On generalization of exponential ray transform in tomography

E. Yu. Derevtsov

Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 18 (2018) no. 4, pp. 29-42

Résumé

A generalization of the operator of exponential ray transform, which values are the initial data for the problem of emission computer tomography, is suggested. The generalization is based on physical facts lying at a foundation of photometry and wave optics, and is realized towards three directions. Namely, an absorption becomes a complex-valued function, integral moments of source distributions with weight are considered, and a dependence of sources on time is introduced. Connections between exponential ray transforms of various orders are established and their differential equations are obtained. Uniqueness theorems for boundary-value and initial-boundary value problems for the derived equations are proved. Close connections of generalized exponential ray transforms with integral geometry of tensor fields and tomography problems are marked.

Détail

Keywords: emission tomography, photometry, wave optics, non-stationary source, exponential ray transform, boundary-value problem.
Mots-clés : absorption, transport equation

Bibliographie

[1] Budinger T. F.,Gullberg G. T., Huesman R. H., “Emission Computed Tomograthy”, Image Reconstruction from Projections, ed. G. T. Herman, Springer, 1979, 147–246 | DOI

[2] Natterer F., The Mathematics of Computerized Tomography, Wiley, Chichester, 1986 | MR | Zbl

[3] Mueller R. K., Kaveh M., Wade G., “Reconstructive Tomography and Applications to Ultrasonic”, Proc. of the IEEE, 67 (1979), 567–587 | DOI

[4] Ball J. S., Johnson S. A., Stenger F., “Explicit Inversion of the Helmholtz Equation for Ultrasound Insonification and Spherical Detection”, Acoustical Imaging, 9, ed. K. Y. Wang, 1980, 451–461 | DOI

[5] V. A. Sharafutdinov, “A problem of integral geometry for generalized tensor fields on $R^n$”, Dokl. Akad. Nauk SSSR, 286:2 (1986), 305–307 (in Russian) | MR | Zbl

[6] Sharafutdinov V. A., Integral Geometry of Tensor Fields, VSP, Urtecht, The Netherlands, 1994 | MR

[7] Schmitt U., Louis A. K., “Efficient Algorithms for the Regularization of Dynamic Inverse Problems: I. Theory”, Inverse Problems, 18 (2002), 645–658 | DOI | MR | Zbl

[8] Schmitt U., Louis A. K, Wolters C., Vauhkonen M., “Efficient Algorithms for the Regularization of Dynamic Inverse Problems: II. Applications”, Inverse Problems, 18 (2002), 659–676 | DOI | MR | Zbl

[9] Hahn B., Louis A. K., “Reconstruction in the Three-Dimensional Parallel Scanning Geometry with Application in Synchrotron-Based X-Ray Tomography”, Inverse Problems, 28 (2012), 045013 | DOI | MR | Zbl

[10] V. R. Kireitov, “On a problem of reconstruction of an optical surface by its images”, Funktsional'ny analiz i ego prilozhenia, 10:3 (1975), 45–54 (in Russian) | MR

[11] Born M., Wolf E., Principles of Optics, Cambridge University Press, 1999 | MR

[12] V. R. Kireitov, Inverse Problems of Photometry, Computing Center of the USSR Acad. Sci., Novosibirsk, 1983 (in Russian)

[13] Case K. M., Zweifel P. F., Linear Transport Theory, Addison-Wesley Publ. Company, 1967 | MR | Zbl

[14] Goodman J. W., Introduction to Fourier Optics, McGraw-Hill Book Company, 1968

[15] Derevtsov E. Yu., Kleshchev A. G., Sharafutdinov V. A., “Numerical Solution of the Emission $2D$-Tomography Problem for a Medium with Absorption and Refraction”, J. Inverse and Ill-Posed Problems, 7:1 (1999), 83–103 | DOI | MR | Zbl