Uniqueness and Nonuniqueness in Inverse Problems for Elliptic Partial Differential Equations and Related Medical Imaging
Unique determination issues about inverse problems for elliptic partial differential equations in divergence form are summarized and discussed. The inverse problems include medical imaging problems including electrical impedance tomography (EIT), diffuse optical tomography (DOT), and inverse scatter...
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| Language: | English |
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Wiley
2015-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2015/908251 |
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| author | Kiwoon Kwon |
| author_facet | Kiwoon Kwon |
| author_sort | Kiwoon Kwon |
| collection | DOAJ |
| description | Unique determination issues about inverse problems for elliptic partial differential equations in divergence form are summarized and discussed. The inverse problems include medical imaging problems including electrical impedance
tomography (EIT), diffuse optical tomography (DOT), and inverse scattering problem (ISP) which is an elliptic inverse problem closely related with DOT and EIT. If the coefficient inside the divergence is isotropic, many uniqueness results are known.
However, it is known that inverse problem with anisotropic coefficients has many possible coefficients giving the same
measured data for the inverse problem. For anisotropic coefficient with anomaly with or without jumps from known or unknown background, nonuniqueness of the inverse problems is discussed and the relation to cloaking or illusion of the anomaly is explained. The uniqueness and nonuniqueness issues are discussed firstly for EIT and secondly for ISP in similar arguments.
Arguing the relation between source-to-detector map and Dirichlet-to-Neumann map in DOT and the uniqueness and nonuniqueness of DOT are also explained. |
| format | Article |
| id | doaj-art-ba5a63a0e34442edaefcb32bcf51350c |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-ba5a63a0e34442edaefcb32bcf51350c2025-02-03T01:31:21ZengWileyAdvances in Mathematical Physics1687-91201687-91392015-01-01201510.1155/2015/908251908251Uniqueness and Nonuniqueness in Inverse Problems for Elliptic Partial Differential Equations and Related Medical ImagingKiwoon Kwon0Department of Mathematics, Dongguk University, Seoul 100-715, Republic of KoreaUnique determination issues about inverse problems for elliptic partial differential equations in divergence form are summarized and discussed. The inverse problems include medical imaging problems including electrical impedance tomography (EIT), diffuse optical tomography (DOT), and inverse scattering problem (ISP) which is an elliptic inverse problem closely related with DOT and EIT. If the coefficient inside the divergence is isotropic, many uniqueness results are known. However, it is known that inverse problem with anisotropic coefficients has many possible coefficients giving the same measured data for the inverse problem. For anisotropic coefficient with anomaly with or without jumps from known or unknown background, nonuniqueness of the inverse problems is discussed and the relation to cloaking or illusion of the anomaly is explained. The uniqueness and nonuniqueness issues are discussed firstly for EIT and secondly for ISP in similar arguments. Arguing the relation between source-to-detector map and Dirichlet-to-Neumann map in DOT and the uniqueness and nonuniqueness of DOT are also explained.http://dx.doi.org/10.1155/2015/908251 |
| spellingShingle | Kiwoon Kwon Uniqueness and Nonuniqueness in Inverse Problems for Elliptic Partial Differential Equations and Related Medical Imaging Advances in Mathematical Physics |
| title | Uniqueness and Nonuniqueness in Inverse Problems for Elliptic Partial Differential Equations and Related Medical Imaging |
| title_full | Uniqueness and Nonuniqueness in Inverse Problems for Elliptic Partial Differential Equations and Related Medical Imaging |
| title_fullStr | Uniqueness and Nonuniqueness in Inverse Problems for Elliptic Partial Differential Equations and Related Medical Imaging |
| title_full_unstemmed | Uniqueness and Nonuniqueness in Inverse Problems for Elliptic Partial Differential Equations and Related Medical Imaging |
| title_short | Uniqueness and Nonuniqueness in Inverse Problems for Elliptic Partial Differential Equations and Related Medical Imaging |
| title_sort | uniqueness and nonuniqueness in inverse problems for elliptic partial differential equations and related medical imaging |
| url | http://dx.doi.org/10.1155/2015/908251 |
| work_keys_str_mv | AT kiwoonkwon uniquenessandnonuniquenessininverseproblemsforellipticpartialdifferentialequationsandrelatedmedicalimaging |