August  2016, 10(3): 711-739. doi: 10.3934/ipi.2016018

Reconstructing a function on the sphere from its means along vertical slices

1. 

Technische Universität Chemnitz, Faculty of Mathematics, D-09107 Chemnitz, Germany, Germany

Received  June 2015 Revised  January 2016 Published  August 2016

We present a novel algorithm for the inversion of the vertical slice transform, i.e. the transform that associates to a function on the two-dimensional unit sphere all integrals along circles that are parallel to one fixed direction. Our approach makes use of the singular value decomposition and resembles the mollifier approach by applying numerical integration with a reconstruction kernel via a quadrature rule. Considering the inversion problem as a statistical inverse problem, we find a family of asymptotically optimal mollifiers that minimize the maximum risk of the mean integrated error for functions within a Sobolev ball. By using fast spherical Fourier transforms and the fast Legendre transform, our algorithm can be implemented with almost linear complexity. In numerical experiments, we compare our algorithm with other approaches and illustrate our theoretical findings.
Citation: Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018
References:
[1]

A. Abouelaz and R. Daher, Sur la transformation de Radon de la sphère $S^d$,, John Wiley & Sons Ltd, (1986).

[2]

M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions,, National Bureau of Standards, (1964). doi: 10.1119/1.1972842.

[3]

H. Bauer, Probability Theory,, De Gruyter, (1996). doi: 10.1515/9783110814668.

[4]

H. Berens, P. L. Butzer and S. Pawelke, Limitierungsverfahren von Reihen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten,, Publ. Res. Inst. Math. Sci., 4 (1968), 201. doi: 10.2977/prims/1195194875.

[5]

L. Cavalier, Nonparametric statistical inverse problems,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/3/034004.

[6]

J. W. Cooley and J. W. Tukey, An algorithm for machine calculation of complex Fourier series,, Math. Comput., 19 (1965), 297. doi: 10.1090/S0025-5718-1965-0178586-1.

[7]

F. Dai and Y. Xu, Approximation Theory and Harmonic Analysis on Spheres and Balls,, Springer Monographs in Mathematics, (2013). doi: 10.1007/978-1-4614-6660-4.

[8]

W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on the Sphere,, Oxford University Press, (1998).

[9]

P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien,, Math. Ann., 74 (1913), 278. doi: 10.1007/BF01456044.

[10]

P. Funk, Beiträge zur Theorie der Kugelfunktionen,, Math. Ann., 77 (1915), 136. doi: 10.1007/BF01456825.

[11]

R. J. Gardner, Geometric Tomography,, 2nd edition, (2006). doi: 10.1017/CBO9781107341029.

[12]

W. Gautschi, Numerical quadrature in the presence of a singularity,, SIAM J. Numer. Anal., 4 (1967), 357. doi: 10.1137/0704031.

[13]

W. Gautschi, Advances in Chebyshev quadrature,, in Numerical Analysis (ed. G. Watson), (1976), 100. doi: 10.1007/BFb0080118.

[14]

S. Gindikin, J. Reeds and L. Shepp, Spherical tomography and spherical integral geometry,, in Tomography, (1994), 83.

[15]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products,, Seventh edition, (2007).

[16]

E. Hecke, Über orthogonal-invariante Integralgleichungen,, Math. Ann., 78 (1917), 398. doi: 10.1007/BF01457114.

[17]

S. Helgason, The Radon Transform,, 2nd edition, (1999). doi: 10.1007/978-1-4757-1463-0.

[18]

K. Hesse and I. H. Sloan, Hyperinterpolation on the sphere,, in Frontiers in Interpolation and Approximation (eds. N. K. Govil, 282 (2007), 213. doi: 10.1201/9781420011388.

[19]

E. Hewitt and R. E. Hewitt, The Gibbs-Wilbraham phenomenon: An episode in Fourier analysis,, Arch. History Exact Sci., 21 (1979), 129. doi: 10.1007/BF00330404.

[20]

R. Hielscher and M. Quellmalz, Optimal mollifiers for spherical deconvolution,, Inverse Problems, 31 (2015). doi: 10.1088/0266-5611/31/8/085001.

[21]

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging,, IEEE Press, (2001). doi: 10.1137/1.9780898719277.

[22]

P. T. Kim and J. Koo, Optimal spherical deconvolution,, J. Multivariate Anal., 80 (2002), 21. doi: 10.1006/jmva.2000.1968.

[23]

S. Kunis and D. Potts, Fast spherical Fourier algorithms,, J. Comput. Appl. Math., 161 (2003), 75. doi: 10.1016/S0377-0427(03)00546-6.

[24]

A. K. Louis and P. Maass, A mollifier method for linear operator equations of the first kind,, Inverse Problems, 6 (1990), 427. doi: 10.1088/0266-5611/6/3/011.

[25]

A. K. Louis, M. Riplinger, M. Spiess and E. Spodarev, Inversion algorithms for the spherical Radon and cosine transform,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/3/035015.

[26]

V. Michel, Lectures on Constructive Approximation: Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball,, Birkhäuser, (2013). doi: 10.1007/978-0-8176-8403-7.

[27]

F. Natterer, The Mathematics of Computerized Tomography,, John Wiley & Sons Ltd, (1986), 978.

[28]

V. P. Palamodov, Reconstructive Integral Geometry, vol. 98 of Monographs in Mathematics,, Birkhäuser, (2004). doi: 10.1007/978-3-0348-7941-5.

[29]

D. Potts, G. Steidl and M. Tasche, Fast algorithms for discrete polynomial transforms,, Math. Comput., 67 (1998), 1577. doi: 10.1090/S0025-5718-98-00975-2.

[30]

D. Potts and G. Steidl, A new linogram algorithm for computerized tomography,, IMA J. Numer. Anal., 21 (2001), 769. doi: 10.1093/imanum/21.3.769.

[31]

J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,, Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math. Nat. Kl., 69 (1917), 262.

[32]

H. Robbins, A remark on Stirling's formula,, Amer. Math. Monthly, 62 (1955), 26. doi: 10.2307/2308012.

[33]

B. Rubin, Generalized Minkowski-Funk transforms and small denominators on the sphere,, Fract. Calc. Appl. Anal., 3 (2000), 177.

[34]

W. Rudin, Uniqueness theory for Laplace series,, Trans. Amer. Math. Soc., 68 (1950), 287. doi: 10.1090/S0002-9947-1950-0033368-1.

[35]

L. A. Shepp and J. B. Kruskal, Computerized tomography: The new medical X-ray technology,, Amer. Math. Monthly, 85 (1978), 420. doi: 10.2307/2320062.

[36]

I. H. Sloan, Polynomial interpolation and hyperinterpolation over general regions,, J. Approx. Theory, 83 (1995), 238. doi: 10.1006/jath.1995.1119.

[37]

M. Storath, A. Weinmann, J. Frikel and M. Unser, Joint image reconstruction and segmentation using the Potts model,, Inverse Problems, 31 (2015). doi: 10.1088/0266-5611/31/2/025003.

[38]

G. Szegő, Orthogonal Polynomials,, 4th edition, (1975).

[39]

N. Thorstensen and O. Scherzer, Convergence of variational regularization methods for imaging on Riemannian manifolds,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/1/015007.

[40]

P. Toft, The Radon Transform - Theory and Implementation,, PhD thesis, (1996).

[41]

A. Tsybakov, Introduction to Nonparametric Estimation,, Springer Verlag, (2009). doi: 10.1007/b13794.

[42]

Y. Xu, A new approach to the reconstruction of images from Radon projections,, Adv. in Appl. Math., 36 (2006), 388. doi: 10.1016/j.aam.2005.08.004.

[43]

Y. Xu and O. Tischenko, Fast OPED algorithm for reconstruction of images from Radon data,, East. J. Approx., 13 (2007), 427.

[44]

G. Zangerl and O. Scherzer, Exact reconstruction in photoacoustic tomography with circular integrating detectors II: Spherical geometry,, Math. Methods Appl. Sci., 33 (2010), 1771. doi: 10.1002/mma.1266.

show all references

References:
[1]

A. Abouelaz and R. Daher, Sur la transformation de Radon de la sphère $S^d$,, John Wiley & Sons Ltd, (1986).

[2]

M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions,, National Bureau of Standards, (1964). doi: 10.1119/1.1972842.

[3]

H. Bauer, Probability Theory,, De Gruyter, (1996). doi: 10.1515/9783110814668.

[4]

H. Berens, P. L. Butzer and S. Pawelke, Limitierungsverfahren von Reihen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten,, Publ. Res. Inst. Math. Sci., 4 (1968), 201. doi: 10.2977/prims/1195194875.

[5]

L. Cavalier, Nonparametric statistical inverse problems,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/3/034004.

[6]

J. W. Cooley and J. W. Tukey, An algorithm for machine calculation of complex Fourier series,, Math. Comput., 19 (1965), 297. doi: 10.1090/S0025-5718-1965-0178586-1.

[7]

F. Dai and Y. Xu, Approximation Theory and Harmonic Analysis on Spheres and Balls,, Springer Monographs in Mathematics, (2013). doi: 10.1007/978-1-4614-6660-4.

[8]

W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on the Sphere,, Oxford University Press, (1998).

[9]

P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien,, Math. Ann., 74 (1913), 278. doi: 10.1007/BF01456044.

[10]

P. Funk, Beiträge zur Theorie der Kugelfunktionen,, Math. Ann., 77 (1915), 136. doi: 10.1007/BF01456825.

[11]

R. J. Gardner, Geometric Tomography,, 2nd edition, (2006). doi: 10.1017/CBO9781107341029.

[12]

W. Gautschi, Numerical quadrature in the presence of a singularity,, SIAM J. Numer. Anal., 4 (1967), 357. doi: 10.1137/0704031.

[13]

W. Gautschi, Advances in Chebyshev quadrature,, in Numerical Analysis (ed. G. Watson), (1976), 100. doi: 10.1007/BFb0080118.

[14]

S. Gindikin, J. Reeds and L. Shepp, Spherical tomography and spherical integral geometry,, in Tomography, (1994), 83.

[15]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products,, Seventh edition, (2007).

[16]

E. Hecke, Über orthogonal-invariante Integralgleichungen,, Math. Ann., 78 (1917), 398. doi: 10.1007/BF01457114.

[17]

S. Helgason, The Radon Transform,, 2nd edition, (1999). doi: 10.1007/978-1-4757-1463-0.

[18]

K. Hesse and I. H. Sloan, Hyperinterpolation on the sphere,, in Frontiers in Interpolation and Approximation (eds. N. K. Govil, 282 (2007), 213. doi: 10.1201/9781420011388.

[19]

E. Hewitt and R. E. Hewitt, The Gibbs-Wilbraham phenomenon: An episode in Fourier analysis,, Arch. History Exact Sci., 21 (1979), 129. doi: 10.1007/BF00330404.

[20]

R. Hielscher and M. Quellmalz, Optimal mollifiers for spherical deconvolution,, Inverse Problems, 31 (2015). doi: 10.1088/0266-5611/31/8/085001.

[21]

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging,, IEEE Press, (2001). doi: 10.1137/1.9780898719277.

[22]

P. T. Kim and J. Koo, Optimal spherical deconvolution,, J. Multivariate Anal., 80 (2002), 21. doi: 10.1006/jmva.2000.1968.

[23]

S. Kunis and D. Potts, Fast spherical Fourier algorithms,, J. Comput. Appl. Math., 161 (2003), 75. doi: 10.1016/S0377-0427(03)00546-6.

[24]

A. K. Louis and P. Maass, A mollifier method for linear operator equations of the first kind,, Inverse Problems, 6 (1990), 427. doi: 10.1088/0266-5611/6/3/011.

[25]

A. K. Louis, M. Riplinger, M. Spiess and E. Spodarev, Inversion algorithms for the spherical Radon and cosine transform,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/3/035015.

[26]

V. Michel, Lectures on Constructive Approximation: Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball,, Birkhäuser, (2013). doi: 10.1007/978-0-8176-8403-7.

[27]

F. Natterer, The Mathematics of Computerized Tomography,, John Wiley & Sons Ltd, (1986), 978.

[28]

V. P. Palamodov, Reconstructive Integral Geometry, vol. 98 of Monographs in Mathematics,, Birkhäuser, (2004). doi: 10.1007/978-3-0348-7941-5.

[29]

D. Potts, G. Steidl and M. Tasche, Fast algorithms for discrete polynomial transforms,, Math. Comput., 67 (1998), 1577. doi: 10.1090/S0025-5718-98-00975-2.

[30]

D. Potts and G. Steidl, A new linogram algorithm for computerized tomography,, IMA J. Numer. Anal., 21 (2001), 769. doi: 10.1093/imanum/21.3.769.

[31]

J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,, Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math. Nat. Kl., 69 (1917), 262.

[32]

H. Robbins, A remark on Stirling's formula,, Amer. Math. Monthly, 62 (1955), 26. doi: 10.2307/2308012.

[33]

B. Rubin, Generalized Minkowski-Funk transforms and small denominators on the sphere,, Fract. Calc. Appl. Anal., 3 (2000), 177.

[34]

W. Rudin, Uniqueness theory for Laplace series,, Trans. Amer. Math. Soc., 68 (1950), 287. doi: 10.1090/S0002-9947-1950-0033368-1.

[35]

L. A. Shepp and J. B. Kruskal, Computerized tomography: The new medical X-ray technology,, Amer. Math. Monthly, 85 (1978), 420. doi: 10.2307/2320062.

[36]

I. H. Sloan, Polynomial interpolation and hyperinterpolation over general regions,, J. Approx. Theory, 83 (1995), 238. doi: 10.1006/jath.1995.1119.

[37]

M. Storath, A. Weinmann, J. Frikel and M. Unser, Joint image reconstruction and segmentation using the Potts model,, Inverse Problems, 31 (2015). doi: 10.1088/0266-5611/31/2/025003.

[38]

G. Szegő, Orthogonal Polynomials,, 4th edition, (1975).

[39]

N. Thorstensen and O. Scherzer, Convergence of variational regularization methods for imaging on Riemannian manifolds,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/1/015007.

[40]

P. Toft, The Radon Transform - Theory and Implementation,, PhD thesis, (1996).

[41]

A. Tsybakov, Introduction to Nonparametric Estimation,, Springer Verlag, (2009). doi: 10.1007/b13794.

[42]

Y. Xu, A new approach to the reconstruction of images from Radon projections,, Adv. in Appl. Math., 36 (2006), 388. doi: 10.1016/j.aam.2005.08.004.

[43]

Y. Xu and O. Tischenko, Fast OPED algorithm for reconstruction of images from Radon data,, East. J. Approx., 13 (2007), 427.

[44]

G. Zangerl and O. Scherzer, Exact reconstruction in photoacoustic tomography with circular integrating detectors II: Spherical geometry,, Math. Methods Appl. Sci., 33 (2010), 1771. doi: 10.1002/mma.1266.

[1]

Sunghwan Moon. Inversion of the spherical Radon transform on spheres through the origin using the regular Radon transform. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1029-1039. doi: 10.3934/cpaa.2016.15.1029

[2]

Leonid Kunyansky. Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries. Inverse Problems & Imaging, 2012, 6 (1) : 111-131. doi: 10.3934/ipi.2012.6.111

[3]

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

[4]

Linh V. Nguyen. Spherical mean transform: A PDE approach. Inverse Problems & Imaging, 2013, 7 (1) : 243-252. doi: 10.3934/ipi.2013.7.243

[5]

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

[6]

Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems & Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721

[7]

Suxiang He, Yunyun Nie. A class of nonlinear Lagrangian algorithms for minimax problems. Journal of Industrial & Management Optimization, 2013, 9 (1) : 75-97. doi: 10.3934/jimo.2013.9.75

[8]

Sanjay Dharmavaram, Timothy J. Healey. Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1669-1684. doi: 10.3934/dcdss.2019112

[9]

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

[10]

Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems & Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649

[11]

Mason A. Porter, Richard L. Liboff. The radially vibrating spherical quantum billiard. Conference Publications, 2001, 2001 (Special) : 310-318. doi: 10.3934/proc.2001.2001.310

[12]

Aravind Asok, James Parson. Equivariant sheaves on some spherical varieties. Electronic Research Announcements, 2011, 18: 119-130. doi: 10.3934/era.2011.18.119

[13]

Robert Schippa. Sharp Strichartz estimates in spherical coordinates. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2047-2051. doi: 10.3934/cpaa.2017100

[14]

Shaobo Lin, Xingping Sun, Zongben Xu. Discretizing spherical integrals and its applications. Conference Publications, 2013, 2013 (special) : 499-514. doi: 10.3934/proc.2013.2013.499

[15]

Peter Boyvalenkov, Maya Stoyanova. New nonexistence results for spherical designs. Advances in Mathematics of Communications, 2013, 7 (3) : 279-292. doi: 10.3934/amc.2013.7.279

[16]

Michael Krause, Jan Marcel Hausherr, Walter Krenkel. Computing the fibre orientation from Radon data using local Radon transform. Inverse Problems & Imaging, 2011, 5 (4) : 879-891. doi: 10.3934/ipi.2011.5.879

[17]

Tapio Helin. On infinite-dimensional hierarchical probability models in statistical inverse problems. Inverse Problems & Imaging, 2009, 3 (4) : 567-597. doi: 10.3934/ipi.2009.3.567

[18]

François Alouges, Sylvain Faure, Jutta Steiner. The vortex core structure inside spherical ferromagnetic particles. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1259-1282. doi: 10.3934/dcds.2010.27.1259

[19]

Marcin Bugdoł, Tadeusz Nadzieja. A nonlocal problem describing spherical system of stars. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2417-2423. doi: 10.3934/dcdsb.2014.19.2417

[20]

C. Bandle, Y. Kabeya, Hirokazu Ninomiya. Imperfect bifurcations in nonlinear elliptic equations on spherical caps. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1189-1208. doi: 10.3934/cpaa.2010.9.1189

2017 Impact Factor: 1.465

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]