Abstract
In this article, we consider the problem of inverting the exponential Radon transform of a function in the presence of noise. We propose a kernel estimator to estimate the true function. Such an estimator is closely related to filtered backprojection type inversion formulas in the noise-less setting. For the estimator proposed in this article, we then show that the convergence to the true function is at a minimax optimal rate.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aguilar, V., Ehrenpreis, L. and Kuchment, P. (1996). Range conditions for the exponential Radon transform. J. Anal. Math. 68, 1–13. https://doi.org/10.1007/BF02790201.
Bal, G. and Moireau, P. (2004). Fast numerical inversion of the attenuated Radon transform with full and partial measurements. Inverse Problems 20, 4, 1137–1164. https://doi.org/10.1088/0266-5611/20/4/009.
Boman, J. and Strömberg, J.-O. (2004). Novikov’s inversion formula for the attenuated Radon transform—a new approach. J. Geom. Anal. 14, 2, 185–198. https://doi.org/10.1007/BF02922067.
Cavalier, L. (1998). Asymptotically efficient estimation in a problem related to tomography. Math. Methods Statist. 7, 4, 445–4561999.
Cavalier, L. (2000). Efficient estimation of a density in a problem of tomography. Ann. Statist. 28, 2, 630–647. https://doi.org/10.1214/aos/1016218233.
Guillement, J. -P., Jauberteau, F., Kunyansky, L., Novikov, R. and Trebossen, R. (2002). On single-photon emission computed tomography imaging based on an exact formula for the nonuniform attenuation correction. Inverse Problems 18, 6, 11–19. https://doi.org/10.1088/0266-5611/18/6/101.
Hahn, M. G. and Quinto, E. T. (1985). Distances between measures from 1-dimensional projections as implied by continuity of the inverse Radon transform. Z. Wahrsch. Verw. Gebiete 70, 3, 361–380. https://doi.org/10.1007/BF00534869.
Hazou, I. A. and Solmon, D. C. (1989). Filtered-backprojection and the exponential Radon transform. J. Math. Anal. Appl. 141, 1, 109–119. https://doi.org/10.1016/0022-247X(89)90209-6.
Holman, S, Monard, F and Stefanov, P (2018). The attenuated geodesic x-ray transform. Inverse Problems 34, 6, 064003–26. https://doi.org/10.1088/1361-6420/aab8bc.
Johnstone, I. M. and Silverman, B. W. (1990). Speed of estimation in positron emission tomography and related inverse problems. Ann. Statist. 18, 1, 251–280. https://doi.org/10.1214/aos/1176347500.
Korostelëv, A. P. and Tsybakov, A. B. (1991). Optimal rates of convergence of estimators in a probabilistic setup of tomography problem. Probl. Inf. Transm. 27, 73–81.
Korostelëv, A.P. and Tsybakov, A.B. (1992). Asymptotically minimax image reconstruction problems. In: Topics in nonparametric estimation. Adv. Soviet math., vol. 12, pp. 45–86. Amer. Math. Soc., providence, RI, ???.
Korostelëv, A.P. and Tsybakov, A. B. (1993). Minimax Theory of Image Reconstruction. Lecture Notes in Statistics, vol. 82, p. 258. Springer ???. doi: https://doi.org/10.1007/978-1-4612-2712-0
Louis, A. K. (1982). Optimal sampling in nuclear magnetic resonance (NMR) tomography. J. Comput. Assist. Tomogr. 6, 2, 334–340. doi: https://doi.org/10.1097/00004728-198204000-00019
Louis, A. K. (1995). Approximate inverse for linear and some nonlinear problems. Inverse Problems 11, 6, 1211–1223.
Louis, A. K. and Maass, P. (1990). A mollifier method for linear operator equations of the first kind. Inverse Problems 6, 3, 427–440.
Monard, F., Nickl, R. and Paternain, G. P. (2019). Efficient nonparametric Bayesian inference for X-ray transforms. Ann. Statist. 47, 2, 1113–1147. https://doi.org/10.1214/18-AOS1708.
Natterer, F. (1979). On the inversion of the attenuated Radon transform. Numer. Math. 32, 4, 431–438. https://doi.org/10.1007/BF01401046.
Natterer, F. (2001a). Inversion of the attenuated Radon transform. Inverse Problems 17, 1, 113–119. https://doi.org/10.1088/0266-5611/17/1/309.
Natterer, F. (2001b). The mathematics of computerized tomography. Society for industrial and applied mathematics ???. https://doi.org/10.1137/1.9780898719284.
Novikov, R.G. (2002). An inversion formula for the attenuated x-ray transformation. Arkiv för Matematik 40, 1, 145–167. https://doi.org/10.1007/bf02384507.
Rigaud, G. and Lakhal, A. (2015). Approximate inverse and Sobolev estimates for the attenuated Radon transform. Inverse Problems 31, 10, 105010–21. https://doi.org/10.1088/0266-5611/31/10/105010.
Rullgå, R.D.H. (2004). An explicit inversion formula for the exponential Radon transform using data from 180∘. Ark. Mat. 42, 2, 353–362. https://doi.org/10.1007/BF02385485.
Salo, M. and Uhlmann, G. (2011). The attenuated ray transform on simple surfaces. J. Differential Geom. 88, 1, 161–187.
Schuster, T. (2007). The Method of Approximate inverse: Theory and Applications. Lecture Notes in Mathematics, vol. 1906, p. 198. Springer ???. https://doi.org/10.1007/978-3-540-71227-5.
Shneı̆berg, I.Y. (1994). Exponential Radon transform. In: Applied problems of radon transform. Amer. Math. Soc. Transl. Ser. 2, vol. 162, pp. 235–245. Amer. Math. Soc., providence, RI, ???. https://doi.org/10.1090/trans2/162/07.
Shneı̆berg, I.Y., Ponomarev, I.V., Dmitrichenko, V.A. and Kalashnikov, S.D. (1994). On a new reconstruction algorithm in emission tomography. In: Applied problems of radon transform. Amer. Math. Soc. Transl. Ser. 2, vol. 162, pp. 247–255. Amer. Math. Soc., providence, RI, ???. https://doi.org/10.1090/trans2/162/08.
Siltanen, S., Kolehmainen, V., rvenp, S.J., Kaipio, J.P., Koistinen, P., Lassas, M., Pirttil, J. and Somersalo, E. (2003). Statistical inversion for medical x-ray tomography with few radiographs: i. general theory. Phys. Med. Biol.48, 10, 1437–1463. https://doi.org/10.1088/0031-9155/48/10/314.
Tretiak, O. and Metz, C. (1980). The exponential Radon transform. SIAM. J. Appl. Math. 39, 2, 341–354. https://doi.org/10.1137/0139029.
Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer Series in Statistics. Springer, Berlin, p. 214. https://doi.org/10.1007/b13794.
Vänskä, S., Lassas, M. and Siltanen, S. (2009). Statistical X-ray tomography using empirical Besov priors. Int. J. Tomogr. Stat. 11, S09, 3–32.
Wen, J. and Liang, Z. (2006). An inversion formula for the exponential radon transform in spatial domain with variable focal-length fan-beam collimation geometry. Med. Phys. 33, 3, 792–798. https://doi.org/10.1118/1.2170596.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interests
There is no conflict of interest or funding information to report for this article.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Abhishek, A. Minimax Optimal Estimator in a Stochastic Inverse Problem for Exponential Radon Transform. Sankhya A 85, 980–998 (2023). https://doi.org/10.1007/s13171-022-00285-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13171-022-00285-4