References on Microlocal Analysis and Related Topics

8 September 2026

• Microlocal Analysis and Integral Geometry
• Applied Mathematics

• Pseudodifferential and Fourier Integral Operators on Manifolds
  • Peter Hintz, "An Introduction to Microlocal Analysis", Springer Cham, 2025, doi.
  • Peter Hintz, Pseudodifferential operators on manifolds with scaled bounded geometry, Communications of the American Mathematical Society, 6 (2026), pp.153-243, doi.
  
  
• X-ray transform, Radon transform, and double fibration transform
  • 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.
  • Marco Mazzucchelli, Mikko Salo, and Leo Tzou, A general support theorem for analytic double fibration transforms, arXiv:2306.05906.
  • 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.
  • 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.
  • 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.
  • P. Stefanov and G. Uhlmann, The geodesic X-ray transform with fold caustics, Analysis & PDE, 5, pp.219-260, doi.
  • András Vasy, A semiclassical approach to geometric X-ray transforms in the presence of convexity, Communications of the American Mathematical Society, 4 (2024), pp.97-116, 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.
  
  
• Limited Data Tomography and 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.
  • 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.
  • 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.
  • 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.
  
  
• 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.
  • G. P. Paternain, M. Salo and G. Uhlmann, "Geometric Inverse Problems: with Emphasis on Two Dimensions", doi, Cambridge University Press, 2023.
  • 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.

• Applied Harmonic Analysis
  • E. J. Candes, J. K. Romberg and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Transactions on Information Theory, 52 (2006), pp.489--509, doi, pdf.
  • E. J. Candes, J. K. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Communications on Pure and Applied Mathematics, 59 (2006), pp.1207--1233, doi, pdf.
  • E. J. Candes and T. Tao, Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?, IEEE Transactions on Information Theory, 52 (2006), pp.5406--5425, doi, pdf.
  • 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.
  • 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.
  • 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.
  
  
• Mathematics of Data Science and AI
  • Michael M. Bronstein, Joan Bruna, Taco Cohen, Petar Veličković, Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges, arXiv:2104.13478, url.
  • Jeff Calder , and Peter J. Olver, "Linear Algebra, Data Science, and Machine Learning", Springer Cham, 2025, doi, GitHub.
  • Gunnar Carlsson, Topology and data, Bulletin of the American Mathematical Society, 46 (2009), pp.255-308, url
  • Mustafa Hajij et al, Topological Deep Learning: Going Beyond Graph Data, arXiv:2206.00606.
  • Philipp Grohs and Gitta Kutyniok eds, "Mathematical Aspects of Deep Learning", Cambridge University Press, 2022, doi.
  • Klaus Hildebrandt, Konrad Polthier, and Max Wardetzky, On the convergence of metric and geometric properties of polyhedral surfaces, Geometriae Dedicata, 123 (2026), pp.89-112, 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.
  • Anthea Monod et al Tropical sufficient statistics for persistent homology, SIAM Journal on Applied Algebra and Geometry, 3 (2019), pp.337-371.
  • Sébastien Roch, "Mathematical Methods in Data Science: Bridging Theory and Applications with Python", Cambridge University Press, 2025, doi, author's page.
  • Richard E. Turner, An Introduction to Transformers, arXiv:2304.10557.
  • Melanie Weber, Manzil Zaheer, Ankit Singh Rawat, Aditya K. Menon, and Sanjiv Kumar, Robust large-margin learning in hyperbolic space, Advances in Neural Information Processing Systems 33 (NeurIPS 2020), url.
  • Nicolás García Trillos and Melanie Weber, Continuum limits of Ollivier's Ricci curvature on data clouds: pointwise consistency and global lower bounds, Discrete & Computational Geometry, (2026), doi.
  • Geordie Williamson, Is deep learning a useful tool for the pure mathematician?, Bulletin of the American Mathematical Society, 61 (2024), pp.271-286, doi.
  
  
• 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.
  • 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.
  • C. L. Epstein, "Introduction to the mathematics of medical imaging, Second edition", SIAM, 2008, doi.
  • 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.
  • 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.