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 and 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 http://eudml.org/doc/87670.

[2]

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

[3]

H. Bauer, Probability Theory, De Gruyter, Berlin, Boston, 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), 201-268. doi: 10.2977/prims/1195194875.

[5]

L. Cavalier, Nonparametric statistical inverse problems, Inverse Problems, 24 (2008), 034004, 19pp. 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), 297-301. 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, Springer, New York, 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, Oxford, 1998.

[9]

P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien, Math. Ann., 74 (1913), 278-300. doi: 10.1007/BF01456044.

[10]

P. Funk, Beiträge zur Theorie der Kugelfunktionen, Math. Ann., 77 (1915), 136-152. doi: 10.1007/BF01456825.

[11]

R. J. Gardner, Geometric Tomography, 2nd edition, no. 58 in Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge; New York, 2006. doi: 10.1017/CBO9781107341029.

[12]

W. Gautschi, Numerical quadrature in the presence of a singularity, SIAM J. Numer. Anal., 4 (1967), 357-362. doi: 10.1137/0704031.

[13]

W. Gautschi, Advances in Chebyshev quadrature, in Numerical Analysis (ed. G. Watson), vol. 506 of Lecture Notes in Mathematics, Springer Berlin Heidelberg, 1976, 100-121. doi: 10.1007/BFb0080118.

[14]

S. Gindikin, J. Reeds and L. Shepp, Spherical tomography and spherical integral geometry, in Tomography, Impedance Imaging, and Integral Geometry (eds. E. T. Quinto, M. Cheney and P. Kuchment), vol. 30 of Lectures in Appl. Math, American Mathematical Society, South Hadley, Massachusetts, 1994, 83-92.

[15]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Seventh edition, Academic Press New York, 2007.

[16]

E. Hecke, Über orthogonal-invariante Integralgleichungen, Math. Ann., 78 (1917), 398-404. doi: 10.1007/BF01457114.

[17]

S. Helgason, The Radon Transform, 2nd edition, Birkhäuser, 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, H. N. Mhaskar, R. N. Mohapatra, Z. Nashed and J. Szabados), Pure and Applied Mathematics, Taylor & Francis Books, Boca Raton, Florida, 282 (2007), 213-248. 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), 129-160. doi: 10.1007/BF00330404.

[20]

R. Hielscher and M. Quellmalz, Optimal mollifiers for spherical deconvolution, Inverse Problems, 31 (2015), 085001, 28pp. doi: 10.1088/0266-5611/31/8/085001.

[21]

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, IEEE Press, New York, NY, USA, 2001. doi: 10.1137/1.9780898719277.

[22]

P. T. Kim and J. Koo, Optimal spherical deconvolution, J. Multivariate Anal., 80 (2002), 21-42. doi: 10.1006/jmva.2000.1968.

[23]

S. Kunis and D. Potts, Fast spherical Fourier algorithms, J. Comput. Appl. Math., 161 (2003), 75-98. 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), 427-440. 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), 035015, 25pp. 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, New York, 2013. doi: 10.1007/978-0-8176-8403-7.

[27]

F. Natterer, The Mathematics of Computerized Tomography, John Wiley & Sons Ltd, 1986, http://link.springer.com/book/10.1007%2F978-3-663-01409-6.

[28]

V. P. Palamodov, Reconstructive Integral Geometry, vol. 98 of Monographs in Mathematics, Birkhäuser, Basel, 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), 1577-1590. 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), 769-782. 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), 262-277.

[32]

H. Robbins, A remark on Stirling's formula, Amer. Math. Monthly, 62 (1955), 26-29. doi: 10.2307/2308012.

[33]

B. Rubin, Generalized Minkowski-Funk transforms and small denominators on the sphere, Fract. Calc. Appl. Anal., 3 (2000), 177-203, URL http://www.mathnet.or.kr/mathnet/paper_file/Louisiana/Boris/min.pdf.

[34]

W. Rudin, Uniqueness theory for Laplace series, Trans. Amer. Math. Soc., 68 (1950), 287-303. 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), 420-439. doi: 10.2307/2320062.

[36]

I. H. Sloan, Polynomial interpolation and hyperinterpolation over general regions, J. Approx. Theory, 83 (1995), 238-254. 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), 025003, 29pp. doi: 10.1088/0266-5611/31/2/025003.

[38]

G. Szegő, Orthogonal Polynomials, 4th edition, Amer. Math. Soc., Providence, RI, USA, 1975.

[39]

N. Thorstensen and O. Scherzer, Convergence of variational regularization methods for imaging on Riemannian manifolds, Inverse Problems, 28 (2012), 015007, 17pp. doi: 10.1088/0266-5611/28/1/015007.

[40]

P. Toft, The Radon Transform - Theory and Implementation, PhD thesis, Technical University of Denmark, 1996, URL http://orbit.dtu.dk/files/5529668/Binder1.pdf.

[41]

A. Tsybakov, Introduction to Nonparametric Estimation, Springer Verlag, Berlin, 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), 388-420. 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), 427-444, arXiv:math/0703617v1.

[44]

G. Zangerl and O. Scherzer, Exact reconstruction in photoacoustic tomography with circular integrating detectors II: Spherical geometry, Math. Methods Appl. Sci., 33 (2010), 1771-1782. 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 http://eudml.org/doc/87670.

[2]

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

[3]

H. Bauer, Probability Theory, De Gruyter, Berlin, Boston, 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), 201-268. doi: 10.2977/prims/1195194875.

[5]

L. Cavalier, Nonparametric statistical inverse problems, Inverse Problems, 24 (2008), 034004, 19pp. 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), 297-301. 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, Springer, New York, 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, Oxford, 1998.

[9]

P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien, Math. Ann., 74 (1913), 278-300. doi: 10.1007/BF01456044.

[10]

P. Funk, Beiträge zur Theorie der Kugelfunktionen, Math. Ann., 77 (1915), 136-152. doi: 10.1007/BF01456825.

[11]

R. J. Gardner, Geometric Tomography, 2nd edition, no. 58 in Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge; New York, 2006. doi: 10.1017/CBO9781107341029.

[12]

W. Gautschi, Numerical quadrature in the presence of a singularity, SIAM J. Numer. Anal., 4 (1967), 357-362. doi: 10.1137/0704031.

[13]

W. Gautschi, Advances in Chebyshev quadrature, in Numerical Analysis (ed. G. Watson), vol. 506 of Lecture Notes in Mathematics, Springer Berlin Heidelberg, 1976, 100-121. doi: 10.1007/BFb0080118.

[14]

S. Gindikin, J. Reeds and L. Shepp, Spherical tomography and spherical integral geometry, in Tomography, Impedance Imaging, and Integral Geometry (eds. E. T. Quinto, M. Cheney and P. Kuchment), vol. 30 of Lectures in Appl. Math, American Mathematical Society, South Hadley, Massachusetts, 1994, 83-92.

[15]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Seventh edition, Academic Press New York, 2007.

[16]

E. Hecke, Über orthogonal-invariante Integralgleichungen, Math. Ann., 78 (1917), 398-404. doi: 10.1007/BF01457114.

[17]

S. Helgason, The Radon Transform, 2nd edition, Birkhäuser, 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, H. N. Mhaskar, R. N. Mohapatra, Z. Nashed and J. Szabados), Pure and Applied Mathematics, Taylor & Francis Books, Boca Raton, Florida, 282 (2007), 213-248. 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), 129-160. doi: 10.1007/BF00330404.

[20]

R. Hielscher and M. Quellmalz, Optimal mollifiers for spherical deconvolution, Inverse Problems, 31 (2015), 085001, 28pp. doi: 10.1088/0266-5611/31/8/085001.

[21]

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, IEEE Press, New York, NY, USA, 2001. doi: 10.1137/1.9780898719277.

[22]

P. T. Kim and J. Koo, Optimal spherical deconvolution, J. Multivariate Anal., 80 (2002), 21-42. doi: 10.1006/jmva.2000.1968.

[23]

S. Kunis and D. Potts, Fast spherical Fourier algorithms, J. Comput. Appl. Math., 161 (2003), 75-98. 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), 427-440. 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), 035015, 25pp. 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, New York, 2013. doi: 10.1007/978-0-8176-8403-7.

[27]

F. Natterer, The Mathematics of Computerized Tomography, John Wiley & Sons Ltd, 1986, http://link.springer.com/book/10.1007%2F978-3-663-01409-6.

[28]

V. P. Palamodov, Reconstructive Integral Geometry, vol. 98 of Monographs in Mathematics, Birkhäuser, Basel, 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), 1577-1590. 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), 769-782. 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), 262-277.

[32]

H. Robbins, A remark on Stirling's formula, Amer. Math. Monthly, 62 (1955), 26-29. doi: 10.2307/2308012.

[33]

B. Rubin, Generalized Minkowski-Funk transforms and small denominators on the sphere, Fract. Calc. Appl. Anal., 3 (2000), 177-203, URL http://www.mathnet.or.kr/mathnet/paper_file/Louisiana/Boris/min.pdf.

[34]

W. Rudin, Uniqueness theory for Laplace series, Trans. Amer. Math. Soc., 68 (1950), 287-303. 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), 420-439. doi: 10.2307/2320062.

[36]

I. H. Sloan, Polynomial interpolation and hyperinterpolation over general regions, J. Approx. Theory, 83 (1995), 238-254. 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), 025003, 29pp. doi: 10.1088/0266-5611/31/2/025003.

[38]

G. Szegő, Orthogonal Polynomials, 4th edition, Amer. Math. Soc., Providence, RI, USA, 1975.

[39]

N. Thorstensen and O. Scherzer, Convergence of variational regularization methods for imaging on Riemannian manifolds, Inverse Problems, 28 (2012), 015007, 17pp. doi: 10.1088/0266-5611/28/1/015007.

[40]

P. Toft, The Radon Transform - Theory and Implementation, PhD thesis, Technical University of Denmark, 1996, URL http://orbit.dtu.dk/files/5529668/Binder1.pdf.

[41]

A. Tsybakov, Introduction to Nonparametric Estimation, Springer Verlag, Berlin, 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), 388-420. 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), 427-444, arXiv:math/0703617v1.

[44]

G. Zangerl and O. Scherzer, Exact reconstruction in photoacoustic tomography with circular integrating detectors II: Spherical geometry, Math. Methods Appl. Sci., 33 (2010), 1771-1782. doi: 10.1002/mma.1266.

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