-
Previous Article
Quantitative photoacoustic tomography with variable index of refraction
- IPI Home
- This Issue
-
Next Article
Recovering boundary shape and conductivity in electrical impedance tomography
Spherical mean transform: A PDE approach
1. | Department of Mathematics, University of Idaho, Moscow, Idaho 83844, United States |
We also discuss how the approach works for the hyperbolic and spherical spaces.
References:
[1] |
M. Agranovsky and P. Kuchment, The support theorem for the single radius spherical mean transform, Memoirs on Differential Equations and Mathematical Physics, 52 (2011), 1-16. |
[2] |
M. Agranovsky and E. Quinto, Injectivity sets for the Radon transform over circles and complete systems of radial functions, J. Funct. Anal., 139 (1996), 383-414.
doi: 10.1006/jfan.1996.0090. |
[3] |
G. Beylkin, The fundamental identity for iterated spherical means and the inversion formula for diffraction tomography and inverse scattering, J. Math. Phys., 24(1983), 1399-1400.
doi: 10.1063/1.525873. |
[4] |
G. Beylkin, Iterated spherical means in linearized inverse problems, in "Conference on Inverse Scattering: Theory and Application" (Tulsa, Okla., 1983), SIAM, Philadelphia, PA, (1983), 112-117. |
[5] |
R. Courant and D. Hilbert, "Methods of Mathematical Physics. Vol. II: Partial Differential Equations," (Vol. II by R. Courant), Interscience Publishers, New York-London, 1962. |
[6] |
M. Courdurier, F. Noo, M. Defrise and H. Kudo, Solving the interior problem of computed tomography using a priori knowledge, Inverse problems, 24 (2008), 065001, 27 pp.
doi: 10.1088/0266-5611/24/6/065001. |
[7] |
A. Cormack and E. Quinto, A Radon transform on spheres through the origin in $R^n$ and applications to the Darboux equation, Trans. Amer. Math. Soc., 260 (1980), 575-581.
doi: 10.2307/1998023. |
[8] |
C. Epstein and B. Kleiner, Spherical means in annular regions, Comm. Pure Appl. Math., 46 (1993), 441-451.
doi: 10.1002/cpa.3160460307. |
[9] |
D. Finch, S. Patch and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240 (electronic).
doi: 10.1137/S0036141002417814. |
[10] |
S. Helgason, "Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions," Pure and Applied Mathematics, 113, Academic Press, Inc., Orlando, FL, 1984. |
[11] |
F. John, "Plane Waves and Spherical Means Applied to Partial Differential Equations," Reprint of the 1955 original, Springer-Verlag, New York-Berlin, 1981. |
[12] |
H. Kudo, M. Courdurier, F. Noo and M. Defrise, Tiny a priori knowledge solves the interior problem in computed tomography, Physics in Medicine and Biology, 53 (2008), 2207. |
[13] |
V. Lin and A. Pinkus, Fundamentality of ridge functions, J. Approx. Theory, 75 (1993), 295-311.
doi: 10.1006/jath.1993.1104. |
[14] |
V. Lin and A. Pinkus, Approximation of multivariate functions, in "Advances in Computational Mathematics" (New Delhi, 1993), Ser. Approx. Decompos., 4, World Sci. Publ., River Edge, NJ, (1994), 257-265. |
[15] |
L. {Nguyen}, Range description for a spherical mean transform on spaces of constant curvatures, arXiv:1107.1746, (2011). |
show all references
References:
[1] |
M. Agranovsky and P. Kuchment, The support theorem for the single radius spherical mean transform, Memoirs on Differential Equations and Mathematical Physics, 52 (2011), 1-16. |
[2] |
M. Agranovsky and E. Quinto, Injectivity sets for the Radon transform over circles and complete systems of radial functions, J. Funct. Anal., 139 (1996), 383-414.
doi: 10.1006/jfan.1996.0090. |
[3] |
G. Beylkin, The fundamental identity for iterated spherical means and the inversion formula for diffraction tomography and inverse scattering, J. Math. Phys., 24(1983), 1399-1400.
doi: 10.1063/1.525873. |
[4] |
G. Beylkin, Iterated spherical means in linearized inverse problems, in "Conference on Inverse Scattering: Theory and Application" (Tulsa, Okla., 1983), SIAM, Philadelphia, PA, (1983), 112-117. |
[5] |
R. Courant and D. Hilbert, "Methods of Mathematical Physics. Vol. II: Partial Differential Equations," (Vol. II by R. Courant), Interscience Publishers, New York-London, 1962. |
[6] |
M. Courdurier, F. Noo, M. Defrise and H. Kudo, Solving the interior problem of computed tomography using a priori knowledge, Inverse problems, 24 (2008), 065001, 27 pp.
doi: 10.1088/0266-5611/24/6/065001. |
[7] |
A. Cormack and E. Quinto, A Radon transform on spheres through the origin in $R^n$ and applications to the Darboux equation, Trans. Amer. Math. Soc., 260 (1980), 575-581.
doi: 10.2307/1998023. |
[8] |
C. Epstein and B. Kleiner, Spherical means in annular regions, Comm. Pure Appl. Math., 46 (1993), 441-451.
doi: 10.1002/cpa.3160460307. |
[9] |
D. Finch, S. Patch and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240 (electronic).
doi: 10.1137/S0036141002417814. |
[10] |
S. Helgason, "Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions," Pure and Applied Mathematics, 113, Academic Press, Inc., Orlando, FL, 1984. |
[11] |
F. John, "Plane Waves and Spherical Means Applied to Partial Differential Equations," Reprint of the 1955 original, Springer-Verlag, New York-Berlin, 1981. |
[12] |
H. Kudo, M. Courdurier, F. Noo and M. Defrise, Tiny a priori knowledge solves the interior problem in computed tomography, Physics in Medicine and Biology, 53 (2008), 2207. |
[13] |
V. Lin and A. Pinkus, Fundamentality of ridge functions, J. Approx. Theory, 75 (1993), 295-311.
doi: 10.1006/jath.1993.1104. |
[14] |
V. Lin and A. Pinkus, Approximation of multivariate functions, in "Advances in Computational Mathematics" (New Delhi, 1993), Ser. Approx. Decompos., 4, World Sci. Publ., River Edge, NJ, (1994), 257-265. |
[15] |
L. {Nguyen}, Range description for a spherical mean transform on spaces of constant curvatures, arXiv:1107.1746, (2011). |
[1] |
Marcus A. Khuri. On the local solvability of Darboux's equation. Conference Publications, 2009, 2009 (Special) : 451-456. doi: 10.3934/proc.2009.2009.451 |
[2] |
Mark Agranovsky, David Finch, Peter Kuchment. Range conditions for a spherical mean transform. Inverse Problems and Imaging, 2009, 3 (3) : 373-382. doi: 10.3934/ipi.2009.3.373 |
[3] |
Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations and Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013 |
[4] |
Thaís Jordão, Xingping Sun. General types of spherical mean operators and $K$-functionals of fractional orders. Communications on Pure and Applied Analysis, 2015, 14 (3) : 743-757. doi: 10.3934/cpaa.2015.14.743 |
[5] |
Torsten Görner, Ralf Hielscher, Stefan Kunis. Efficient and accurate computation of spherical mean values at scattered center points. Inverse Problems and Imaging, 2012, 6 (4) : 645-661. doi: 10.3934/ipi.2012.6.645 |
[6] |
Jan Haskovec, Nader Masmoudi, Christian Schmeiser, Mohamed Lazhar Tayeb. The Spherical Harmonics Expansion model coupled to the Poisson equation. Kinetic and Related Models, 2011, 4 (4) : 1063-1079. doi: 10.3934/krm.2011.4.1063 |
[7] |
Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control and Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017 |
[8] |
Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223 |
[9] |
Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159 |
[10] |
Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control and Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73 |
[11] |
Djemaa Messaoudi, Osama Said Ahmed, Komivi Souley Agbodjan, Ting Cheng, Daijun Jiang. Numerical recovery of magnetic diffusivity in a three dimensional spherical dynamo equation. Inverse Problems and Imaging, 2020, 14 (5) : 797-818. doi: 10.3934/ipi.2020037 |
[12] |
Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001 |
[13] |
Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099 |
[14] |
Julia Calatayud, Juan Carlos Cortés, Marc Jornet. On the random wave equation within the mean square context. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 409-425. doi: 10.3934/dcdss.2021082 |
[15] |
Alexander Barg, Oleg R. Musin. Codes in spherical caps. Advances in Mathematics of Communications, 2007, 1 (1) : 131-149. doi: 10.3934/amc.2007.1.131 |
[16] |
Susanna V. Haziot. On the spherical geopotential approximation for Saturn. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022035 |
[17] |
Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and transition point formulas. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 131-155. doi: 10.3934/dcds.2019006 |
[18] |
Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9 |
[19] |
Sergey A. Suslov. Two-equation model of mean flow resonances in subcritical flow systems. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 165-176. doi: 10.3934/dcdss.2008.1.165 |
[20] |
Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]