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Integral transform in mathematics
In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional)
Radon_transform
Method of detecting shapes within images
computing the Hough transform. Mathematically it is simply the Radon transform in the plane, known since at least 1917, but the Hough transform refers to its
Hough_transform
Integral transform
geometry, the Funk transform (also known as Minkowski–Funk transform, Funk–Radon transform or spherical Radon transform) is an integral transform defined by integrating
Funk_transform
Austrian mathematician (1887–1956)
so-called Radon–Riesz property. Radon spaces Radonifying function Brigitte Bukovics: Biography of Johann Radon, in: 75 Years of Radon Transform, S. Gindikin
Johann_Radon
Integral transform used in various branches of mathematics
dimensions. Abel transform can be viewed as the Radon transform of an isotropic 2D function f(r). As f(r) is isotropic, its Radon transform is the same at
Abel_transform
Medical diagnostic method
(s,θ) obtained by performing radon transform to μ(x, y), and (2)μ(x, y) is restored by performing inverse radon transform to measurement results. Consider
Operation of computed tomography
Operation_of_computed_tomography
History of radiation-based medical imaging
Radon in 1917 who worked on integral transforms without having a certain practical application in mind. He became the eponym of the Radon transform and
History of computed tomography
History_of_computed_tomography
Estimate object properties from a finite number of projections
obtained in non-invasive manner. Recent developments have seen the Radon transform and its inverse used for tasks related to realistic object insertion
Tomographic_reconstruction
Topics referred to by the same term
buildings Radon may also refer to: Radon, Orne, a town in France Johann Radon, Austrian mathematician Radon transform, in mathematics Radon measure, in
Radon_(disambiguation)
Laplace–Stieltjes transform Legendre transform Linear canonical transform Mellin transform Inverse Mellin transform Poisson–Mellin–Newton cycle Radon transform X-ray
List_of_transforms
Mapping involving integration between function spaces
In mathematics, an integral transform is a type of transformation that maps a function from its original function space into another function space via
Integral_transform
Integral transform
very closely related to the Radon transform, and coincides with it in two dimensions. In higher dimensions, the X-ray transform of a function is defined
X-ray_transform
Generalized function whose value is zero everywhere except at zero
equation). This is essentially a form of the inversion formula for the Radon transform because it recovers the value of φ(x) from its integrals over hyperplanes
Dirac_delta_function
The Mojette transform is an application of discrete geometry. More specifically, it is a discrete and exact version of the Radon transform, thus a projection
Mojette_transform
Topics referred to by the same term
logograph used to write several different languages Radon transform, a type of integral transform in mathematics A visual representation of the raw data
Sinogram
Concept in mathematics
space. Such transformations often take the form of integral transforms such as the Radon transform and its generalizations. Integral geometry as such first
Integral_geometry
Theorem in mathematics
function f(r), project (e.g. using the Radon transform) it onto a (one-dimensional) line, and do a Fourier transform of that projection. Take that same function
Projection-slice_theorem
Mathematical operation
transform. These have been generalized into a supersymmetric FRFT, and a supersymmetric Radon transform. There is also a fractional Radon transform,
Fractional_Fourier_transform
theoretical physics, the Penrose transform, introduced by Roger Penrose (1967, 1968, 1969), is a complex analogue of the Radon transform that relates massless fields
Penrose_transform
German-American mathematician (1910–1994)
differential equations and ill-posed problems. His early work was on the Radon transform and he is remembered for John's equation. He was a 1984 MacArthur Fellow
Fritz_John
Mathematical transform that expresses a function of time as a function of frequency
called the Fourier–Stieltjes transform due to its connection with the Riemann-Stieltjes integral representation of (Radon) measures. If μ {\displaystyle
Fourier_transform
Imaging technique using inaudible air blasts
systems for transmission information based imaging are classical inverse radon transform and fourier slice theorem and derived algorithms (cone beam etc.).
Ultrasound computer tomography
Ultrasound_computer_tomography
Modification using the principle of template matching
transform Randomized Hough transform Radon transform Template matching Outline of object recognition D.H. Ballard, "Generalizing the Hough Transform to
Generalised_Hough_transform
Process of calculating the causal factors that produced a set of observations
Radon transform. Although from a theoretical point of view many linear inverse problems are well understood, problems involving the Radon transform and
Inverse_problem
Icelandic mathematician (1927–2023)
(1964). "A duality in integral geometry; some generalizations of the Radon transform". Bulletin of the American Mathematical Society. 70 (4): 435–446. doi:10
Sigurður Helgason (mathematician)
Sigurður_Helgason_(mathematician)
Imaging by sections or sectioning using a penetrative wave
tomography Johann Radon Medical imaging Network tomography Nonogram – a type of puzzle based on a discrete model of tomography Radon transform Tomographic reconstruction
Tomography
Estimate of the spectral density of a signal
Matched filter Filtered backprojection (Radon transform) Welch's method Bartlett's method Discrete-time Fourier transform Least-squares spectral analysis, for
Periodogram
Result in integral geometry
Barbier's theorem for bodies of constant brightness. Buffon's noodle The Radon transform can be viewed as a measure-theoretic generalization of the Crofton
Crofton_formula
singularity theorem in general relativity Penrose transform, a complex analogue of the Radon transform in theoretical physics Penrose–Terrell effect, visual
List of things named after Roger Penrose
List_of_things_named_after_Roger_Penrose
Four-dimensional number system
2004). "The Bingham distribution of quaternions and its spherical radon transform in texture analysis". Mathematical Geology. 36 (8): 917–943. Bibcode:2004MatGe
Quaternion
Filtered back-projection: efficiently computes the inverse 2-dimensional Radon transform. Level set method (LSM): a numerical technique for tracking interfaces
List_of_algorithms
Group of differential equations
2025-05-21. Retrieved 2026-05-06. L. Ehrenpreis, The Universality of the Radon Transform, Oxford Univ. Press, 2003. Gromov, M. (1986), Partial differential
System of differential equations
System_of_differential_equations
Correspondence between quaternions and 3D rotations
2004). "The Bingham Distribution of Quaternions and Its Spherical Radon Transform in Texture Analysis". Mathematical Geology. 36 (8): 917–943. doi:10
Quaternions and spatial rotation
Quaternions_and_spatial_rotation
Medical imaging technique
CT reconstructions, in that they are based on performing an inverse Radon transform. Due to partial data sampling with very few projections, approximation
Tomosynthesis
Concept in mathematics
the Bochner integral is that the Radon–Nikodym theorem fails to hold in general, and instead is a property (the Radon–Nikodym property) defining an important
Bochner_integral
Coordinate system in two dimensions
Log-polar coordinates can also be used to construct fast methods for the Radon transform and its inverse. Polar coordinates Cartesian coordinates Cylindrical
Log-polar_coordinates
components as well. The τ-p transform is a special case of the Radon transform, and is simpler to apply than the Fourier transform. It allows one to study
Multidimensional seismic data processing
Multidimensional_seismic_data_processing
Integral transform type in mathematics
function from knowledge of its orbital integrals. The Funk transform and Radon transform are two special cases. When G/K is a Riemannian symmetric space
Orbital_integral
theorem – Hans Adolph Rademacher and Dmitrii Menshov Radon transform – Johann Karl August Radon Raman scattering – Chandrasekhara Venkata Raman Ramsauer–Townsend
Scientific phenomena named after people
Scientific_phenomena_named_after_people
Medical imaging procedure
tomography goes back to at least 1917 with the mathematical theory of the Radon transform. In October 1963, William H. Oldendorf received a U.S. patent for a
CT_scan
Technique and process of creating visual representations of the interior of a body
and its contents using the mathematical principles laid out in the Radon transform. It has a greater ionizing radiation dose burden than projection radiography;
Medical_imaging
Set of techniques to measure and visualize aspects of the nervous system
program that performs a numerical integral calculation (the inverse Radon transform) on the measured X-ray series to estimate how much of an X-ray beam
Neuroimaging
Representation theory
for U/K0. An equivalent computation that arises in the theory of the Radon transform has been discussed by Beerends (1987), Stade (1999) and Gindikin (2008)
Plancherel theorem for spherical functions
Plancherel_theorem_for_spherical_functions
Vietnamese-American mathematician
PMID 16593153. Phong DH, Stein EM (1983). "Singular integrals related to the Radon transform and boundary value problems". Proc Natl Acad Sci U S A. 80 (24): 7697–701
Duong_Hong_Phong
Filters used in signal processing that are optimal in some sense
fulfilling a particular task." Periodogram Filtered backprojection (Radon transform) Digital filter Statistical signal processing Whittle likelihood Profile
Matched_filter
more general cases, in mathematical analysis, some analogue of the Radon transform applies. In algebraic geometry, assuming therefore that X is V, a subvariety
Hyperplane_section
slopes have a transcendental cross ratio. Radon transform Funk transform (a.k.a. spherical Radon transform) Tomography Tomographic reconstruction Discrete
Geometric_tomography
Topics referred to by the same term
refer to: Mojette beans, a kind of French beans Mojette transform, an exact discrete Radon transform This disambiguation page lists articles associated with
Mojette
Surface homeomorphic to a sphere
74: 278–300, doi:10.1007/BF01456044 Guillemin, Victor (1976), "The Radon transform on Zoll surfaces", Advances in Mathematics, 22 (1): 85–119, doi:10
Zoll_surface
Ultrahyperbolic partial differential equation
MR 1546052, Zbl 0019.02404 Á. Kurusa, A characterization of the Radon transform's range by a system of PDEs, J. Math. Anal. Appl., 161(1991), 218--226
John's_equation
Subject in mathematics
{\displaystyle f} on E c ′ {\displaystyle E'_{c}} is the Fourier transform of a Radon measure on E {\displaystyle E} if and only if the function is continuous
Measure theory in topological vector spaces
Measure_theory_in_topological_vector_spaces
Method of utilizing water in magnetic resonance imaging
tomography and vector mathematics developed nearly 100 years ago—the Funk Radon Transform. Note, there is ongoing debate about the best way to preprocess DW-MRI
Diffusion-weighted magnetic resonance imaging
Diffusion-weighted_magnetic_resonance_imaging
American mathematician (born 1940)
study of the singularities of the Radon transform is given, a complete description of the asymptotics of the Radon transform near a point of its singular support
Alexander_Ramm
Overview of and topical guide to computer vision
(Perona–Malik equation) Affine transform Homography (computer vision) Hough transform Radon transform Walsh–Hadamard transform Image compression Filter bank
Outline_of_computer_vision
Book by Max Born and Emil Wolf
revolutionised diagnosis in medicine. There is also an account of the Radon transform developed in 1917, which underlies the theory of CAT. An account of
Principles_of_Optics
Soviet Armenian astrophysicist (1908–1996)
tomography, noted that it is the "first numerical inversion of the Radon transform" and suggested that it disproves the "often made statement that computed
Viktor_Ambartsumian
Parameter-free superresolution algorithm
density of a signal Filtered backprojection – Integral transform in mathematics (Radon transform) MUltiple SIgnal Classification – Algorithm used for frequency
SAMV_(algorithm)
Seismological method
wave-field transform base on frequency decomposition and slant stacking, performed by Xia et al. (2007). The fifth is a high-resolution Linear Radon transformation
Surface-wave_inversion
reconstruction algorithms used for CT and OPT are similar; for example, radon transform or iterative reconstruction based on projection data are used in both
Optical_projection_tomography
American mathematician
theory of singular Radon transforms, in various stages of development. He is also well known for a fundamental estimate on Fourier transforms of surface carried
Alexander_Nagel
Anglo-Canadian mathematician
the Sidon set union problem, worked on restrictions of Fourier and Radon transforms to curves, and generalized von Neumann's inequality. In operator theory
Stephen_Drury_(mathematician)
cylindrical measure on the dual of a nuclear space is a Radon measure if its Fourier transform is continuous. It is named after Robert Adol'fovich Minlos
Minlos's_theorem
The input data of a reconstruction algorithm may be given as the Radon transform or sinogram ( Y ) {\displaystyle \left(Y\right)} of the data to reconstruct
Fly_algorithm
Group in group theory and physics
Robert S. (1991). " L p {\displaystyle L^{p}} harmonic analysis and Radon transforms on the Heisenberg group". Journal of Functional Analysis. 96 (2): 350–406
Heisenberg_group
Japanese light novel series
he can enter freely, but others require his permission to enter. Radon (ラードーン, Rādōn) Voiced by: Tomokazu Sugita (Dragon form), Rie Kugimiya (Human form)
I'm a Noble on the Brink of Ruin, So I Might as Well Try Mastering Magic
I'm_a_Noble_on_the_Brink_of_Ruin,_So_I_Might_as_Well_Try_Mastering_Magic
series See also list of transforms, list of Fourier-related transforms Kernel (integral operator) Convolution Radon transform Buffon's needle Hadwiger's
List of integration and measure theory topics
List_of_integration_and_measure_theory_topics
American mathematician
Archived 2006-12-06 at the Wayback Machine Guillemin, Victor (1976). "The Radon transform on Zoll surfaces". Advances in Mathematics. 22 (1): 85–119. doi:10
Victor_Guillemin
Measure defined on all open sets of a topological space
finite, it is called a Radon measure. Alternatively, if a regular Borel measure μ {\displaystyle \mu } is tight, it is a Radon measure. If X {\displaystyle
Borel_measure
Armenian mathematician (born 1941)
Cormack wrote: "Ambartsumian gave the first numerical inversion of the Radon transform and it gives the lie to the often made statement that computed tomography
Rouben_V._Ambartzumian
Integral expressing the amount of overlap of one function as it is shifted over another
analysis that depend on the Fourier transform. Let G be a (multiplicatively written) topological group. If μ and ν are Radon measures on G, then their convolution
Convolution
discrete Fourier slice theorem, which is a generalization of the discrete Radon transform, to compute refocused image. Assume that a lightfield L ¯ f {\displaystyle
Light_field_microscopy
American mathematician
ISBN 978-3-030-67427-4. ISSN 0075-8434. Kuchment, Peter (2013-12-27). The Radon Transform and Medical Imaging. Philadelphia, PA: Society for Industrial and Applied
Peter_Kuchment
Theorem relating stationary processes' autocorrelations and power spectra
< ∞ {\displaystyle -\infty <f<\infty } , or equivalently a non negative Radon measure μ {\displaystyle \mu } on the frequency domain, such that r x x
Wiener–Khinchin_theorem
Objects that generalize functions
are straightforward mappings from both locally integrable functions and Radon measures to corresponding distributions, but not all distributions can be
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
Type of plane curve
MR 2721878 Ricci, Fulvio; Travaglini, Giancarlo (2001), "Convex curves, Radon transforms and convolution operators defined by singular measures", Proceedings
Convex_curve
Russian mathematician and mathematical economist
integral formulas, Akademie Verlag 1988, Birkhäuser 1988 The Abel-Radon transform and several complex variables, in: Annals of Mathematics Studies No
Gennadi_Henkin
Adam Radoń (born July 8, 1971, in Brodnica) is a graphic designer, cultural manager, and long-time director of the International Festival of Comics and
Adam_Radoń
Spanish mathematician
integral geometry of Gelfand” (p. 345) which involves inverting the Radon transform. Includes standard vector algebra, vector analysis, introduction to
Luis_Santaló
Binder used as basic ingredient of concrete
such as radon, with their effectiveness depending on the thickness of the concrete and its diffusion properties. Direct measurement of the radon diffusion
Portland_cement
Unproven hypothesis regarding low doses of ionizing radiation on health
an average annual dose from terrestrial and cosmic radiation, excluding radon which primarily doses the lung). There is limited evidence that exposures
Radiation_hormesis
Integral geometry on manifolds with boundary and applications." The Radon Transform: The First 100 Years and Beyond 22 (2019): 43. Santaló, Luis Antonio
Santaló's_formula
Fundamental Principle". From Fourier Analysis and Number Theory to Radon Transforms and Geometry. Developments in Mathematics. Vol. 28. pp. 491–507. doi:10
Ehrenpreis's fundamental principle
Ehrenpreis's_fundamental_principle
3D model's surface projected to a 2D image
Geodesic Least squares conformal map Mesh parameterization NURBS Polygon mesh Radon transformation Lightmap Digital sculpting Barycentric coordinate system
UV_mapping
Topological vector spaces
necessarily a Radon measure). Lebesgue measure is an example of a positive Radon measure. One particularly important class of Radon measures are those
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Unification of discrete and continuous theories of calculus
s)}f(t)d\mu ^{\Delta }(t)} and the delta derivative turns out to be the Radon–Nikodym derivative with respect to this measure f Δ ( t ) = d f d μ Δ (
Time-scale_calculus
f\colon X\to Y} be a measurable function. Let μ {\displaystyle \mu } be a Radon measure on X {\displaystyle X} and assume that the pushforward measure ν
Mapping theorem (point process)
Mapping_theorem_(point_process)
Distance function defined between probability distributions
z {\displaystyle f(x,y,z)=z} . Compare this with the definition of the Radon metric: ρ ( μ , ν ) := sup { ∫ M f ( x ) d ( μ − ν ) ( x ) | continuous
Wasserstein_metric
Mathematical technique for simplification
_{E}|{\text{det}}D_{x}G|dx} . As a corollary of this theorem, we may compute the Radon–Nikodym derivatives of both the pullback and pushforward measures of m {\displaystyle
Change_of_variables
2008 video game
Schwarze Auge: Drakensang) is a 2008 role-playing video game developed by Radon Labs and published by dtp. It is the first video game in The Dark Eye universe
Drakensang:_The_Dark_Eye
Decomposition of periodic functions
referred to as a Fourier-Stieltjes transform. This follows from an earlier and more concrete representation of a Radon measure (i.e. a locally finite Borel
Fourier_series
Graduate-level textbooks in mathematics
Ito's Perspective Daniel W. Stroock 2003-05-26 288 978-0691115436 156 Radon Transforms and the Rigidity of the Grassmannians Jacques Gasqui, Hubert Goldschmidt
Annals_of_Mathematics_Studies
have a paper recorded in Mathematical Reviews (Moedomo and J. J. Uhl, Jr. "Radon-Nikodym theorems for the Bochner and Pettis integrals" published in the
Moedomo_Soedigdomarto
Generalization of basis splines (B-splines) to multiple variables
spaces generated by box splines spaces are closed under X-ray and Radon transforms. In this application while the signal is represented in shift-invariant
Box_spline
Type of igneous rock
into soils over granite can become a trap for radon gas, which is formed by the decay of uranium. Radon gas poses significant health concerns and is the
Granite
Town in Imereti, Georgia
the Tsqaltubo Municipality of the Imereti province. It is known for its radon-carbonate mineral springs, whose natural temperature of 33–35 °C (91–95 °F)
Tsqaltubo
Process of extraction of uranium ore from the ground
miners to radon as uranium (and its decay product radon) are often found in and near coal deposits and can accumulate underground as radon is denser than
Uranium_mining
Inventiones mathematicae, doi:10.1007/s002220200213 Helgason (2010), "Integral Geometry and Radon Transforms", Springer, doi:10.1007/978-1-4419-6055-9_9
Harish-Chandra's Schwartz space
Harish-Chandra's_Schwartz_space
Theorem of Fourier transforms of Borel measures
theorem (named for Salomon Bochner) characterizes the Fourier-Stieltjes transform of a positive finite Borel measure on the real line. More generally in
Bochner's_theorem
Group of diseases involving cell growth
Sources of ionizing radiation include medical imaging and radon gas. Residential exposure to radon gas has similar cancer risks as passive smoking. Radiation
Cancer
Type of radioactive decay
public radiation dose is radon, a naturally occurring, radioactive gas found in soil and rock. If the gas is inhaled, some of the radon particles may attach
Alpha_decay
RADON TRANSFORM
RADON TRANSFORM
Boy/Male
Hebrew American
Jehovah has heard. A Biblical name.
Boy/Male
Biblical
A chiding; a garment; his measure.
Boy/Male
American, British, Christian, Danish, Dutch, English, Finnish, French, German, Hindu, Indian, Japanese, Spanish, Swedish, Teutonic
Wisely; Advice; Decision Protection; Counselor; Protector; Form of Raymond Guards Wisely; Mighty Protector
Boy/Male
English
From the heath.
Boy/Male
Muslim
Male
Greek
(Λαδων) Greek name of unknown LADON means. In mythology, this is the name of a river god, and the name of a hundred-headed dragon who guarded the garden of the Hesperides. The Ladon was supposed to be a cleansing river.
Male
English
Anglicized form of Hebrew Yadown, JADON means "judge," "thankful," or "whom God has judged." In the bible, this is the name of a man who helped Nehemiah rebuild the walls of Jerusalem.
Boy/Male
Biblical
His redemption; ox-yoke.
Boy/Male
Teutonic
From the hill.
Male
Croatian
, happy.
Surname or Lastname
English
English : variant of Rand 1, from the Old French oblique case.
Boy/Male
Greek
Dragon of Hera.
Boy/Male
English French
Counselor.
Male
Hebrew
Variant spelling of Hebrew Yadown, YADON means "judge," "thankful," or "whom God has judged."
Surname or Lastname
English
English : habitational name from the place so called in West Yorkshire, named with Old Norse rauðr ‘red’ + dūn ‘hill’.
Boy/Male
English
Variants of Randolph 'Wolf's shield.' Surname.
Boy/Male
Teutonic American German Spanish
Wise protector.
Boy/Male
Biblical Hebrew
Ruling; a judgment of malediction.
Male
English
Anglicized form of Hebrew Ardown, ARDON means "fugitive." In the bible, this is the name of a son of Caleb.
Boy/Male
American, British, English
From the Heather Covered Hill; From the Hill of Heather
RADON TRANSFORM
RADON TRANSFORM
Girl/Female
Tamil
Boy/Male
British, English
Place Name; The Awesome One's Meadow
Girl/Female
Australian, Turkish
Moon Stream
Girl/Female
Indian
Pleasant smell, Sweet smell, Fragrance
Girl/Female
Latin
Daughter of Cebren.
Male
Icelandic
Icelandic form of Greek SimÅn, SÃMON means "hearkening."
Boy/Male
Gaelic
Ancient.
Boy/Male
Polish
God is great.
Boy/Male
American, Australian, Greek, Spanish
With Christ Inside; Christ Bearer
Male
English
Variant spelling of English Hughie, HEWIE means "heart," "mind," or "spirit."Â
RADON TRANSFORM
RADON TRANSFORM
RADON TRANSFORM
RADON TRANSFORM
RADON TRANSFORM
n.
Ray; beam.
n.
One of the changes of assimilation, in which proteid matter which has been transformed, and made a part of the tissue or tissue cells, is endowed with life, and thus enabled to manifest the phenomena of irritability, contractility, etc.
n.
Random.
v. t.
To change the form of; to change in shape or appearance; to metamorphose; as, a caterpillar is ultimately transformed into a butterfly.
a.
Capable of being transformed or changed.
v. t.
To change; to transform; to invert.
n.
A small rat.
a.
Having power, or a tendency, to transform.
v. t.
To change into another substance; to transmute; as, the alchemists sought to transform lead into gold.
n.
One who, or that which, transforms. Specif. (Elec.), an apparatus for producing from a given electrical current another current of different voltage.
n.
The act of transforming, or the state of being transformed; change of form or condition.
p. pr. & vb. n.
of Transform
v. t.
To change the form, quality, aspect, or effect of; to alter; to metamorphose; to convert; to transform; -- often with to or into before the word denoting the effect or product of the change; as, to turn a worm into a winged insect; to turn green to blue; to turn prose into verse; to turn a Whig to a Tory, or a Hindu to a Christian; to turn good to evil, and the like.
v. t.
To change from one species to another; to transform.
imp. & p. p.
of Transform
v. t.
To change into another shape or form; to transform.
v. i.
To go or stray at random.
n.
A change of form, direction, or the like; transformation; conversion; turning.
n.
A deity among the ancient Syrians, in honor of whom the Hebrew idolatresses held an annual lamentation. This deity has been conjectured to be the same with the Phoenician Adon, or Adonis.
v. i.
To be changed, altered, or transformed; to become transmuted; also, to become by a change or changes; to grow; as, wood turns to stone; water turns to ice; one color turns to another; to turn Mohammedan.