{ "id": "ep7x34s6hm", "slug": "radon-transform", "title": "Radon transform", "updated": "2026-10-21", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.analysis", "label": "Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis" }, { "id": "physical.math.analysis.harmonic_analysis_and_transforms", "label": "Harmonic analysis, transforms, and integral equations", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis.harmonic_analysis_and_transforms" } ], "geo": [ { "id": "geo.nongeo.t1800.physical.math.analysis", "label": "Non-geographic · 1800 to 1945: Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.analysis", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1800", "label": "Non-geographic · 1800 to 1945", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800" }, { "id": "geo.nongeo.t1800.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical" }, { "id": "geo.nongeo.t1800.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math" }, { "id": "geo.nongeo.t1800.physical.math.analysis", "label": "Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.analysis" } ] }, { "id": "geo.nongeo.t1946.physical.math.analysis.harmonic_analysis_and_transforms", "label": "Non-geographic · 1946 to 2000: Harmonic analysis, transforms, and integral equations", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis.harmonic_analysis_and_transforms", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.analysis", "label": "Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis" }, { "id": "geo.nongeo.t1946.physical.math.analysis.harmonic_analysis_and_transforms", "label": "Harmonic analysis, transforms, and integral equations", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis.harmonic_analysis_and_transforms" } ] } ], "excerpt": "The Radon transform is an integral transform mapping a function to its integrals over lines or hyperplanes, introduced by Johann Radon in 1917 and central to CT image reconstruction.", "snippet": "The Radon transform is an integral transform mapping a function to its integrals over lines or hyperplanes, introduced by Johann Radon in 1917 and central to CT image reconstruction.", "node": "physical.math.analysis.harmonic_analysis_and_transforms", "markdown": "# Radon transform\n\nThe Radon transform is an [integral transform](https://www.edgepedia.org/integral-transform) that maps a function to its integrals over lines or hyperplanes, so that a two-dimensional image is replaced by its projections at every angle. Inverting it is the mathematical core of computed tomography (CT): the scanning process measures line integrals of the attenuation coefficient, since the transmitted intensity satisfies \\( I_1/I_0 = \\exp(-\\int_L f\\,dx) \\), and reconstruction means recovering \\( f \\) from those integrals.[1](https://epubs.siam.org/doi/book/10.1137/1.9780898719284) For a 2D image, the output is the sinogram: a collection of 1D projections at multiple angles, each projection assigning the integral of the object's contrast along parallel rays to a single pixel.[2](https://scikit-image.org/docs/0.26.x/auto_examples/transform/plot_radon_transform.html)\n\n| Key fact | Value | Source |\n|---|---|---|\n| Definition (2D) | \\( Rf(\\theta, s) = \\int_{\\Theta^\\perp} f(x + s\\theta)\\,dx \\), integrals over lines orthogonal to \\(\\theta\\) at signed distance \\( s \\) | [3](https://library.slmath.org/books/Book47/files/faridani.pdf) |\n| Output for a 2D image | The sinogram, one 1D projection per angle | [2](https://scikit-image.org/docs/0.26.x/auto_examples/transform/plot_radon_transform.html) |\n| Fourier slice theorem | \\( (Rf)\\tilde{\\ }(\\sigma,\\omega) = \\hat{f}(\\sigma\\omega) \\): the 1D Fourier transform of a projection is a slice of the 2D Fourier transform | [4](https://assets.cambridge.org/97813165/10872/excerpt/9781316510872_excerpt.pdf) |\n| FBP inversion formula | \\( f = \\frac{1}{4\\pi} R^* \\lvert D_s \\rvert Rf \\) | [4](https://assets.cambridge.org/97813165/10872/excerpt/9781316510872_excerpt.pdf) |\n| FBP time complexity | \\( N^3 \\) on an \\( N \\times N \\) lattice; fast \\( O(N^2 \\log N) \\) backprojection methods exist | [5](https://arxiv.org/pdf/2402.12141) |\n| Ill-posedness | Singular values tend to zero; inversion loses about half the accurate digits | [6](https://candes.su.domains/teaching/math262/Lectures/Lecture11.pdf) • [3](https://library.slmath.org/books/Book47/files/faridani.pdf) |\n| Introduced by | J. Radon, 1917, Ber. Verh. Sächs. Akad. 69, pp. 262–277 | [7](https://encyclopediaofmath.org/wiki/Radon_transform) |\n\n## How it works\n\nIn two dimensions the Radon transform of \\( f \\) is defined by \\( Rf(\\theta, s) = R_\\theta f(s) = \\int_{\\Theta^\\perp} f(x + s\\theta)\\,dx \\), with \\( s \\in \\mathbb{R} \\): the integral of \\( f \\) over the line orthogonal to the unit vector \\( \\theta \\) at signed distance \\( s \\).[3](https://library.slmath.org/books/Book47/files/faridani.pdf) Equivalently, \\( Rf(\\theta, s) = \\int_{x \\cdot \\theta = s} f(x)\\,dx \\).[8](https://staff.fnwi.uva.nl/c.c.stolk/FourierAnalysis2013/notes_Radon1.pdf) In \\( n \\) dimensions the lines become hyperplanes. In two dimensions the Radon transform coincides with the [X-ray transform](https://www.edgepedia.org/x-ray-transform), which integrates along straight lines, apart from the parameterization.[3](https://library.slmath.org/books/Book47/files/faridani.pdf)\n\nUniqueness comes from the Fourier slice theorem: \\( (Rf)\\tilde{\\ }(\\sigma,\\omega) = \\hat{f}(\\sigma\\omega) \\), so the 1D [Fourier transform](https://www.edgepedia.org/fourier-transform) of a projection equals the 2D Fourier transform of the image along a line through the origin.[4](https://assets.cambridge.org/97813165/10872/excerpt/9781316510872_excerpt.pdf) • [9](https://gray.mgh.harvard.edu/media/com_dpattachments/attachments/com_content.article/Image-Reconstruction-Part-I-Planar-Reconstruction-from-Projections.pdf) It follows that the Radon transform completely determines any Fourier-transformable object.[10](https://cw.fel.cvut.cz/b232/_media/courses/zsl/c-tomo.pdf)\n\nThe normal operator satisfies \\( R^*R = 4\\pi \\lvert D \\rvert^{-1} \\), giving \\( f = \\frac{1}{4\\pi \\lvert D \\rvert} R^* R f \\).[4](https://assets.cambridge.org/97813165/10872/excerpt/9781316510872_excerpt.pdf) The filtered backprojection form is \\( f = \\frac{1}{4\\pi} R^* \\lvert D_s \\rvert Rf \\), where \\( \\lvert D_s \\rvert Rf \\) is the inverse Fourier transform of \\( \\lvert \\sigma \\rvert (Rf)\\tilde{\\ } \\) with respect to \\( \\sigma \\).[4](https://assets.cambridge.org/97813165/10872/excerpt/9781316510872_excerpt.pdf) In Radon's own formulation, Theorem III is understood as a Stieltjes integral over tangents of circles about \\( P \\).[11](https://websites.umich.edu/~ners580/ners-bioe_481/lectures/pdfs/1917-Radon_inverseTransform%28translation1986%29.pdf) In \\( n \\) dimensions the formula splits by parity: for odd \\( n \\) only local information near the point is needed, while for even \\( n \\) integrals over all hyperplanes meeting the support are required.[12](https://encyclopediaofmath.org/wiki/Tomography) The convolution identity \\( f \\star (R^\\# g) = R^\\#(g \\star Rf) \\) is the basis of filtered backprojection.[12](https://encyclopediaofmath.org/wiki/Tomography)\n\n## How it is done\n\nNo practical exact implementation of the inverse Radon transform exists; only good approximate algorithms are available.[2](https://scikit-image.org/docs/0.26.x/auto_examples/transform/plot_radon_transform.html) As a sampling rule of thumb, the number of projections should be about the same as the number of pixels across the object.[2](https://scikit-image.org/docs/0.26.x/auto_examples/transform/plot_radon_transform.html)\n\nFiltered backprojection (FBP) is the most widely used reconstruction method in tomography[10](https://cw.fel.cvut.cz/b232/_media/courses/zsl/c-tomo.pdf) and is viewed as a computer implementation of the Radon inversion formula.[1](https://epubs.siam.org/doi/book/10.1137/1.9780898719284) Its four steps are: (1) Fourier transform each projection \\( \\lambda_\\phi(p) \\to \\Lambda_\\phi(\\nu) \\); (2) multiply by \\( \\lvert \\nu \\rvert \\); (3) inverse Fourier transform; (4) backproject and integrate over \\( \\phi \\).[9](https://gray.mgh.harvard.edu/media/com_dpattachments/attachments/com_content.article/Image-Reconstruction-Part-I-Planar-Reconstruction-from-Projections.pdf) Unfiltered backprojection alone yields a blurred laminogram that would need deconvolution; applying the 1D ramp filter to each projection first avoids that.[10](https://cw.fel.cvut.cz/b232/_media/courses/zsl/c-tomo.pdf) The discrete ramp filter bandlimited to \\( 1/(2\\Delta p) \\) is known as the Ram-Lak filter; Shepp-Logan, cosine, Hamming, and Hann are smoothed variants.[9](https://gray.mgh.harvard.edu/media/com_dpattachments/attachments/com_content.article/Image-Reconstruction-Part-I-Planar-Reconstruction-from-Projections.pdf) • [2](https://scikit-image.org/docs/0.26.x/auto_examples/transform/plot_radon_transform.html)\n\nFBP needs only 1D Fourier transforms, whereas direct Fourier and backproject-filter methods require 2D transforms and suffer interpolation artifacts and wrap-around effects.[10](https://cw.fel.cvut.cz/b232/_media/courses/zsl/c-tomo.pdf) On an \\( N \\times N \\) lattice with \\( O(N) \\) samples in \\( s \\) and \\( \\theta \\), FBP costs \\( N^3 \\) operations, while fast \\( O(N^2 \\log N) \\) backprojection methods exist.[5](https://arxiv.org/pdf/2402.12141)\n\n## Origin\n\nThe paper solves the inversion of the linear functional transformation taking a point function in the plane to its straight-line integral values, answering whether every suitable line function arises this way, whether \\( f \\) is unique, and how to compute it.[11](https://websites.umich.edu/~ners580/ners-bioe_481/lectures/pdfs/1917-Radon_inverseTransform%28translation1986%29.pdf)\n\nDetermining a function on the sphere from its great-circle integrals was realized via Abelian integral equations, relating to a geometric theorem.[11](https://websites.umich.edu/~ners580/ners-bioe_481/lectures/pdfs/1917-Radon_inverseTransform%28translation1986%29.pdf) • [13](https://math.mit.edu/~helgason/integral-geometry.pdf) • [14](https://bpb-us-e1.wpmucdn.com/sites.tufts.edu/dist/e/6333/files/2021/01/sc-article.pdf) • [13](https://math.mit.edu/~helgason/integral-geometry.pdf) It was not until the transform was reinvented in 1963 that it was used in tomography; Cormack shared the 1979 Nobel Prize in Medicine with Hounsfield, who built the first medical CT scanner.[14](https://bpb-us-e1.wpmucdn.com/sites.tufts.edu/dist/e/6333/files/2021/01/sc-article.pdf)\n\n## Variants\n\nIn 2D the X-ray transform coincides with the Radon transform up to parameterization.[3](https://library.slmath.org/books/Book47/files/faridani.pdf) FBP extends to fan-beam sampling, and cone-beam geometry handled reconstruction of data from large-area detectors.[3](https://library.slmath.org/books/Book47/files/faridani.pdf) • [15](https://rcastoragev2.blob.core.windows.net/de8e8cb411b59498acaa8d4420d84812/PMC6443602.pdf)\n\nRadon himself proved in 1917 that a differentiable function on \\( \\mathbb{R}^3 \\) is determined explicitly by its integrals over planes, and extended his results to hyperplanes in higher dimensions and to non-Euclidean (elliptic and hyperbolic) planes, also discussing determining a function on the hyperbolic plane from its geodesic integrals.[13](https://math.mit.edu/~helgason/integral-geometry.pdf) • [11](https://websites.umich.edu/~ners580/ners-bioe_481/lectures/pdfs/1917-Radon_inverseTransform%28translation1986%29.pdf) • [12](https://encyclopediaofmath.org/wiki/Tomography) Later generalizations include the k-plane (Radon–John) transform integrating over k-dimensional planes, the attenuated Radon transform, and generalized Radon transforms defined via double fibrations, with inversion formulas on \\( L^2 \\) and \\( L^p \\) spaces.[16](https://www.cambridge.org/core/books/analytic-tomography/BC6F12F55FFFA09931B6379103EDEFC2) Sigurdur Helgason developed the transform on Euclidean spaces, compact two-point [homogeneous spaces](https://www.edgepedia.org/homogeneous-space), and Grassmann manifolds in a 1965 Acta Mathematica paper.[17](https://doi.org/10.1007/bf02391776) Important generalizations are also credited to John, Gel'fand, Helgason, and Strichartz.[18](https://home.ba.infn.it/~facchi/papers/131%20tomomumford.pdf)\n\nAmong iterative variants, SART, an implementation of ART, was introduced by A. H. Andersen and A. C. Kak in 1984 in Ultrasonic Imaging.[15](https://rcastoragev2.blob.core.windows.net/de8e8cb411b59498acaa8d4420d84812/PMC6443602.pdf) • [19](https://doi.org/10.1177/016173468400600107) Differentiable software implementations have followed: TorchRadon, an open-source CUDA PyTorch library with differentiable forward and backward projections, was released by Matteo Ronchetti in 2020 on arXiv,[20](https://doi.org/10.48550/arxiv.2009.14788) and Pyro-NN provides Python reconstruction operators in neural networks, published by Syben and colleagues in 2019 on arXiv.[21](https://doi.org/10.48550/arxiv.1904.13342)\n\n## Applications\n\nIn medical CT, reconstruction is exactly the inversion of the 2D Radon transform.[1](https://epubs.siam.org/doi/book/10.1137/1.9780898719284) Beyond CT, a technique for using Radon transforms to reconstruct a map of a planet's polar regions from a spacecraft in a polar orbit exists.[22](https://mathworld.wolfram.com/RadonTransform.html) [Digital breast tomosynthesis](https://www.edgepedia.org/digital-breast-tomosynthesis) reconstructs 3D slices from a few limited-angle projection images, with FBP, MITS, MLEM, and SART among the algorithms investigated.[23](https://www.mdpi.com/1999-4893/17/2/71) Iterative methods built on the X-ray transform and its adjoint underpin statistical reconstruction in PET and CT, though their cost is dominated by repeatedly applying those operators.[24](https://pmc.ncbi.nlm.nih.gov/articles/PMC3505201/)\n\n## Limitations and alternatives\n\nThe singular values of the Radon transform are \\( \\sqrt{2\\pi/\\lVert \\omega \\rVert} \\) and tend to zero, so the inverse is unbounded.[6](https://candes.su.domains/teaching/math262/Lectures/Lecture11.pdf) The factor \\( \\lvert \\sigma \\rvert \\) in the inverse Fourier integral grows arbitrarily large, making the inversion unstable; the problem is moderately ill-posed, with a loss of about half the number of accurate digits.[3](https://library.slmath.org/books/Book47/files/faridani.pdf) The FBP filter, whose Fourier transform is \\( \\lvert \\omega \\rvert \\), acts like a derivative and amplifies noise.[6](https://candes.su.domains/teaching/math262/Lectures/Lecture11.pdf) A theorem of Smith and colleagues states that a compactly supported planar function is uniquely determined by any infinite set but by no finite set of its tomograms, so uniqueness must be abandoned in applications and regularization introduced.[18](https://home.ba.infn.it/~facchi/papers/131%20tomomumford.pdf)\n\nAll problems in CT are ill-posed to varying degrees, and incomplete-data problems such as the limited angle problem tend to be severely ill-posed.[1](https://epubs.siam.org/doi/book/10.1137/1.9780898719284) In the limited angle problem, the Fourier transform of \\( f \\) is only determined in a cone, which explains missing-wedge instability.[3](https://library.slmath.org/books/Book47/files/faridani.pdf) The instability of limited-angle inversion has been characterized via the singular value decomposition of the operator, and mollification methods delay but do not prevent the onset of instability as the angular range decreases.[25](https://epubs.siam.org/doi/10.1137/0143028) Microlocal analysis predicts which singularities are stably reconstructed from limited data (exterior, interior, limited-angle regions of interest), and Lambda tomography is a singularity-detection alternative using interior data.[14](https://bpb-us-e1.wpmucdn.com/sites.tufts.edu/dist/e/6333/files/2021/01/sc-article.pdf)\n\n[Iterative reconstruction](https://www.edgepedia.org/iterative-reconstruction) is preferable when data for some angles are missing, for artifact suppression, or when noise information can improve results over FBP.[5](https://arxiv.org/pdf/2402.12141) Regularized inversion minimizes \\( \\frac{1}{2}\\lVert g - Rf \\rVert^2 + \\lambda P(f) \\); total variation uses \\( P(f) = \\lVert f \\rVert_{TV} \\), and L1 regularization better preserves edges.[6](https://candes.su.domains/teaching/math262/Lectures/Lecture11.pdf) [Radiation](https://www.edgepedia.org/radiation) dose can be reduced with iterative reconstruction by 23 to 76% without compromising image quality compared to FBP, and the first FDA-cleared iterative algorithm, IRIS (Siemens), was cleared in 2009.[15](https://rcastoragev2.blob.core.windows.net/de8e8cb411b59498acaa8d4420d84812/PMC6443602.pdf) Statistical regularization methods surpass traditional FBP by incorporating system models and physical constraints.[23](https://www.mdpi.com/1999-4893/17/2/71) The Hough transform maps an image to a parameter space much like the Radon transform; the rescaled Hough sinogram converges to the Radon sinogram as the discretization step tends to zero, which may help denoise Radon inversion in PET.[26](https://ar5iv.labs.arxiv.org/html/1605.09201)\n\n## References\n\n1. [The Mathematics of Computerized Tomography (Frank Natterer, SIAM)](https://epubs.siam.org/doi/book/10.1137/1.9780898719284)\n2. [Radon transform, skimage 0.26.0 documentation](https://scikit-image.org/docs/0.26.x/auto_examples/transform/plot_radon_transform.html)\n3. [Introduction to the Mathematics of Computed Tomography (Faridani)](https://library.slmath.org/books/Book47/files/faridani.pdf)\n4. [The Radon Transform in the Plane (Geometric Inverse Problems, Paternain, Salo, Uhlmann)](https://assets.cambridge.org/97813165/10872/excerpt/9781316510872_excerpt.pdf)\n5. [Fast algorithms and efficient GPU implementations for the Radon transform and the back-projection operator represented as convolution operators](https://arxiv.org/pdf/2402.12141)\n6. [Ill-posedness of the inverse problem (Candès lecture notes, Stanford)](https://candes.su.domains/teaching/math262/Lectures/Lecture11.pdf)\n7. [Radon transform - Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Radon_transform)\n8. [The Radon transform (Stolk, UvA lecture notes)](https://staff.fnwi.uva.nl/c.c.stolk/FourierAnalysis2013/notes_Radon1.pdf)\n9. [Image Reconstruction 1 – Planar reconstruction from projections](https://gray.mgh.harvard.edu/media/com_dpattachments/attachments/com_content.article/Image-Reconstruction-Part-I-Planar-Reconstruction-from-Projections.pdf)\n10. [Analytical Tomographic Image Reconstruction (chapter)](https://cw.fel.cvut.cz/b232/_media/courses/zsl/c-tomo.pdf)\n11. [1917 Radon inverseTransform(translation1986) (websites.umich.edu)](https://websites.umich.edu/~ners580/ners-bioe_481/lectures/pdfs/1917-Radon_inverseTransform%28translation1986%29.pdf)\n12. [Tomography - Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Tomography)\n13. [Geometric Analysis on Symmetric Spaces / The Radon Transform (Sigurdur Helgason, MIT)](https://math.mit.edu/~helgason/integral-geometry.pdf)\n14. [An Introduction to X-ray Tomography and Radon Transforms (Eric Todd Quinto)](https://bpb-us-e1.wpmucdn.com/sites.tufts.edu/dist/e/6333/files/2021/01/sc-article.pdf)\n15. [The evolution of image reconstruction for CT, from filtered back projection to artificial intelligence (PMC copy)](https://rcastoragev2.blob.core.windows.net/de8e8cb411b59498acaa8d4420d84812/PMC6443602.pdf)\n16. [Analytic Tomography (Gardner, Cambridge University Press, 2006)](https://www.cambridge.org/core/books/analytic-tomography/BC6F12F55FFFA09931B6379103EDEFC2)\n17. [Sigurdur Helgason (1965). The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds. Acta Mathematica.](https://doi.org/10.1007/bf02391776)\n18. [Tomography: mathematical aspects and applications (review)](https://home.ba.infn.it/~facchi/papers/131%20tomomumford.pdf)\n19. [A. H. Andersen, A. C. Kak (1984). Simultaneous Algebraic Reconstruction Technique (SART): A Superior Implementation of the Art Algorithm. Ultrasonic Imaging.](https://doi.org/10.1177/016173468400600107)\n20. [Ronchetti, Matteo (2020). TorchRadon: Fast Differentiable Routines for Computed Tomography. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.2009.14788)\n21. [Syben, Christopher and colleagues (2019). PYRO-NN: Python Reconstruction Operators in Neural Networks. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1904.13342)\n22. [Radon Transform -- from Wolfram MathWorld](https://mathworld.wolfram.com/RadonTransform.html)\n23. [Algorithms in Tomography and Related Inverse Problems, A Review (MDPI Algorithms, 2024)](https://www.mdpi.com/1999-4893/17/2/71)\n24. [Fast parallel algorithms for the x-ray transform and its adjoint](https://pmc.ncbi.nlm.nih.gov/articles/PMC3505201/)\n25. [The Ill-Conditioned Nature of the Limited Angle Tomography Problem (SIAM)](https://epubs.siam.org/doi/10.1137/0143028)\n26. [The Radon transform and the Hough transform: a unifying perspective](https://ar5iv.labs.arxiv.org/html/1605.09201)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Harmonic analysis, transforms, and integral equations*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026; Oct 21, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://websites.umich.edu/~ners580/ners-bioe_481/lectures/pdfs/1917-Radon_inverseTransform%28translation1986%29.pdf", "https://math.mit.edu/~helgason/integral-geometry.pdf" ], "url": "https://www.edgepedia.org/radon-transform", "markdown_url": "https://www.edgepedia.org/radon-transform.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Radon transform\", Edgepedia (EdgeChat), https://www.edgepedia.org/radon-transform. Edgepedia Community License 1.0.", "credit_md": "\"[Radon transform](https://www.edgepedia.org/radon-transform)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/radon-transform](https://www.edgepedia.org/radon-transform). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Radon transform\", Edgepedia (EdgeChat), https://www.edgepedia.org/radon-transform. Edgepedia Community License 1.0.", "speakable": "The Radon transform is an integral transform mapping a function to its integrals over lines or hyperplanes, introduced by Johann Radon in 1917 and central to CT image reconstruction." }