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

Spherical mean transform: A PDE approach

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.
Citation: Linh V. Nguyen. Spherical mean transform: A PDE approach. Inverse Problems and 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.

[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).

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