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
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show all references

References:
[1]

John Wiley & Sons Ltd, 1986 http://eudml.org/doc/87670.  Google Scholar

[2]

National Bureau of Standards, Washington, DC, 1964. doi: 10.1119/1.1972842.  Google Scholar

[3]

De Gruyter, Berlin, Boston, 1996. doi: 10.1515/9783110814668.  Google Scholar

[4]

Publ. Res. Inst. Math. Sci., 4 (1968), 201-268. doi: 10.2977/prims/1195194875.  Google Scholar

[5]

Inverse Problems, 24 (2008), 034004, 19pp. doi: 10.1088/0266-5611/24/3/034004.  Google Scholar

[6]

Math. Comput., 19 (1965), 297-301. doi: 10.1090/S0025-5718-1965-0178586-1.  Google Scholar

[7]

Springer Monographs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-6660-4.  Google Scholar

[8]

Oxford University Press, Oxford, 1998.  Google Scholar

[9]

Math. Ann., 74 (1913), 278-300. doi: 10.1007/BF01456044.  Google Scholar

[10]

Math. Ann., 77 (1915), 136-152. doi: 10.1007/BF01456825.  Google Scholar

[11]

2nd edition, no. 58 in Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge; New York, 2006. doi: 10.1017/CBO9781107341029.  Google Scholar

[12]

SIAM J. Numer. Anal., 4 (1967), 357-362. doi: 10.1137/0704031.  Google Scholar

[13]

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

[14]

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.  Google Scholar

[15]

Seventh edition, Academic Press New York, 2007.  Google Scholar

[16]

Math. Ann., 78 (1917), 398-404. doi: 10.1007/BF01457114.  Google Scholar

[17]

2nd edition, Birkhäuser, 1999. doi: 10.1007/978-1-4757-1463-0.  Google Scholar

[18]

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.  Google Scholar

[19]

Arch. History Exact Sci., 21 (1979), 129-160. doi: 10.1007/BF00330404.  Google Scholar

[20]

Inverse Problems, 31 (2015), 085001, 28pp. doi: 10.1088/0266-5611/31/8/085001.  Google Scholar

[21]

IEEE Press, New York, NY, USA, 2001. doi: 10.1137/1.9780898719277.  Google Scholar

[22]

J. Multivariate Anal., 80 (2002), 21-42. doi: 10.1006/jmva.2000.1968.  Google Scholar

[23]

J. Comput. Appl. Math., 161 (2003), 75-98. doi: 10.1016/S0377-0427(03)00546-6.  Google Scholar

[24]

Inverse Problems, 6 (1990), 427-440. doi: 10.1088/0266-5611/6/3/011.  Google Scholar

[25]

Inverse Problems, 27 (2011), 035015, 25pp. doi: 10.1088/0266-5611/27/3/035015.  Google Scholar

[26]

Birkhäuser, New York, 2013. doi: 10.1007/978-0-8176-8403-7.  Google Scholar

[27]

John Wiley & Sons Ltd, 1986, http://link.springer.com/book/10.1007%2F978-3-663-01409-6.  Google Scholar

[28]

Birkhäuser, Basel, 2004. doi: 10.1007/978-3-0348-7941-5.  Google Scholar

[29]

Math. Comput., 67 (1998), 1577-1590. doi: 10.1090/S0025-5718-98-00975-2.  Google Scholar

[30]

IMA J. Numer. Anal., 21 (2001), 769-782. doi: 10.1093/imanum/21.3.769.  Google Scholar

[31]

Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math. Nat. Kl., 69 (1917), 262-277. Google Scholar

[32]

Amer. Math. Monthly, 62 (1955), 26-29. doi: 10.2307/2308012.  Google Scholar

[33]

Fract. Calc. Appl. Anal., 3 (2000), 177-203, URL http://www.mathnet.or.kr/mathnet/paper_file/Louisiana/Boris/min.pdf.  Google Scholar

[34]

Trans. Amer. Math. Soc., 68 (1950), 287-303. doi: 10.1090/S0002-9947-1950-0033368-1.  Google Scholar

[35]

Amer. Math. Monthly, 85 (1978), 420-439. doi: 10.2307/2320062.  Google Scholar

[36]

J. Approx. Theory, 83 (1995), 238-254. doi: 10.1006/jath.1995.1119.  Google Scholar

[37]

Inverse Problems, 31 (2015), 025003, 29pp. doi: 10.1088/0266-5611/31/2/025003.  Google Scholar

[38]

4th edition, Amer. Math. Soc., Providence, RI, USA, 1975.  Google Scholar

[39]

Inverse Problems, 28 (2012), 015007, 17pp. doi: 10.1088/0266-5611/28/1/015007.  Google Scholar

[40]

PhD thesis, Technical University of Denmark, 1996, URL http://orbit.dtu.dk/files/5529668/Binder1.pdf. Google Scholar

[41]

Springer Verlag, Berlin, 2009. doi: 10.1007/b13794.  Google Scholar

[42]

Adv. in Appl. Math., 36 (2006), 388-420. doi: 10.1016/j.aam.2005.08.004.  Google Scholar

[43]

East. J. Approx., 13 (2007), 427-444, arXiv:math/0703617v1.  Google Scholar

[44]

Math. Methods Appl. Sci., 33 (2010), 1771-1782. doi: 10.1002/mma.1266.  Google Scholar

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