References on Geometric Tomography and Related Topics

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References on Geometric and Microlocal Analysis

23 November 2025

Microlocal Analysis in Integral Geometry
Microlocal Analysis and Applied Mathematics
Introductry References
Other Topics

Microlocal Analysis in Integral Geometry

X-ray transform
- Nikolas Eptaminitakis, Stability estimates for the X-ray transform on simple asymptotically hyperbolic manifolds, Pure and Applied Analysis, 4(3) (2022), pp.487-516, doi.
- S. Gouezel and T. Lefeuvre, Analysis & PDE 14 (2021), pp.301-322, Classical and microlocal analysis of the X-ray transform on Anosov manifolds, doi.
- Colin Guillarmou, Matti Lassas, and Leo Tzou, X-ray Transform in Asymptotically Conic Spaces, International Mathematics Research Notices, 2022(5) (2022), pp.3918-3976, doi.
- S. Holman and G. Uhlmann, On the microlocal analysis of the geodesic X-ray transform with conjugate points, Journal of Differential Geometry, 108 (2018), pp.459-494, doi.
- V. P. Krishnan and E. T. Quinto, Microlocal analysis in tomography, Handbook of mathematical methods in imaging. Vol. 1, 2, 3, pp.847--902, Springer, New York, 2015, pdf.
- E. T. Quinto, An introduction to X-ray tomography and Radon transforms, "The Radon Transform, Inverse Problems, and Tomography", Proceedings of Symposia in Applied Mathematics, 63, pp.1--23, American Mathematical Society, Providence, RI, 2006, url.
- P. Stefanov and G. Uhlmann, The geodesic X-ray transform with fold caustics, Analysis & PDE, 5, pp.219-260, doi.
Inversion formula
- M. V. de Hoop, G. Uhlmann and J. Zhai, Inverting the local geodesic ray transform of higher rank tensors, Inverse Problems, 35 (2019), 115009, doi.
- C. Guillarmou and F. Monard, Reconstruction formulas for X-ray transforms in negative curvature, Annles de L'Institut Fourier (Grenoble), 67 (2017), pp.1353–1392, doi.
- L. Pestov and G. Uhlmann, On characterization of the range and inversion formulas for the geodesic X-ray transform, International Mathematics Research Notices, 2004, no. 80, pp.4331–4347, doi.
Analytic microlocal analysis and tomography
- A. Homan and H. Zhou, Injectivity and stability for a generic class of generalized Radon transforms, The Journal of Geometric Analysis, 27 (2017), pp.1515–1529, doi.
- P. Stefanov, Support theorems for the light ray transform on analytic Lorentzian manifolds, Proceeding of the American Mathematical Society, 145 (2017), pp.1259-1274, doi.
- E. T. Quinto, Radon transforms satisfying the Bolker assumption, pp. 263-270, Proceedings of conference "Seventy-five Years of Radon Transforms", International Press Co. Ltd., Hong Kong, 1994.
- E. T. Quinto, Support theorems for the spherical Radon transform on manifolds, International Mathematics Research Notices, Volume 2006 (2006), article ID 67205, doi.
Microlocal Artifacts
- J. K. Choi, H. S. Park, S. Wang, Y. Wang, and J. K. Seo, Inverse problem in quantitative susceptibility mapping, SIAM Jornal of Imaging Science, 7 (2014), pp.1669–1689, doi.
- H. S. Park, J. K. Choi, and J. K. Seo, Characterization of metal artifacts in X-ray computed tomography, Communications on Pure and Applied Mathematics, 70 (2017), pp.2191–2217, doi.
- B. Palacios, G. Uhlmann and Y. Wang, Quantitative analysis of metal artifacts in X-ray tomography, SIAM Journal on Mathematical Analysis, 50 (2018), pp.4914--4936, doi.
- Y. Wang and Y. Zou, Streak artifacts from non-convex metal objects in X-ray tomography, Pure and Applied Analysis, 3 (2021), pp.295-318, doi.
Spiral CT
- A. Katsevich, Theoretically exact filtered backprojection-type inversion algorithm for spiral CT, SIAM Journal of Applied Mathematics, 62 (2002), pp.2012–2026, doi.
- A. Katsevich, Microlocal analysis of an FBP algorithm for truncated spiral cone beam data, Journal of Fourier Analysis and Applications, 8 (2002), pp.407–425, doi.
- A. Katsevich, An improved exact filtered backprojection algorithm for spiral computed tomography, Advances in Applied Mathematics 32 (2004), pp.681–697, doi.
- A. Katsevich, Stability estimates for helical computer tomography, Journal of Fourier Analysis and Applications, 11 (2005), pp.85–105, doi.
- A. Katsevich, 3PI algorithms for helical computer tomography, Advances in Applied Mathematics 36 (2006), pp.213–250, doi.
- M. Kapralov and A. Katsevich, A one-PI algorithm for helical trajectories that violate the convexity condition, Inverse Problems 22 (2006), pp.2123–2143, doi.
- A. Katsevich and M. Kapralov, Filtered backprojection inversion of the cone beam transform for a general class of curves, SIAM Journal Applied Mathematics, 68 (2007), pp.334–353, doi.
- M. Kapralov and A. Katsevich, A study of 1PI algorithms for a general class of curves, SIAM Journal Imaging Science, 1 (2008), pp.418–459, doi.
Limied Data
- L. Borg, J. Frikel, J. S. Jørgensen and E. T. Quinto, Analyzing reconstruction artifacts from arbitrary incomplete X-ray CT data, SIAM Journal on Imaging Sciences, 11 (2018), pp.2786--2814, doi.
- D. Finch, I.-R. Lan and G. Uhlmann, Microlocal analysis of the x-ray transform with sources on a curve, Inside out: inverse problems and applications, Mathematical Sciences Research Institute Publications, 47 (2003), pp.193--218, pdf.
- J. Frikel and E. T. Quinto, Limited Data Problems for the Generalized Radon Transform in Rⁿ, SIAM Journal on Mathematical Analysis, 48 (2016), pp.2301--2318, doi.
- C. Mathison, Sampling in thermoacoustic tomography, Journal of Inverse and Ill-posed Problems, 28 (2020), pp.881–897, doi.
- E. T. Quinto, Singularities of the X-Ray Transform and Limited Data Tomography in R² and R³, SIAM Journal on Mathematical Analysis, 24 (1993), pp.1215--1225, doi.
Sampling
- P. Stefanov, Semiclassical sampling and discretization of certain linear inverse problems, SIAM Journal on Mathematical Analysis, 52 (2020), pp.5554–5597, doi.
- P. Stefanov and S. Tindel, Sampling linear inverse problems with noise, Asymptotic Analysis, in press, doi.
- P. Stefanov, The Radon transform with finitely many angles, arXiv:2208.05936 .
- F. Monard and P. Stefanov, Sampling the X-ray transform on simple surfaces, SIAM Journal on Mathematical Analysis, 55(3) (2023), pp.1707-1736, doi.
Tensor Tomography
- G. P. Paternain, M. Salo and G. Uhlmann, Tensor tomography: Progress and challenges, Chinese Annales of Mathematics, Series B, 35 (2014), pp.399--428, doi.
- G. P. Paternain, M. Salo and G. Uhlmann, Invariant distributions, Beurling transforms and tensor tomography in higher dimensions, Mathematische Annalen, 363 (2015), pp.305--362, doi.
- P. Stefanov, G. Uhlmann, A. Vasy and H. Zhou, Travel time tomography, Acta Mathematica Sinica, English Series, 35 (2019), pp.1085--1114, doi.
- P. Stefanov, G. Uhlmann and A. Vasy, Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge, Annales of Mathematics, 194 (2021), pp.1--95, doi.

Microlocal Analysis and Applied Mathematics

Imaging Science
- Bin Dong, Ting Lin, Zuowei Shen, and Peichu Xie, Analysis of a wavelet frame based two-scale model for enhanced edges, SIAM Journal on Imaging Sciences, {\bf 18} (2025), pp.2008-2057, doi.
Deep Learning
- Daniel Freeman, and Daniel Haider, Optimal lower Lipschitz bounds for ReLU layers, saturation, and phase retrieval, Applied and Computational Harmonic Analysis, {\bf 80} (2026), pp.2008-2057, 101801, doi.
Sampling and Approximation of PsDOs and FIOs
- Maarten V. de Hoop, Gunther Uhlmann, András Vasy, and Herwig Wendt, Multiscale Discrete Approximations of Fourier Integral Operators Associated with Canonical Transformations and Caustics, Multiscale Modeling & Simulation 11(2) (2013), pp. 566–585, doi.
- Emmanuel Candè&s, Laurent Demanet, and Lexing Ying, A Fast Butterfly Algorithm for the Computation of Fourier Integral Operators, Multiscale Modeling & Simulation, 7(4) (2009), pp. 1727–1750, doi.
- Emmanuel Candè&s, Laurent Demanet, and Lexing Ying, Fast Computation of Fourier Integral Operators, SIAM Journal on Scientific Computing, 29(6) (2007), pp. 2464–249, doi.
Descrete Differential Geometry
- Alexander I. Bobenko and Yuri B. Suris, "Discrete Differential Geometry: Integrable Structure", Graduate Studies in Mathematics, 98, American Mathematical Society, Providence, RI, 2008, url.
- Iasonas Kokkinos, Michael M. Bronstein, Roee Litman, and Alex M. Bronstein, Intrinsic shape context descriptors for deformable shapes, 2012 IEEE Conference on Computer Vision and Pattern Recognition, Providence, RI, USA, 2012, pp.159-166, doi.
- Jonathan Masci, Davide Boscaini, Michael M. Bronstein, and Pierre Vandergheynst, Geodesic Convolutional Neural Networks on Riemannian Manifolds, 2015 IEEE International Conference on Computer Vision Workshop (ICCVW), (2015), pp.832-840, doi.
Finite Difference Scheme on Surface Mesh
- Qiang Du and Lili Ju, Finite volume methods on spheres and spherical centroidal voronoi meshes, SIAM Journal on Numerical Analysis, 43(4) (2005), pp.1673-1692, doi.
- Do Y. Kwak and Jun S. Lee, Comparison of V-cycle multigrid method for cell-centered finite difference on triangular meshes, Numerical Methods for Partial Differential Equations, 22(5) (2006), pp.1080-1089, doi.
- Wei Liu and Yirang Yuan, Finite difference schemes for two-dimensional miscible displacement flow in porous media on composite triangular grids, Computers and Mathematics with Applications, 55(3) (2008), pp.470-484, doi.
- P. Salinas, C. Rodrigo, F. J. Gaspar and F. J. Lisbona, Multigrid methods for cell-centered discretizations on triangular meshes, Numerical Linear Algebra with Applications, 20(4) (20), pp.626-644, doi.
- C. Rodrigo ∗, P. Salinas, F.J. Gaspar and F.J. Lisbona, Local Fourier analysis for cell-centered multigrid methods on triangular grids, Journal of Computational and Applied Mathematics, 259 (2014), pp.35-47, doi.
- Pratik Suchde and Jörg Kuhnert, A meshfree generalized finite difference method for surface PDEs, Computers & Mathematics with Applications, 78(8) (2019), pp.2789-2805, doi.
- Andrew M. Jones, Peter A. Bosler, Paul A. Kuberry and Grady B. Wrigh, Generalized moving least squares vs. radial basis function finite difference methods for approximating surface derivatives, Computers & Mathematics with Applications, 147 (2023), pp.1-13, doi.
- Oleg Davydov, Error bounds for a least squares meshless finite difference method on closed manifolds, Advances in Computational Mathematics, 49 (2023), Article Number 48, 42 papes, doi.
- Grady B. Wright, Andrew Jones and Varun Shankar, MGM: a meshfree geometric multilevel method for systems arising from elliptic equations on point cloud surfaces, SIAM Journal on Scientific Computiong, 45(2) (2023), A312-A337, doi.
- Qile Yan, Shixiao W. Jiang and John Harlim, Kernel-based methods for solving time-dependent advection-diffusion equations on manifolds, Journal of Scientific Computiong, 94(5) (2023), doi.
Computer Vision
- Ke Chen, Carola-Bibiane Schönlieb, Xue-Cheng Tai, and Laurent Younes eds, "Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging: Mathematical Imaging and Vision", Springer Cham, 2023, url.
- Jonathan Masci, Davide Boscaini, Michael M. Bronstein, and Pierre Vandergheynst, Geodesic convolutional neural networks on Riemannian manifolds, ICCVW 2015 (2015 IEEE International Conference on Computer Vision Workshop), pp.832-840, doi.
- Federico Monti, Davide Boscaini, Jonathan Masci, Emanuele Rodolà, Jan Svoboda, and Michael M. Bronstein, Geometric deep learning on graphs and manifolds using mixture model CNNs, CVPR 2017 (2017 IEEE Conference on Computer Vision and Pattern Recognition), pp.5425-5434, doi.
- Simone Melzi, Riccardo Spezialetti, Federico Tombari, Michael M. Bronstein, Luigi Di Stefano, and Emanuele Rodolà, GFrames: gradient-based local reference frame for 3D shape matching, CVPR 2019 (2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition), pp.4624-4633, doi.
- Taco Cohen, Maurice Weiler, Berkay Kicanaoglu, and Max Welling, Gauge equivariant convolutional networks and the icosahedral CNN, PMLR (Proceedings of Machine Learning Research) 97 (2019), pp.1321-1330, url.
- Pim de Haan, Maurice Weiler, Taco Cohen, and Max Welling, Gauge equivariant mesh CNNs: anisotropic convolutions on geometric graphs, ICLR 2021 (International Conference on Learning Representations 2021), url.
Microlocal Analysis and Deep Learning
- H. Andrade-Loarca, G. Kutyniok, O. Öktem, and P. C. Petersen, Extraction of digital wavefront sets using applied harmonic analysis and deep neural network, SIAM Journal on Imaging Sciences, 12 (2019), pp.1936--1966, doi.
- H. Andrade-Loarca, G. Kutyniok and O. Öktem Shearlets as feature extractor for semantic edge detection: the model-based and data-driven realm, Proceedings of the Royal Society A: Mathematical, Phisical and Engineering Science, 25 November 2020, doi.
- H. Andrade-Loarca, G. Kutyniok, O. Öktem and P. Petersen, Deep microlocal reconstruction for limited-angle tomography, Applied and Computational Harmonic Analysis, 59 (2022), pp.155-197, doi.
- T. A. Bubba, M. Galinier, M. Lassas, M. Prato, L. Ratti and S. Siltanen, Deep neural networks for inverse problems with pseudodifferential operators: an application to limited-angle tomography, SIAM Journal of Imaging Science, 14 (2021), pp.470-505, doi.
- T. A. Bubba, G. Kutyniok, M. Lassas, M. Marz, W. Samek, S. Siltanen and V. Srinivasan, Learning the invisible: a hybrid deep learning-shearlet framework for limited angle computed tomography, Inverse Problems, 35 (2019), 064002, doi.
- Siiri Rautio, Rashmi Murthy, Tatiana A. Bubba, Matti Lassas, and Samuli Siltanen, Learning a microlocal prior for limited-angle tomography, IMA Journal of Applied Mathematics, 88 (2023), pp.888-916, doi.
- P. Grohs, Continuous shearlet frames and resolution of the wavefront set, Monatshefte für Mathematik, 164 (2011), pp.393--426, doi.
- Gunther Uhlmann and Yiran Wang, Convolutional neural networks in phase space and inverse problems, SIAM Journal on Applied Mathematics, 80(6) (2020), pp.2560–2585, doi.
- B. Han, S. Paul and N. K. Shukla, Microlocal analysis and characterization of Sobolev wavefront sets using shearlets, Constructive Approximation, (2021), doi.
- G. Kutyniok and D. Labate, Resolution of the wavefront set using continuous shearlets, Transactions of the American Mathematical Society, 361 (2009), pp.2719--2754, doi.
- J. Fell, H. Führ, and F. Voigtlaender, Resolution of the wavefront set using general continuous wavelet transforms, Journal of Fourier Analysis and Applications, 22 (2016), pp.997--1058, doi.

Introductry References

Deep Learning and Neural Networks
- Philipp Grohs and Gitta Kutyniok eds, "Mathematical Aspects of Deep Learning", Cambridge University Press, 2022, doi.
- Gitta Kutyniok, The Mathematics of Artificial Intelligence, arXiv:2203.08890.
- Benoit Liquet, Sarat Moka and Yoni Nazarathy, "The Mathematical Engineering of Deep Learning", CRC Data Science Series, Chapman & Hall, 2024, url, support page, examples.
- Bastian Bohn, Jochen Garcke, and Michael Griebel, "Algorithmic Mathematics in Machine Learning", SIAM, 2024, url.
- Michael M. Bronstein, Joan Bruna, Taco Cohen, Petar Veličković, Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges, arXiv:2104.13478, url.
- Anthea Monod et al Tropical sufficient statistics for persistent homology, SIAM Journal on Applied Algebra and Geometry, 3 (2019), pp.337-371.
- Stéphane Mallat, Understanding deep convolutional networks, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 374 (2016), Issue 2065, arXiv:2305.05601.
- Ashish Vaswani et al, Attention is all you need, Advances in Neural Information Processing Systems 30 (NIPS 2017), url.
- Geordie Williamson, Is deep learning a useful tool for the pure mathematician?, Bulletin of the American Mathematical Society, 61 (2024), pp.271-286, doi.
- Borjan Geshkovski, Cyril Letrouit, Yury Polyanskiy, and Philippe Rigollet, A mathematical perspective on transformers, Bulletin of the American Mathematical Society, 62 (2025), pp.427-479, doi.
Toplogy and Data
- Mustafa Hajij et al, Topological Deep Learning: Going Beyond Graph Data, arXiv:2206.00606.
- Gunnar Carlsson, Topology and data, Bulletin of the American Mathematical Society, 46 (2009), pp.255-308, url
- Herbert Edelsbrunner and John L. Harer, "Computational Topology: An Introduction", American Mathematical Society, 2010, url.
- Robert Ghrist, "Elementary Applied Topology", Createspace, 2014, url.
Wasserstein Geometry
- Cédric Villani, "Topics in Optimal Transportation", American Mathematical Society, 2003, url.
- Filippo Santambrogio, "Optimal Transport for Applied Mathematicians", Springer, 2015, doi.
- Gabriel Peyré and Marco Cuturi, "Computational Optimal Transport: With Applications to Data Science", Now Foundations and Trends, 2019, doi.
Some Papers on Integral Geometry and Geometric Tomography
- C. L. Epstein, Introduction to magnetic resonance imaging for mathematicians, Annales de l'institut Fourier (Grenoble), 54 (2004), pp.1697--1716, doi.
- T. Kaasalainen, M. Ekholm, T. Siiskonen, and M. Kortesniemi, Dental cone beam CT: An updated review, European Journal of Medical Physics, 88 (2021), pp.193--217, doi.
- P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, European Journal of Applied Mathematics, 19 (2008), pp.191--224, doi.
- P. Kuchment and L. Kunyansky, Mathematics of Photoacoustic and Thermoacoustic Tomography, Handbook of mathematical methods in imaging. Vol. 1, 2, 3, pp.817--867, Springer, New York, 2015, doi.
- R. G. Novikov, An inversion formula for the attenuated X-ray transformation, Arkiv för Matematik, 40 (2002), pp.145--167, doi.
Some Books on Integral Geometry and Geometric Tomography
- C. L. Epstein, "Introduction to the mathematics of medical imaging, Second edition", SIAM, 2008, doi.
- R. J. Gardner, "Geometric Tomograhy" Second Edition, Encyclopedia of Mathematics and its Applications, 58, Cambridge University Press, 1995, 2006, url.
- S. Helgason, "Integral Geometry and Radon Transforms", Springer, 2011, doi.
- P. Kuchment, "The Radon Transform and Medical Imaging", CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 2014, doi.
- F. Natterer, "The Mathematics of Computerized Tomography", SIAM, 2001, doi.
- F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction", SIAM monographs on mathematical modeling and computation, SIAM, 2001, doi.
- G. Olafsson and E. T. Quinto eds, "The Radon Transform, Inverse Problems, and Tomography", Proceedings of Symposia in Applied Mathematics, 63, American Mathematical Society, Providence, RI, 2006, url.
- R. Ramlau and O. Scherzer eds, "The Radon Transform - The First 100 Years and Beyond", Radon Series on Computational and Applied Mathematics 22, de Gruyter, 2019, url.
- B. Rubin, "Introduction to Radon Transforms, With Elements of Fractional Calculus and Harmonic Analysis", Encyclopedia of Mathematics and its Applications, 160, Cambridge University Press, 2015, url.
- O. Scherzer eds, "Handbook of Mathematical methods in Imaging", Springer, 2015, url, contents.
- V. A. Sharafutdinov, "Integral Geometry of Tensor Fields", Inverse and Ill-Posed Problems Series, 1, De Gruyter, 1994, doi.
Original Papers or Surveys on Basic Ideas
- A. M. Cormack, Representation of a function by its line integrals, with some radiological applications, Journal of Applied physics, 34 (1963), pp.2722, doi.
- A. M. Cormack, Representation of a function by its line integrals, with some radiological applications. II, Journal of Applied physics, 35 (1964), pp.2908, doi.
- P. C. Lauterbur, Image formation by induced local interactions: Examples employing nuclear magnetic resonance, Nature, 242 (1973), pp.190--191, doi.
- G. Beylkin, The inversion problem and applications of the generalized radon transform, Communications on Pure and Applied Mathematics, 37 (1984), pp.579--599, doi.
- M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Review, 41 (1999), pp.85--101, doi.
- L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), pp.R99--R136, doi.
Calculus, Linear Algebra, Scientific Programming and etc
- Alan Edelman, David P. Sanders and Charles E. Leiserson, MIT 18.S191 Introduction to Computational Thinking, Spring 2021, url
- G. Strang, "Linear Algebra and Learning from Data", Wellesley Publishers, 2018, support page.
- G. Strang, MIT 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, Spring 2018, YouTube Playlist.
- C. Eckart and G. Young, The approximation of one matrix by another of lower rank, Psychometrika, 1 (1936), pp.211–218, doi.
- M. Udell and A. Townsend, Why are big data matrices approximately low rank?, SIAM Journal on Mathematics of Data Science, 1, (2019), pp.144-160, doi.