On the inverse problem and Sobolev estimates of the generalized X-ray transform

References

• Natterer F. The mathematics of computerized tomography. Philadelphia: SIAM; 2001. (Classics in Applied Mathematics; vol. 32).
   Google Scholar
• Wang JP, Du JY. A note on singular value decomposition for Radon transform in Rn. Acta Math Sci. 2002;22B(3):311–318.
   Google Scholar
• Ding T, Ren K, Valllian S. A one-step reconstruction algorithm for quantitative photoacoustic imaging. Inverse Probl. 2015;31:095005.
   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
• Aldroubi A, Gröchenig K. Nonuniform sampling and reconstruction in shift-invariant spaces. Siam Rev. 2001;43(4):585–620.
   Web of Science ®Google Scholar
• Aldroubi A. Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces. Appl Comput Harmon Anal. 2002;13(2):151–161.
   Web of Science ®Google Scholar
• Solmon DC. The x-ray transform. J Math Anal Appl. 1976;56:61–83.
   Google Scholar
• Louis KA. Combining image reconstruction and image analysis with an application to 2D-tomography. SIAM J Imag Sci. 2008;1(2):188–208.
   Web of Science ®Google Scholar
• Gordon R, Bender R, Herman GT. Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography. J Theor Biol. 1970;29:471–481.
   PubMed Web of Science ®Google Scholar
• Hart RG, Yoshiyama JM. Polytropic montage: ribosomes examined in a tissue section. Proc Natl Acad Sci. 1970;65:402–408.
   PubMedGoogle Scholar
• Hunt BR. The inverse problem of radiography. Math Biosci. 1970;8:161–179.
   Google Scholar
• John F. Plane waves and spherical means applied to partial differential equations. New York: Wiley-Interscience; 1975.
   Google Scholar
• Gelfand IM, Graev MI, Vilenkin NY. Generalized functions. New York: Academic; 1966.
   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
• Shneiberg L. Some inversion formulas in the single photon emission computed tomography. Appl Anal. 1994;53:221–231.
   Google Scholar
• Finch DV. Uniqueness for the attenuated x-ray transform in the physical range. Inverse Probl. 1986;2:197–203.
   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.
   Web of Science ®Google Scholar
• Xian J. Weighted sampling and reconstruction in weighted reproducing kernel spaces. J Math Anal Appl. 2010;367:34–42.
   Web of Science ®Google Scholar
• Xian J, Sun WC. Local sampling and reconstruction in shift-invariant spaces and their applications in spline subspaces. Numer Func Anal Optim. 2010;31:366–386.
   Web of Science ®Google Scholar
• Xian J, Li S. General A-P iterative algorithm in shift-invariant spaces. Acta Math Sci. 2009;25:545–552.
   Google Scholar
• Xian J, Luo SP, Lin W. Weighted sampling and signal reconstruction in spline subspaces. Signal Process. 2006;86:331–340.
   Web of Science ®Google Scholar
• Geng CX, Wang JP. Convergence analysis of an accelerated expectation-maximization algorithm for ill-posed integral equations. Math Method Appl Sci. 2016;39:668–675.
   Web of Science ®Google Scholar
• Tretiak O, Metz C. The exponential Radon transform. SIAM J Appl Math. 1980;39:341–354.
   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 RG. An inversion formula for the attenuated X-ray transform. Ark Mat. 2002;40:145–167.
   Web of Science ®Google Scholar
• Novikov RG. On the range characterization for the two-dimensional attenuated x-ray transformation. Inverse Probl. 2002;18:677–700.
   Web of Science ®Google Scholar
• Boman J, Stromberg JO. Novikov's inversion formula for the attenuated Radon transform – a new approach. J Geo Anal. 2004;14:185–198.
   Web of Science ®Google Scholar
• Quinto ET. The invertibility of rotation invariant Radon transforms. J Math Anal Appl. 1983;91(2):510–522.
   Web of Science ®Google Scholar
• Boman J. An example of non-uniqueness for a generalized Radon transform. J Anal Math. 1993;61(1):395–401.
   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
• 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
• Lu JK. Boundary value problems for analytic functions. Singapore: World Scientific; 1993.
   Google Scholar
• Wang YF, Han PJ, Wang YJ. On Riemann problem of automorphic polyanalytic functions connected with a rotation group. Complex Var Elliptic Equ. 2015;60:1033–1057.
   Web of Science ®Google Scholar
• Begehr H, Du JY, Wang YF. A Dirichlet problem for polyharmonic functions. Ann Mat Pura Appl. 2008;187(3):435–457.
   Web of Science ®Google Scholar
• Rigaud G, Lakhal A. Approximate inverse and Sobolev estimates for the attenuated Radon transform. Inverse Probl. 2015;31:105010.
   Web of Science ®Google Scholar
• Rigaud G, Lakhal A. Image and feature reconstruction for the attenuated Radon transform via circular harmonic decomposition of the kernel. Inverse Probl. 2015;31(2):025007.
   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
• Li W, Wang JP. A new complex approach towards approximate inverse and generalized Radon transform. Complex Var Elliptic Equ. 2018;63(9):1258–1270.
   Web of Science ®Google Scholar
• Li W, Wang JP. On inversion and Sobolev estimate results about the weighted Radon transform. Math Methods Appl Sci. 2019;42(4):1227–1235.
   Web of Science ®Google Scholar
• Li W, Wang JP. New approximate inverse results for the attenuated Radon transform with complex-valued coefficients. Complex Var Elliptic Equ. 2019;64(6):1009–1024.
   Web of Science ®Google Scholar
• Luo S, Qian J, Stefanov P. Adjoint state method for the identification problem in SPECT: recovery of both the source and the attenuation in the attenuated X-ray transform. SIAM J Imaging Sci. 2014;7(2):696–715.
   Web of Science ®Google Scholar
• Smith KT, Keinert F. Mathematical foundation of computed tomography. Appl Optics. 1985;24(23):3950–3957.
   PubMed Web of Science ®Google Scholar