American Institute of Mathematical Sciences

Advanced
February  2013, 7(1): 243-252. doi: 10.3934/ipi.2013.7.243

Spherical mean transform: A PDE approach

Linh V. Nguyen 1,

1. 

Department of Mathematics, University of Idaho, Moscow, Idaho 83844, United States

Received  January 2012 Revised  June 2012 Published  February 2013

We study the spherical mean transform on $\mathbb{R}^n$. The transform is characterized by the Euler-Poisson-Darboux equation. By looking at the spherical harmonic expansions, we obtain a system of $1+1$-dimension hyperbolic equations. Using these equations, we discuss two known problems. The first one is a local uniqueness problem investigated by M. Agranovsky and P. Kuchment, [ Memoirs on Differential Equations and Mathematical Physics, 52 (2011), 1--16]. We present a proof which only involves simple energy arguments. The second problem is to characterize the kernel of spherical mean transform on annular regions, which was studied by C. Epstein and B. Kleiner [ Comm. Pure Appl. Math., 46(3) (1993), 441--451]. We present a short proof that simultaneously provides the necessity and sufficiency for the characterization. As a consequence, we derive a reconstruction procedure for the transform with additional interior (or exterior) information.
    We also discuss how the approach works for the hyperbolic and spherical spaces.
Keywords: Spherical mean, Darboux equation..
Mathematics Subject Classification: Primary: 53C65; Secondary: 35L1.
Citation: Linh V. Nguyen. Spherical mean transform: A PDE approach. Inverse Problems & Imaging, 2013, 7 (1) : 243-252. doi: 10.3934/ipi.2013.7.243
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

C. Epstein and B. Kleiner, Spherical means in annular regions, Comm. Pure Appl. Math., 46 (1993), 441-451. doi: 10.1002/cpa.3160460307.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

F. John, "Plane Waves and Spherical Means Applied to Partial Differential Equations," Reprint of the 1955 original, Springer-Verlag, New York-Berlin, 1981.  Google Scholar

[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. Google Scholar

[13]

V. Lin and A. Pinkus, Fundamentality of ridge functions, J. Approx. Theory, 75 (1993), 295-311. doi: 10.1006/jath.1993.1104.  Google Scholar

[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.  Google Scholar

[15]

L. {Nguyen}, Range description for a spherical mean transform on spaces of constant curvatures, arXiv:1107.1746, (2011). Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

C. Epstein and B. Kleiner, Spherical means in annular regions, Comm. Pure Appl. Math., 46 (1993), 441-451. doi: 10.1002/cpa.3160460307.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

F. John, "Plane Waves and Spherical Means Applied to Partial Differential Equations," Reprint of the 1955 original, Springer-Verlag, New York-Berlin, 1981.  Google Scholar

[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. Google Scholar

[13]

V. Lin and A. Pinkus, Fundamentality of ridge functions, J. Approx. Theory, 75 (1993), 295-311. doi: 10.1006/jath.1993.1104.  Google Scholar

[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.  Google Scholar

[15]

L. {Nguyen}, Range description for a spherical mean transform on spaces of constant curvatures, arXiv:1107.1746, (2011). Google Scholar

[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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Imaging, 2020, 14 (5) : 797-818. doi: 10.3934/ipi.2020037
[12]

Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099
[13]

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
[14]

Julia Calatayud, Juan Carlos Cortés, Marc Jornet. On the random wave equation within the mean square context. Discrete & Continuous Dynamical Systems - S, 2021  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]

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 & Continuous Dynamical Systems, 2019, 39 (1) : 131-155. doi: 10.3934/dcds.2019006
[17]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9
[18]

Sergey A. Suslov. Two-equation model of mean flow resonances in subcritical flow systems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 165-176. doi: 10.3934/dcdss.2008.1.165
[19]

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
[20]

Chiara Corsato, Colette De Coster, Pierpaolo Omari. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. Conference Publications, 2015, 2015 (special) : 297-303. doi: 10.3934/proc.2015.0297

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (128)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences