In mathematics, the Radon transform in two dimensions, named after the Austrian
mathematician Johann Radon, is the integral transform consisting of the
integral of a function over straight lines. The transform was introduced in
1917 by Radon, who also provided a formula for the inverse transform. Radon
further included formulas for the transform in three dimensions, in which
the integral is taken over planes. It was later generalised to
higher-dimensional Euclidean spaces, and more broadly in the context of integral
geometry. The complex analog of the Radon transform is known as the Penrose
transform. The Radon transform is widely applicable
to tomography, the creation of an image from the projection data associated with
cross-sectional scans of an object. If a function ƒ represents an unknown
density, then the Radon transform represents the projection data obtained
as the output of a tomographic scan. Hence the inverse of the Radon transform
can be used to reconstruct the original density from the projection data, and
thus it forms the mathematical underpinning for tomographic
reconstruction, also known as image reconstruction. The Radon transform data
is often called a sinogram because the Radon transform of a Dirac delta
function is a distribution supported on the graph of a sine wave. Consequently
the Radon transform of a number of small objects appears graphically as a number
of blurred sine waves with different amplitudes and phases. The Radon
transform is useful in computed axial tomography, barcode scanners, electron
microscopy of macromolecular assemblies like viruses and protein complexes,
reflection seismology and in the solution of hyperbolic partial
differential equations. Definition
Let ƒ(x) = ƒ(x,y) be a compactly supported continuous function on R2. The
Radon transform, Rƒ, is a function defined on the space of straight lines L
in R2 by the line integral along each such line:
Concretely, the parametrization of any straight line L with respect to arc
length z can always be written where s is the distance of L from the
origin and is the angle the normal vector to L makes with the x axis. It
follows that the quantities can be considered as coordinates on the space
of all lines in R2, and the Radon transform can be expressed in these
coordinates by More generally, in the n-dimensional
Euclidean space Rn, the Radon transform of a compactly supported continuous
function ƒ is a function Rƒ on the space Σn of all hyperplanes in Rn. It is
defined by for ξ ∈Σn, where the integral is taken
with respect to the natural hypersurface measure, dσ. Observe that any element of
Σn is characterized as the solution locus of an equation
where α ∈ Sn−1 is a unit vector and s ∈ R. Thus the n-dimensional Radon
transform may be rewritten as a function on Sn−1×R via
It is also possible to generalize the Radon transform still further by
integrating instead over k-dimensional affine subspaces of Rn. The X-ray
transform is the most widely used special case of this construction, and
is obtained by integrating over straight lines.
Generalized Radon Transform Let s ∈ R, φ ∈[0,2π]. We let μ(φ,x) be a
smooth function, 2π periodic in φ defined on Rx R 2 . Then, the
generalized radon transform is defined as follows.
Where is the arc length measure on the line L(φ, s). Note, that the transform
integrates the weighted function f(x) along the lines L(φ,s).
Relationship with the Fourier transform The Radon transform is closely related
to the Fourier transform. For a function of one variable the Fourier transform is
defined by and for a function of a 2-vector ,
For convenience define as it is only meaningful to take the Fourier transform
in the variable. The Fourier slice theorem then states
where Thus the two-dimensional Fourier
transform of the initial function is the one variable Fourier transform of the
Radon transform of that function. More generally, one has the result valid in n
dimensions Indeed, the result follows at once by
computing the two variable Fourier integral along appropriate slices:
An application of the Fourier inversion formula also gives an explicit inversion
formula for the Radon transform, and thus shows that it is invertible on
suitably chosen spaces of functions. However this form is not particularly
useful for numerical inversion, and faster discrete inversion methods exist.
Dual transform The dual Radon transform is a kind of
adjoint to the Radon transform. Beginning with a function g on the space
Σn, the dual Radon transform is the function R∗g on Rn defined by
The integral here is taken over the set of all lines incident with the point x ∈
Rn, and the measure dμ is the unique probability measure on the set
invariant under rotations about the point x.
Concretely, for the two-dimensional Radon transform, the dual transform is
given by In the context of image processing, the
dual transform is commonly called backprojection as it takes a function
defined on each line in the plane and 'smears' or projects it back over the
line to produce an image. Computationally efficient inversion
formulas reconstruct the image from the points where the back-projection lines
meet. = Intertwining property =
Let Δ denote the Laplacian on Rn: This is a natural rotationally invariant
second-order differential operator. On Σn, the "radial" second derivative
is also rotationally invariant. The Radon transform and its dual are
intertwining operators for these two differential operators in the sense that
Inversion formulas Explicit and computationally efficient
inversion formulas for the Radon transform and its dual are available.
The Radon transform in n dimensions can be inverted by the formula
where and the power of the Laplacian(n−1)/2 is
defined as a pseudodifferential operator if necessary by the Fourier transform
For computational purposes, the power of the Laplacian is commuted with the dual
transform R* to give where Hs is the Hilbert transform with
respect to the s variable. In two dimensions, the operator Hsd/ds appears
in image processing as a ramp filter. One can prove directly from the Fourier
slice theorem and change of variables for integration that for a compactly
supported continuous function ƒ of two variables
Thus in an image processing context the original image ƒ can be recovered from
the 'sinogram' data Rƒ by applying a ramp filter and then back-projecting. As
the filtering step can be performed efficiently and the back projection step
is simply an accumulation of values in the pixels of the image, this results in
a highly efficient, and hence widely used, algorithm.
Explicitly, the inversion formula obtained by the latter method is
if n is odd, and if n is even.
The dual transform can also be inverted by an analogous formula:
See also Deconvolution
Funk transform The Hough transform, when written in a
continuous form, is very similar, if not equivalent, to the Radon transform.
Cauchy-Crofton theorem is a closely related formula for computing the length
of curves in space. Fourier transform
Notes ^ Radon 1917.
^ a b Roerdink 2001 ^ Helgason 1984, Lemma I.2.1
^ Helgason 1984, Theorem I.2.13 ^ Helgason 1984, Theorem I.2.16
^ Filtered Back Projection nary=mvanginkel_radonandhough_tr2004.pdf
References Deans, Stanley R., The Radon Transform
and Some of Its Applications, New York: John Wiley & Sons .
Helgason, Sigurdur, Geometric analysis on symmetric spaces, Mathematical
Surveys and Monographs 39, Providence, R.I.: American Mathematical Society,
ISBN 978-0-8218-4530-1, MR 2463854 . Helgason, Sigurdur, Groups and Geometric
Analysis: Integral Geometry, Invariant Differential Operators, and Spherical
Functions, Academic Press, ISBN 0-12-338301-3 .
Herman, Gabor T., Fundamentals of Computerized Tomography: Image
Reconstruction from Projections, Springer, ISBN 978-1-85233-617-2 .
Minlos, R.A., "Radon transform", in Hazewinkel, Michiel, Encyclopedia of
Mathematics, Springer, ISBN 978-1-55608-010-4 .
Natterer, Frank, The Mathematics of Computerized Tomography, Classics in
Applied Mathematics 32, Society for Industrial and Applied Mathematics, ISBN
0-89871-493-1 Natterer, Frank; Wübbeling, Frank,
Mathematical Methods in Image Reconstruction, Society for Industrial
and Applied Mathematics, ISBN 0-89871-472-9 .
Radon, Johann, "Über die Bestimmung von Funktionen durch ihre Integralwerte
längs gewisser Mannigfaltigkeiten", Berichte über die Verhandlungen der
Königlich-Sächsischen Akademie der Wissenschaften zu Leipzig,
Mathematisch-Physische Klasse [Reports on the proceedings of the Royal Saxonian
Academy of Sciences at Leipzig, mathematical and physical section]:
262–277 ; Translation: Radon, J.; Parks, P.C., "On the determination of functions
from their integral values along certain manifolds", IEEE Transactions on Medical
Imaging 5: 170–176, doi:10.1109/TMI.1986.4307775, PMID
18244009 . Roerdink, J.B.T.M., "Tomography", in
Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN
978-1-55608-010-4 . Weisstein, Eric W., "Radon transform",
MathWorld..