{"title":"以 C(R2)${cal C}(\\mathbf {R}^2)$ 上的曲线为中心的双数据循环拉顿变换的反演","authors":"Rafik Aramyan","doi":"10.1111/sapm.12722","DOIUrl":null,"url":null,"abstract":"<p>More often, in the mathematical literature, the injectivity of the spherical Radon transform (SRT) for compactly supported functions is considered. In this article, an additional condition, for the reconstruction of an unknown function <span></span><math>\n <semantics>\n <mrow>\n <mi>f</mi>\n <mo>∈</mo>\n <mi>C</mi>\n <mo>(</mo>\n <msup>\n <mi>R</mi>\n <mn>2</mn>\n </msup>\n <mo>)</mo>\n </mrow>\n <annotation>$f\\in C(\\mathbf {R}^2)$</annotation>\n </semantics></math> (the support can be noncompact) using the circular Radon transform (CRT) over circles centered on a smooth simple curve is found. It is proved that this problem is equivalent to the injectivity of a so-called two-data CRT over circles centered on a smooth curve (can be a segment). Also, we present an inversion formula of the transform that uses the local data of the circular integrals to reconstruct the unknown function. Such inversions are the mathematical base of modern modalities of imaging, such as thermo- and photoacoustic tomography and radar imaging, and have theoretical significance.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"153 2","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Inversion of the two-data circular Radon transform centered on a curve on \\n \\n \\n C\\n (\\n \\n R\\n 2\\n \\n )\\n \\n ${\\\\cal C}(\\\\mathbf {R}^2)$\",\"authors\":\"Rafik Aramyan\",\"doi\":\"10.1111/sapm.12722\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>More often, in the mathematical literature, the injectivity of the spherical Radon transform (SRT) for compactly supported functions is considered. In this article, an additional condition, for the reconstruction of an unknown function <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>f</mi>\\n <mo>∈</mo>\\n <mi>C</mi>\\n <mo>(</mo>\\n <msup>\\n <mi>R</mi>\\n <mn>2</mn>\\n </msup>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$f\\\\in C(\\\\mathbf {R}^2)$</annotation>\\n </semantics></math> (the support can be noncompact) using the circular Radon transform (CRT) over circles centered on a smooth simple curve is found. It is proved that this problem is equivalent to the injectivity of a so-called two-data CRT over circles centered on a smooth curve (can be a segment). Also, we present an inversion formula of the transform that uses the local data of the circular integrals to reconstruct the unknown function. Such inversions are the mathematical base of modern modalities of imaging, such as thermo- and photoacoustic tomography and radar imaging, and have theoretical significance.</p>\",\"PeriodicalId\":51174,\"journal\":{\"name\":\"Studies in Applied Mathematics\",\"volume\":\"153 2\",\"pages\":\"\"},\"PeriodicalIF\":2.6000,\"publicationDate\":\"2024-05-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Studies in Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1111/sapm.12722\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studies in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1111/sapm.12722","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Inversion of the two-data circular Radon transform centered on a curve on
C
(
R
2
)
${\cal C}(\mathbf {R}^2)$
More often, in the mathematical literature, the injectivity of the spherical Radon transform (SRT) for compactly supported functions is considered. In this article, an additional condition, for the reconstruction of an unknown function (the support can be noncompact) using the circular Radon transform (CRT) over circles centered on a smooth simple curve is found. It is proved that this problem is equivalent to the injectivity of a so-called two-data CRT over circles centered on a smooth curve (can be a segment). Also, we present an inversion formula of the transform that uses the local data of the circular integrals to reconstruct the unknown function. Such inversions are the mathematical base of modern modalities of imaging, such as thermo- and photoacoustic tomography and radar imaging, and have theoretical significance.
期刊介绍:
Studies in Applied Mathematics explores the interplay between mathematics and the applied disciplines. It publishes papers that advance the understanding of physical processes, or develop new mathematical techniques applicable to physical and real-world problems. Its main themes include (but are not limited to) nonlinear phenomena, mathematical modeling, integrable systems, asymptotic analysis, inverse problems, numerical analysis, dynamical systems, scientific computing and applications to areas such as fluid mechanics, mathematical biology, and optics.