The stability estimates of the inverse problem for the weighted Radon transform

References

• Natterer F. The mathematics of computerized tomography. New York: John Wiley and Sons; 1986.
   Google Scholar
• Tretiak O, Metz C. The exponential Radon transform. Society for industrial and applied mathematics. J Appl Math. 1980;39(2):341–354.
   Google Scholar
• Novikov R G. An inversion formula for the attenuated x-ray transformation. Ark Mat. 2002;40:145–167.
   Web of Science ®Google Scholar
• Natterer F. Inversion of the attenuated Radon transform. Inverse Probl. 2001;17:113–119.
   Web of Science ®Google Scholar
• Novikov R G. On the range characterization for the two-dimensional attenuated x-ray transformation. Inverse Probl. 2002;18:677–700.
   Web of Science ®Google Scholar
• Arbuzov E V, Bukhgeim A L, Kazantsev S G. Two-dimensional tomography problems and the theory of A-analytic functions. Siber Adv Math. 1998;8:1–20.
   Google Scholar
• Rigaud G, Lakhal A. Approximate inverse and Sobolev estimates for the attenuated Radon transform. Inverse Probl. 2015;31(10):105010.
   Web of Science ®Google Scholar
• Wang JP, Du JY. A note on singular value decomposition for Radon transform in RnRn. Acta Math Sci. 2002;22B(3):311–318. https://doi.org/10.1016/S0252-9602(17)30300-4.
   Google Scholar
• Wang JP. Inversion and property characterization of generalized transforms of Radon type. Acta Math Sci. 2011;31(3):636–643.
   Google Scholar
• Bal G, Moireau P. Fast numerical inversion of the attenuated Radon transform with full and partial measurements. Inverse Probl. 2004;20:1137–1164.
   Web of Science ®Google Scholar
• Courdurier M, Monard F, Osses A, et al. Simultaneous source and attenuation reconstruction in SPECT using ballistic and single scattering data. Inverse Probl. 2015;31(9):681–695.
   Web of Science ®Google Scholar
• Kunyansky L A. A new SPECT reconstruction algorithm based on Novikov's explicit inversion formula. Inverse Probl. 2001;17:293–306.
   Web of Science ®Google Scholar
• Boman J. An example of non-uniqueness for a generalized Radon transform. J D'Anal Math. 1993;61(1):395–401.
   Google Scholar
• Markoe A, Quinto E T. An elementary proof of local invertibility for generalized and attenuated Radon transforms. SIAM J Math Anal. 1985;16(5):1114–1119.
   Web of Science ®Google Scholar
• Shen ZJ, Wang JP. The research of complex analytic method in spect image reconstruction. Appl Anal. 2014;93(11):2451–2461.
   Web of Science ®Google Scholar
• Coroianu L, Costarelli D, Gal GS, et al. The max-product generalized sampling operators: convergence and quantitative estimates. Appl Math Comput. 2019;355:173–183.
   Web of Science ®Google Scholar
• Xian J, Li S. Sampling set conditions in weighted multiply generated shift-invariant spaces and their applications. Appl Comput Harmon Anal. 2007;23(2):171–180.
   Web of Science ®Google Scholar
• Xian J, Luo SP, Lin W. Weighted sampling and signal reconstruction in spline subspaces. Signal Process. 2006;86(2):331–340.
   Web of Science ®Google Scholar
• Führ H, Xian J. Quantifying invariance properties of shift-invariant spaces. Appl Comput Harmon Anal. 2014;36(3):514–521.
   Web of Science ®Google Scholar
• Sun QY, Xian J. Rate of innovation for (non-)periodic signals and optimal lower stability bound for filtering.J Fourier Anal Appl. 2014;20(1):119–134.
   Web of Science ®Google Scholar
• Geng CX, Wang JP. Convergence analysis of an accelerated expectation-maximization algorithm for ill-posed integral equations. Math Methods Appl Sci. 2016;39(4):668–675.
   Web of Science ®Google Scholar
• Xu YZ, Mawata C, Lin W. Generalized dual space indicator method for underwater imaging. Inverse Probl. 2000;16(6):1761–1776.
   Web of Science ®Google Scholar
• Gilbert RP, Xu YZ. An inverse problem for harmonic acoustics in stratified oceans. J Math Anal Appl. 1993;176(1):121–137.
   Web of Science ®Google Scholar
• Liu KJ, Xu YZ, Zou J. A parallel radial bisection algorithm for inverse scattering problems. Inverse Probl Sci Eng. 2013;21(2):197–209.
   Web of Science ®Google Scholar
• Lange K. A quasi-Newton acceleration of the EM algorithm. Stat Sin. 1995;5:1–18.
   Web of Science ®Google Scholar
• Louis AK. Combining image reconstruction and image analysis with an application to 2D-tomography. SIAMJ Imaging Sci. 2008;1(2):188–208.
   Web of Science ®Google Scholar
• Sharafutdinov VA. The Reshetnyak formula and Natterer stability estimates in tensor tomography. Inverse Probl. 2017;33(2):025002.
   Google Scholar
• Stein EM, Weiss D. Introduction to Fourier analysis on Euclidean spaces. Princeton, NJ: Princeton University Press; 1971.
   Google Scholar
• Smith TK, Keinert F. Mathematical foundation of computed tomography. Appl Opt. 1985;24(23):3950–3957.
   PubMed Web of Science ®Google Scholar
• Aldroubi A, Gröchenig K. Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 2001;43(4):585–620.
   Web of Science ®Google Scholar
• Rullgård H. Stability of the inverse problem for the attenuated Radon transform with 180∘ data. Inverse Probl. 2004;20:781–797. https://doi.org/10.1088/0266-5611/20/3/008.
   Web of Science ®Google Scholar