# Radon transform

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 analogue 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.

## Explanation

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.

## Definition

Let be a function that satisfies the three regularity conditions:

- is continuous;
- the double integral , extending over the whole plane, converges;
- for any arbitrary point on the plane it holds that

The Radon transform, , is a function defined on the space of straight lines by the line integral along each such line as:

*with respect to arc length can always be written:*

*makes with the -axis. It follows that the quantities can be considered as coordinates on the space of all lines in , and the Radon transform can be expressed in these coordinates by:*

*on the space of all hyperplanes in . It is defined by:*

## Relationship with the Fourier transform

The Radon transform is closely related to the Fourier transform. We define the univariate Fourier transform here as:

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:

## Dual transform

The dual Radon transform is a kind of adjoint to the Radon transform. Beginning with a function *g* on the space , the dual Radon transform is the function on **R**^{n} defined by:

Concretely, for the two-dimensional Radon transform, the dual transform is given by:

*back-projection*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.

### Intertwining property

Let denote the Laplacian on defined by:

## Reconstruction approaches

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 two-dimensional case, the most commonly used analytical formula to recover from its Radon transform is the *Filtered Back-projection Formula* or *Radon Inversion Formula*:

### Ill-posedness

Intuitively, in the *filtered back-projection* 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:

### Iterative reconstruction methods

Compared with the *Filtered Back-projection* method, iterative reconstruction costs large computation time, limiting its practical use. However, due to the ill-posedness of Radon Inversion, the *Filtered Back-projection* method may be infeasible in the presence of discontinuity or noise. Iterative reconstruction methods (*e.g.* iterative Sparse Asymptotic Minimum Variance) could provide metal artefact reduction, noise and dose reduction for the reconstructed result that attract much research interest around the world.

## Inversion formulas

Explicit and computationally efficient inversion formulas for the Radon transform and its dual are available. The Radon transform in dimensions can be inverted by the formula:

*s*variable. In two dimensions, the operator 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 of two variables:

Explicitly, the inversion formula obtained by the latter method is:

## Radon transform in algebraic geometry

In algebraic geometry, a Radon transform (also known as the *Brylinski–Radon transform*) is constructed as follows.

Write

for the universal hyperplane, i.e., *H* consists of pairs (*x*, *h*) where *x* is a point in *d*-dimensional projective space and *h* is a point in the dual projective space (in other words, *x* is a line through the origin in (*d*+1)-dimensional affine space, and *h* is a hyperplane in that space) such that *x* is contained in *h*.

Then the Brylinksi–Radon transform is the functor between appropriate derived categories of étale sheaves

The main theorem about this transform is that this transform induces an equivalence of the categories of perverse sheaves on the projective space and its dual projective space, up to constant sheaves.

## See also

- Periodogram
- Matched filter
- Deconvolution
- 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.
- Fast Fourier transform

## Notes

**^**Radon 1917.**^**Radon, J. (December 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. S2CID 26553287.- ^
^{a}^{b}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. doi:10.1090/s0002-9904-1964-11051-x.**^**Bonneel, N.; Rabin, J.; Peyre, G.; Pfister, H. (2015). "Sliced and Radon Wasserstein Barycenters of Measures".*Journal of Mathematical Imaging and Vision*.**51**(1): 22–25. doi:10.1007/s10851-014-0506-3. S2CID 1907942.**^**Rim, D. (2018). "Dimensional Splitting of Hyperbolic Partial Differential Equations Using the Radon Transform".*SIAM J. Sci. Comput*.**40**(6): A4184–A4207. arXiv:1705.03609. doi:10.1137/17m1135633. S2CID 115193737.- ^
^{a}^{b}^{c}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. arXiv:1802.03070. Bibcode:2013ITSP...61..933A. doi:10.1109/tsp.2012.2231676. ISSN 1053-587X. S2CID 16276001.**^**Helgason 1984, Theorem I.2.13.**^**Helgason 1984, Theorem I.2.16.**^**Nygren 1997.**^**Kiehl & Weissauer (2001, Ch. IV, Cor. 2.4)**^**van Ginkel, Hendricks & van Vliet 2004.