Sharp adaptation for spherical inverse problems with applications to medical imaging

Journal of Multivariate Analysis

Volumn 99, Issue 2, 2008, Pages 165-190

  Koo, Ja Yong ^a   Kim, Peter T ^b  

^a Korea University   (South Korea)

^b UNIVERSITY OF GUELPH   (Canada)

Author keywords

Deconvolution; Klein Nishina distribution; Mixtures; Pinsker theory; Spherical harmonics

Indexed keywords

EID: 35848965495     PISSN: 0047259X     EISSN: 10957243     Source Type: Journal    
DOI: 10.1016/j.jmva.2006.06.007     Document Type: Article

References (35)

1. Basko R., Zeng G., and Gullberg G. Application of spherical harmonics to image reconstruction for the Compton camera. Phys. Med. Biol. 43 (1998) 887-894
2. Belitser E.N., and Levit B.Y. On minimax filtering on ellipsoids. Math. Methods Statist. 4 (1995) 259-273
3. Beran R., Feuerverger A., and Hall P. On nonparametric estimation of intercept and slope distributions in random coefficient regression. Ann. Statist. 24 (1996) 2569-2592
4. Bröcker T., and Tom Diek T. Representations of Compact Lie Groups (1985), Springer, New York
5. Cavalier L., Golubev G., Picard D., and Tsybakov A. Oracle inequalities for inverse problems. Ann. Statist. 30 (2002) 843-874
6. Cavalier L., and Tsybakov A. Sharp adaptation for inverse problems with random noise. Probab. Theory Related Fields. 123 (2002) 323-354
7. Chang T. Spherical regression. Ann. Statist. 14 (1986) 907-924
8. Efromovich S. On sharp adaptive estimation of multivariate curves. Math. Methods Statist. 9 (2000) 117-139
9. Fan J. On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 (1991) 1257-1272
10. Feuerverger A., and Vardi Y. On some connections between positron emission tomography and random coefficients regression. In: Huskova M., Lachout P., and Visek J.A. (Eds). Proceedings of the Sixth Prague Symposium on Asymptotic Statistics: Volume 1 (1998), Charles University, Prague 153-157
11. Feuerverger A., and Vardi Y. Positron emission tomography and random coefficients regression. Ann. Inst. Statist. Math. 52 (2000) 123-138
12. Goldenshluger A. Density deconvolution in the circular structural model. J. Multivariate Anal. 81 (2002) 360-375
13. Gradshteyn I.S., and Ryzhik I.M. Table of Integrals, Series, and Products (1980), Academic Press, New York
14. Grangeat P. Mathematical framework of cone-beam 3D reconstruction via the first derivative of the radon transform. Mathematical methods in tomography. In: Heamn G.T., Louis A.K., and Natterer F. (Eds). Lecture Notes in Mathematics vol. 1497 (1991), Springer, Berlin 66-97
15. Gullberg G.T., Christian P.E., Zeng G.L., Datz F.L., and Morgan H.T. Cone beam tomography of the heart using single-photon emission computed tomography. IEEE Trans. Med. Imaging 11 (1991) 91-101
16. Healy D.M., Hendriks H., and Kim P.T. Spherical deconvolution. J. Multivariate Anal. 67 (1998) 1-22
17. Hendriks H. Nonparametric estimation of a probability density on a Riemannian manifold using Fourier expansions. Ann. Statist. 18 (1990) 832-849
18. Hubbell J.H., Gimm H.A., and Øverbø I. Pair triplet and total atomic cross sections (and mass attenuation coefficients) for 1 MeV-100 GeV photons in elements Z = 1 to 100. J. Phys. Chem. Ref. Data 9 (1980) 1023-1145
19. Ibragimov I.A., and Khasminskii R.Z. On nonparametric estimation of the value of a linear functional in Gaussian white noise. Theory Probab. Appl. 29 (1984) 18-34
20. Jackson J.D. Classical Electrodynamics (1975), Wiley, New York
21. Jansons K.M., and Alexander D.C. Persistent angular structure: new insights from diffusion magnetic resonance imaging data. Inverse Problems 19 (2003) 1031-1046
22. Jaszczak R.J., Floyd C.E., Manglos S.H., Green K.L., and Coleman R.E. Three-dimensional single photon emission computed tomography using cone-beam collimation (CB-SPECT). Proc. SPIE 671 (1986) 193-199
23. Kim P.T., and Koo J.-Y. Directional mixture models and optimal estimation of the mixing density. Canad. J. Statist. 28 (2000) 383-398
24. Kim P.T., and Koo J.-Y. Optimal spherical deconvolution. J. Multivariate Anal. 80 (2002) 21-42
25. Klemelä J. Asymptotic minimax risk for the white noise model on the sphere. Scand. J. Statist. 26 (1999) 465-473
26. Mair B.A., and Ruymgaart F.H. Statistical inverse estimation in Hilbert scales. SIAM J. Appl. Math. 56 (1996) 1424-1444
27. Marinucci D., and Piccioni M. The empirical process on Gaussian spherical harmonics. Ann. Statist. 32 (2004) 1261-1288
28. Parra L. Reconstruction of cone-beam projections from Compton scattered data. IEEE Trans. Nucl. Sci. 47 (2000) 1543-1550
29. Pinsker M.S. Optimal filtering of square integrable signals in Gaussian white noise. Problems Inform. Transmission 16 (1980) 120-133
30. van Rooij A.C.M., and Ruymgaart F.H. Regularized deconvolution on the circle and the sphere. In: Roussas G. (Ed). Nonparametric Functional Estimation and Related Topics (1991), Kluwer, Amsterdam 679-690
31. Ruymgaart F.H. A unified approach to inversion problems in statistics. Math. Methods Statist. 2 (1993) 130-146
32. Speckman P. Spline smoothing and optimal rates of convergence in nonparametric regression models. Ann. Statist. 13 (1984) 970-983
33. Taguchi K., Zeng G., and Gulberg G. Cone-beam image reconstruction using spherical harmonics. Phys. Med. Biol. 46 (2001) N127-N138
34. Tournier J.D., Calamante F., Gadian D.G., and Connelly A. Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution. NeuroImage 23 (2004) 1176-1185
35. Wahba G. Spline interpolation and smoothing on the sphere. SIAM J. Sci. Comput. 2 (1981) 5-16 (Erratum 3 (1982) 385-386)