Figure 1.
Illustration of complex $\mathscr{C}_0$
-
Previous Article
A scaled gradient method for digital tomographic image reconstruction
- IPI Home
- This Issue
-
Next Article
Scattering problems for perturbations of the multidimensional biharmonic operator
February
2018, 12(1): 229-237.
doi: 10.3934/ipi.2018009
Parametrices for the light ray transform on Minkowski spacetime
Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, USA |
We consider restricted light ray transforms arising from an inverse problem of finding cosmic strings. We construct a relative left parametrix for the transform on two tensors, which recovers the space-like and some light-like singularities of the two tensor.
Keywords:
Light ray transforms,
parametrix,
space-times,
Lagrangian distributions,
inverse problems.
Mathematics Subject Classification:
Primary: 53C65; Secondary: 35S30.
Citation:
Yiran Wang. Parametrices for the light ray transform on Minkowski spacetime. Inverse Problems and Imaging,
2018, 12
(1)
: 229-237.
doi: 10.3934/ipi.2018009
![]()
References:
| [1] |
J. Antoniano and G. Uhlmann,
A functional calculus for a class of pseudodifferential operators with singular symbols, Proc. Symp. Pure Math., 43 (1985), 5-16.
|
| [2] |
M. de Hoop, G. Uhlmann and A. Vasy,
Diffraction from conormal singularities, Annales Scientifiques de l'École Normale Supérieure, 4e serie, 48 (2015), 351-408.
doi: 10.24033/asens.2247. |
| [3] |
A. Greenleaf and A. Seeger,
Fourier integral operators with fold singularities, J. reine angew. Math., 455 (1994), 35-56.
|
| [4] |
A. Greenleaf and G. Uhlmann,
Nonlocal inversion formulas for the X-ray transform, Duke Math. J., 58 (1989), 205-240.
doi: 10.1215/S0012-7094-89-05811-0. |
| [5] |
A. Greenleaf and G. Uhlmann,
Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms, Annales de l'institut Fourier, 40 (1990), 443-466.
doi: 10.5802/aif.1220. |
| [6] |
A. Greenleaf and G. Uhlmann,
Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, Journal of Functional Analysis, 89 (1990), 202-232.
doi: 10.1016/0022-1236(90)90011-9. |
| [7] |
A. Greenleaf and G. Uhlmann,
Microlocal techniques in integral geometry, Contemporary Mathematics, 113 (1990), 121-135.
|
| [8] |
A. Greenleaf and G. Uhlmann,
Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms. Ⅱ, Duke Math. J., 64 (1991), 415-444.
doi: 10.1215/S0012-7094-91-06422-7. |
| [9] |
A. Greenleaf and G. Uhlmann,
Recovering singularities of a potential from singularities of scattering data, Communications in Mathematical Physics, 157 (1993), 549-572.
doi: 10.1007/BF02096882. |
| [10] |
V. Guillemin,
Cosmology in $(2+1) $-Dimensions, Cyclic Models, and Deformations of $M_{2, 1} $ Annals of Mathematics Studies, No. 121, Princeton University Press, 1989. |
| [11] |
V. Guillemin and G. Uhlmann,
Oscillatory integrals with singular symbols, Duke Math. J., 48 (1981), 251-267.
doi: 10.1215/S0012-7094-81-04814-6. |
| [12] |
L. Hörmander,
Fourier integral operators. Ⅰ, Acta Mathematica, 127 (1971), 79-183.
doi: 10.1007/BF02392052. |
| [13] |
L. Hörmander,
The Analysis of Linear Partial Differential Operators Ⅳ: Fourier Integral Operators Springer-Verlag, Berlin, Heidelberg, 2009. |
| [14] |
M. Lassas, L. Oksanen, P. Stefanov and G. Uhlmann, On the inverse problem of finding cosmic strings and other topological defects, preprint, arXiv: 1505.03123. |
| [15] |
R. Melrose and G. Uhlmann,
Lagrangian intersection and the Cauchy problem, Communications on Pure and Applied Mathematics, 32 (1979), 483-519.
doi: 10.1002/cpa.3160320403. |
| [16] |
B. Palacios, G. Uhlmann and Y. Wang,
Reducing streaking artifacts in quantitative susceptibility mapping, SIAM Journal of Imaging Sciences, 10 (2017), 1921-1934.
|
| [17] |
P. Stefanov, Support theorems for the light ray transform on analytic Lorentzian manifolds, Proc. Amer. Math. Soc., 145 (2017), 1259–1274. arXiv: 1504.01184. |
show all references
References:
| [1] |
J. Antoniano and G. Uhlmann,
A functional calculus for a class of pseudodifferential operators with singular symbols, Proc. Symp. Pure Math., 43 (1985), 5-16.
|
| [2] |
M. de Hoop, G. Uhlmann and A. Vasy,
Diffraction from conormal singularities, Annales Scientifiques de l'École Normale Supérieure, 4e serie, 48 (2015), 351-408.
doi: 10.24033/asens.2247. |
| [3] |
A. Greenleaf and A. Seeger,
Fourier integral operators with fold singularities, J. reine angew. Math., 455 (1994), 35-56.
|
| [4] |
A. Greenleaf and G. Uhlmann,
Nonlocal inversion formulas for the X-ray transform, Duke Math. J., 58 (1989), 205-240.
doi: 10.1215/S0012-7094-89-05811-0. |
| [5] |
A. Greenleaf and G. Uhlmann,
Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms, Annales de l'institut Fourier, 40 (1990), 443-466.
doi: 10.5802/aif.1220. |
| [6] |
A. Greenleaf and G. Uhlmann,
Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, Journal of Functional Analysis, 89 (1990), 202-232.
doi: 10.1016/0022-1236(90)90011-9. |
| [7] |
A. Greenleaf and G. Uhlmann,
Microlocal techniques in integral geometry, Contemporary Mathematics, 113 (1990), 121-135.
|
| [8] |
A. Greenleaf and G. Uhlmann,
Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms. Ⅱ, Duke Math. J., 64 (1991), 415-444.
doi: 10.1215/S0012-7094-91-06422-7. |
| [9] |
A. Greenleaf and G. Uhlmann,
Recovering singularities of a potential from singularities of scattering data, Communications in Mathematical Physics, 157 (1993), 549-572.
doi: 10.1007/BF02096882. |
| [10] |
V. Guillemin,
Cosmology in $(2+1) $-Dimensions, Cyclic Models, and Deformations of $M_{2, 1} $ Annals of Mathematics Studies, No. 121, Princeton University Press, 1989. |
| [11] |
V. Guillemin and G. Uhlmann,
Oscillatory integrals with singular symbols, Duke Math. J., 48 (1981), 251-267.
doi: 10.1215/S0012-7094-81-04814-6. |
| [12] |
L. Hörmander,
Fourier integral operators. Ⅰ, Acta Mathematica, 127 (1971), 79-183.
doi: 10.1007/BF02392052. |
| [13] |
L. Hörmander,
The Analysis of Linear Partial Differential Operators Ⅳ: Fourier Integral Operators Springer-Verlag, Berlin, Heidelberg, 2009. |
| [14] |
M. Lassas, L. Oksanen, P. Stefanov and G. Uhlmann, On the inverse problem of finding cosmic strings and other topological defects, preprint, arXiv: 1505.03123. |
| [15] |
R. Melrose and G. Uhlmann,
Lagrangian intersection and the Cauchy problem, Communications on Pure and Applied Mathematics, 32 (1979), 483-519.
doi: 10.1002/cpa.3160320403. |
| [16] |
B. Palacios, G. Uhlmann and Y. Wang,
Reducing streaking artifacts in quantitative susceptibility mapping, SIAM Journal of Imaging Sciences, 10 (2017), 1921-1934.
|
| [17] |
P. Stefanov, Support theorems for the light ray transform on analytic Lorentzian manifolds, Proc. Amer. Math. Soc., 145 (2017), 1259–1274. arXiv: 1504.01184. |
| [1] |
Wei Dai, Daoyuan Fang, Chengbo Wang. Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4985-4999. doi: 10.3934/dcds.2020208 |
| [2] |
Venkateswaran P. Krishnan, Ramesh Manna, Suman Kumar Sahoo, Vladimir A. Sharafutdinov. Momentum ray transforms. Inverse Problems and Imaging, 2019, 13 (3) : 679-701. doi: 10.3934/ipi.2019031 |
| [3] |
Nicholas Hoell, Guillaume Bal. Ray transforms on a conformal class of curves. Inverse Problems and Imaging, 2014, 8 (1) : 103-125. doi: 10.3934/ipi.2014.8.103 |
| [4] |
Siamak RabieniaHaratbar. Support theorem for the Light-Ray transform of vector fields on Minkowski spaces. Inverse Problems and Imaging, 2018, 12 (2) : 293-314. doi: 10.3934/ipi.2018013 |
| [5] |
Sari Lasanen. Non-Gaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems and Imaging, 2012, 6 (2) : 215-266. doi: 10.3934/ipi.2012.6.215 |
| [6] |
Jan Boman. Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform. Inverse Problems and Imaging, 2010, 4 (4) : 619-630. doi: 10.3934/ipi.2010.4.619 |
| [7] |
Alexander Balandin. The localized basis functions for scalar and vector 3D tomography and their ray transforms. Inverse Problems and Imaging, 2016, 10 (4) : 899-914. doi: 10.3934/ipi.2016026 |
| [8] |
Xiantao Xiao, Liwei Zhang, Jianzhong Zhang. On convergence of augmented Lagrangian method for inverse semi-definite quadratic programming problems. Journal of Industrial and Management Optimization, 2009, 5 (2) : 319-339. doi: 10.3934/jimo.2009.5.319 |
| [9] |
Azmy S. Ackleh, Ben G. Fitzpatrick, Horst R. Thieme. Rate distributions and survival of the fittest: a formulation on the space of measures. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 917-928. doi: 10.3934/dcdsb.2005.5.917 |
| [10] |
Seung Jun Chang, Jae Gil Choi. Generalized transforms and generalized convolution products associated with Gaussian paths on function space. Communications on Pure and Applied Analysis, 2020, 19 (1) : 371-389. doi: 10.3934/cpaa.2020019 |
| [11] |
T. J. Sullivan. Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors. Inverse Problems and Imaging, 2017, 11 (5) : 857-874. doi: 10.3934/ipi.2017040 |
| [12] |
Kaitlyn (Voccola) Muller. A reproducing kernel Hilbert space framework for inverse scattering problems within the Born approximation. Inverse Problems and Imaging, 2019, 13 (6) : 1327-1348. doi: 10.3934/ipi.2019058 |
| [13] |
María Guadalupe Morales, Zuzana Došlá, Francisco J. Mendoza. Riemann-Liouville derivative over the space of integrable distributions. Electronic Research Archive, 2020, 28 (2) : 567-587. doi: 10.3934/era.2020030 |
| [14] |
Evgeny L. Korotyaev. Estimates for solutions of KDV on the phase space of periodic distributions in terms of action variables. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 219-225. doi: 10.3934/dcds.2011.30.219 |
| [15] |
Wenqing Bao, Xianyi Wu, Xian Zhou. Optimal stopping problems with restricted stopping times. Journal of Industrial and Management Optimization, 2017, 13 (1) : 399-411. doi: 10.3934/jimo.2016023 |
| [16] |
Ji-Bo Wang, Bo Zhang, Hongyu He. A unified analysis for scheduling problems with variable processing times. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1063-1077. doi: 10.3934/jimo.2021008 |
| [17] |
Omer Gursoy, Kamal Adli Mehr, Nail Akar. Steady-state and first passage time distributions for waiting times in the $ MAP/M/s+G $ queueing model with generally distributed patience times. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021078 |
| [18] |
Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic and Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042 |
| [19] |
Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems and Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059 |
| [20] |
Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems and Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]