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) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. The transform was introduced in 1917 by Johann 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 (integrating over lines is known as the X-ray transform). It was later generalized 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.
- 1 Explanation
- 2 Definition
- 3 Relationship with the Fourier transform
- 4 Dual transform
- 5 Reconstruction approaches
- 6 Inversion formulas
- 7 See also
- 8 Notes
- 9 References
- 10 Further reading
- 11 External links
If a function 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 iterative reconstruction.
The Radon transform data is often called a sinogram because the Radon transform of an off-center point source is a sinusoid. 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 (CAT scan), barcode scanners, electron microscopy of macromolecular assemblies like viruses and protein complexes, reflection seismology and in the solution of hyperbolic partial differential equations.
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 (α,s) can be considered as coordinates on the space of all lines in R2, and the Radon transform can be expressed in these coordinates by
for ξ ∈Σn, where the integral is taken with respect to the natural hypersurface measure, dσ (generalizing the |dx| term from the 2-dimensional case). 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.
Relationship with the Fourier transform
The Radon transform is closely related to the Fourier transform. We define the one variable Fourier transform here as
and for a function of a 2-vector ,
For convenience, denote . The Fourier slice theorem then states
Thus the two-dimensional Fourier transform of the initial function along a line at the inclination angle is the one variable Fourier transform of the Radon transform (acquired at angle ) of that function. This fact can be used to compute both the Radon transform and its inverse.
The result can be generalized into n dimensions
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 on Rn defined by
The integral here is taken over the set of all hyperplanes 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.
Let Δ denote the Laplacian on Rn:
This is a natural rotationally invariant second-order differential operator. On Σn, the "radial" second derivative
In analyzing the solutions to the wave equation in multiple spatial dimensions, the intertwining property leads to the translational representation of Lax and Philips. In imaging and numerical analysis this is exploited to reduce multi-dimensional problems into single-dimensional ones, as a dimensional splitting method.
The process of reconstruction produces the image (or function in the previous section) from its projection data. Reconstruction is an inverse problem.
Radon inversion formula
In the 2D case, the most commonly used analytical formula to recover from its Radon transform is the Filtered Backprojection Formula or Radon Inversion Formula:
where is such that .
The convolution kernel is referred to as Ramp filter in some literature.
Intuitively, in the filtered backprojection formula, by analogy with differentiation, for which , we see that the filter performs an operation similar to a derivative. Roughly speaking, then, the filter makes objects more singular.
A quantitive statement of the ill-posedness of Radon Inversion goes as follows:
where is the previously defined adjoint to the Radon Transform.
Thus for ,
The complex exponential is thus an eigenfunction of with eigenvalue . Thus the singular values of are . Since these singular values tend to 0, is unbounded.
Iterative reconstruction methods
Compared with the Filtered Backprojection method, iterative reconstruction costs large computation time, limiting its practical use. However, due to the ill-posedness of Radon Inversion, the Filtered Backprojection method may be infeasible in the presence of discontinuity or noise. Iterative reconstruction methods (e.g., iterative Sparse Asymptotic Minimum Variance) could provide metal artifact reduction, noise and dose reduction for the reconstructed result that attract much research interest around the world.
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
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 (in the variable) and then back-projecting. As the filtering step can be performed efficiently (for example using digital signal processing techniques) 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:
- Matched filter
- X-ray transform
- 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
- Fast Fourier transform
- Radon 1917.
- Roerdink 2001.
- Helgason 1984, Lemma I.2.1.
- Lax, P. D.; Philips, R. S. (1964). "Scattering theory". Bull. Amer. Math. Soc. 70 (1): 130–142.
- Bonneel, N.; Rabin, J.; Peyre, G.; Pfister, H. (2015). "Sliced and Radon Wasserstein Barycenters of Measures". 51 (1): 22–25.
- Rim, D. (2018). "Dimensional Splitting of Hyperbolic Partial Differential Equations Using the Radon Transform". SIAM J. Sci. Comput. 40 (6): A4184–A4207.
- Candès 2016a.
- Candès 2016b.
- Abeida, Habti; Zhang, Qilin; Li, Jian; Merabtine, Nadjim (2013). "Iterative Sparse Asymptotic Minimum Variance Based Approaches for Array Processing" (PDF). IEEE Transactions on Signal Processing. IEEE. 61 (4): 933–944. doi:10.1109/tsp.2012.2231676. ISSN 1053-587X.
- Helgason 1984, Theorem I.2.13.
- Helgason 1984, Theorem I.2.16.
- Nygren 1997.
- van Ginkel, Hendricks & van Vliet 2004.
- Radon, Johann (1917), "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], Leipzig: Teubner (69): 262–277; Translation: Radon, J.; Parks, P.C. (translator) (1986), "On the determination of functions from their integral values along certain manifolds", IEEE Transactions on Medical Imaging, 5 (4): 170–176, doi:10.1109/TMI.1986.4307775, PMID 18244009.
- Roerdink, J.B.T.M. (2001) , "Tomography", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4.
- Helgason, Sigurdur (1984), Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Academic Press, ISBN 0-12-338301-3.
- Candès, Emmanuel (February 2, 2016a). "Applied Fourier Analysis and Elements of Modern Signal Processing - Lecture 9" (PDF).
- Candès, Emmanuel (February 4, 2016b). "Applied Fourier Analysis and Elements of Modern Signal Processing - Lecture 10" (PDF).
- Nygren, Anders J. (1997). "Filtered Back Projection". Tomographic Reconstruction of SPECT Data.
- van Ginkel, M.; Hendricks, C.L. Luengo; van Vliet, L.J. (2004). "A short introduction to the Radon and Hough transforms and how they relate to each other" (PDF). Archived (PDF) from the original on 2016-07-29.
- Deans, Stanley R. (1983), The Radon Transform and Some of Its Applications, New York: John Wiley & Sons
- Helgason, Sigurdur (2008), Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, 39 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4530-1, MR 2463854
- Herman, Gabor T. (2009), Fundamentals of Computerized Tomography: Image Reconstruction from Projections (2nd ed.), Springer, ISBN 978-1-85233-617-2
- Minlos, R.A. (2001) , "Radon transform", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, 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