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).   Google Scholar

[2]

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

[3]

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

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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.  Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

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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.  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[32]

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

[33]

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

[34]

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

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[40]

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

[41]

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

[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.  Google Scholar

[43]

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

[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.  Google Scholar

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).   Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[32]

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

[33]

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

[34]

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

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[40]

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

[41]

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

[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.  Google Scholar

[43]

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

[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.  Google Scholar

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