springer, New York, 2006."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1515\/jiip.2000.8.2.161"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"V. Guillemin and S. Sternberg,\n                      Geometric Asymptotics\n                      , Math. Surveys 14, AMS, Providence, RI, 1977.","DOI":"10.1090\/surv\/014"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.3934\/ipi.2010.4.111"},{"key":"R9","unstructured":"P. Juhlin,\n                      Principles of Doppler Tomography\n                      , Technical report SE-221 00, Center for Mathematical Sciences, Lund Institute of Technology, Lund, Sweden, 1992."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/j.optcom.2005.04.070"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1111\/j.1365-246X.1989.tb00491.x"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/16\/3\/311"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/17\/4\/312"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"T. Schuster,\n                      The Method of Approximate Inverse: Theory and Applications\n                      , Lecture Notes in Math. 1906, Springer, Berlin, Heidelberg, New York, 2007.","DOI":"10.1007\/978-3-540-71227-5"},{"key":"R15","first-page":"articl","volume":"2008","author":"Schuster T.","year":"2008","journal-title":"J. Biomed. Imaging"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"V. A. Sharafutdinov,\n                      Integral Geometry of Tensor Fields\n                      , VSP, Utrecht, The Netherlands, 1994.","DOI":"10.1515\/9783110900095"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/11\/5\/008"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/BF02930983"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/11\/5\/009"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"P. Stefanov and G. Uhlmann,\n                      Boundary and lens rigidity, tensor tomography and analytic microlocal analysis\n                      , in Algebraic Analysis of Differential Equations, Springer, Tokyo, Berlin, Heidelberg, New York, 2008, pp. 275\u2013293.","DOI":"10.1007\/978-4-431-73240-2_23"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"F. Tr\u00e8ves,\n                      Introduction to Pseudodifferential and Fourier Integral Operators\n                      , Vols. 1 and 2, Plenum Press, New York, London, 1980.","DOI":"10.1007\/978-1-4684-8780-0_1"},{"key":"R22","unstructured":"A. Wernsd\u00f6rfer,\n                      Complete reconstruction of three-dimensional vector fields\n                      , in Proceedings of ECAPT 93, Karlsruhe, Germany, 1993, pp. 132\u2013135."}],"container-title":["SIAM Journal on Imaging Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/100786770","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:42:56Z","timestamp":1787316176000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/100786770"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1]]},"references-count":22,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2011,1]]}},"alternative-id":["10.1137\/100786770"],"URL":"https:\/\/doi.org\/10.1137\/100786770","relation":{},"ISSN":["1936-4954"],"issn-type":[{"value":"1936-4954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1]]}}}