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 & 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. 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. 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. doi: 10.1063/1.525873. Google Scholar

[4]

G. Beylkin, Iterated spherical means in linearized inverse problems,, in, (1983), 112. Google Scholar

[5]

R. Courant and D. Hilbert, "Methods of Mathematical Physics. Vol. II: Partial Differential Equations,", (Vol. II by R. Courant), (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). 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. doi: 10.2307/1998023. Google Scholar

[8]

C. Epstein and B. Kleiner, Spherical means in annular regions,, Comm. Pure Appl. Math., 46 (1993), 441. 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. 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 (1984). Google Scholar

[11]

F. John, "Plane Waves and Spherical Means Applied to Partial Differential Equations,", Reprint of the 1955 original, (1955). 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). Google Scholar

[13]

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

[14]

V. Lin and A. Pinkus, Approximation of multivariate functions,, in, 4 (1994), 257. Google Scholar

[15]

L. {Nguyen}, Range description for a spherical mean transform on spaces of constant curvatures,, , (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. 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. 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. doi: 10.1063/1.525873. Google Scholar

[4]

G. Beylkin, Iterated spherical means in linearized inverse problems,, in, (1983), 112. Google Scholar

[5]

R. Courant and D. Hilbert, "Methods of Mathematical Physics. Vol. II: Partial Differential Equations,", (Vol. II by R. Courant), (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). 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. doi: 10.2307/1998023. Google Scholar

[8]

C. Epstein and B. Kleiner, Spherical means in annular regions,, Comm. Pure Appl. Math., 46 (1993), 441. 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. 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 (1984). Google Scholar

[11]

F. John, "Plane Waves and Spherical Means Applied to Partial Differential Equations,", Reprint of the 1955 original, (1955). 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). Google Scholar

[13]

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

[14]

V. Lin and A. Pinkus, Approximation of multivariate functions,, in, 4 (1994), 257. Google Scholar

[15]

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

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