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,^[1] 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${\displaystyle f}$ 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 ${\displaystyle f({\textbf {x)))=f(x,y)}$ be a function that satisfies the three regularity conditions:^[3]

1. ${\displaystyle f({\textbf {x)))}$ is continuous;
2. the double integral ${\displaystyle \iint {\dfrac {\vert f({\textbf {x)))\vert }{\sqrt {x^{2}+y^{2))))\,dx\,dy}$, extending over the whole plane, converges;
3. for any arbitrary point ${\displaystyle (x,y)}$ on the plane it holds that ${\displaystyle \lim _{r\to \infty }\int _{0}^{2\pi }f(x+r\cos \varphi ,y+r\sin \varphi )\,d\varphi =0.}$

The Radon transform, ${\displaystyle Rf}$, is a function defined on the space of straight lines ${\displaystyle L\subset \mathbb {R} ^{2))$ by the line integral along each such line as:

$${\displaystyle Rf(L)=\int _{L}f(\mathbf {x} )\vert d\mathbf {x} \vert .}$$

${\displaystyle L}$

${\displaystyle z}$

$${\displaystyle (x(z),y(z))={\Big (}(z\sin \alpha +s\cos \alpha ),(-z\cos \alpha +s\sin \alpha ){\Big )}\,}$$

${\displaystyle s}$

${\displaystyle L}$

${\displaystyle \alpha }$

${\displaystyle L}$

${\displaystyle X}$

${\displaystyle (\alpha ,s)}$

${\displaystyle \mathbb {R} ^{2))$

$${\displaystyle {\begin{aligned}Rf(\alpha ,s)&=\int _{-\infty }^{\infty }f(x(z),y(z))\,dz\\&=\int _{-\infty }^{\infty }f{\big (}(z\sin \alpha +s\cos \alpha ),(-z\cos \alpha +s\sin \alpha ){\big )}\,dz.\end{aligned))}$$

${\displaystyle n}$

${\displaystyle \mathbb {R} ^{n))$

${\displaystyle f}$

${\displaystyle Rf}$

${\displaystyle \Sigma _{n))$

${\displaystyle \mathbb {R} ^{n))$

Shepp Logan phantom

Radon transform

Inverse Radon transform

$${\displaystyle Rf(\xi )=\int _{\xi }f(\mathbf {x} )\,d\sigma (\mathbf {x} ),\quad \forall \xi \in \Sigma _{n))$$

${\displaystyle d\sigma }$

${\displaystyle \vert d\mathbf {x} \vert }$

${\displaystyle 2}$

${\displaystyle \Sigma _{n))$

${\displaystyle \mathbf {x} \cdot \alpha =s}$

${\displaystyle \alpha \in S^{n-1))$

${\displaystyle s\in \mathbb {R} }$

${\displaystyle n}$

${\displaystyle S^{n-1}\times \mathbb {R} }$

$${\displaystyle Rf(\alpha ,s)=\int _{\mathbf {x} \cdot \alpha =s}f(\mathbf {x} )\,d\sigma (\mathbf {x} ).}$$

${\displaystyle k}$

${\displaystyle \mathbb {R} ^{n))$

Relationship with the Fourier transform

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

$${\displaystyle {\hat {f))(\omega )=\int _{-\infty }^{\infty }f(x)e^{-2\pi ix\omega }\,dx.}$$

${\displaystyle 2}$

${\displaystyle \mathbf {x} =(x,y)}$

$${\displaystyle {\hat {f))(\mathbf {w} )=\iint _{\mathbb {R} ^{2))f(\mathbf {x} )e^{-2\pi i\mathbf {x} \cdot \mathbf {w} }\,dx\,dy.}$$

${\displaystyle {\mathcal {R))_{\alpha }[f](s)={\mathcal {R))[f](\alpha ,s)}$

$${\displaystyle {\widehat ((\mathcal {R))_{\alpha }[f]))(\sigma )={\hat {f))(\sigma \mathbf {n} (\alpha ))}$$

${\displaystyle \mathbf {n} (\alpha )=(\cos \alpha ,\sin \alpha ).}$

Thus the two-dimensional Fourier transform of the initial function along a line at the inclination angle ${\displaystyle \alpha }$ is the one variable Fourier transform of the Radon transform (acquired at angle ${\displaystyle \alpha }$) 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:

$${\displaystyle {\hat {f))(r\alpha )=\int _{\mathbb {R} }{\mathcal {R))f(\alpha ,s)e^{-2\pi isr}\,ds.}$$

Dual transform

The dual Radon transform is a kind of adjoint to the Radon transform. Beginning with a function g on the space ${\displaystyle \Sigma _{n))$, the dual Radon transform is the function ${\displaystyle {\mathcal {R))^{*}g}$ on Rⁿ defined by:

$${\displaystyle {\mathcal {R))^{*}g(\mathbf {x} )=\int _{\mathbf {x} \in \xi }g(\xi )\,d\mu (\xi ).}$$

${\displaystyle {\textbf {x))\in \mathbb {R} ^{n))$

${\displaystyle d\mu }$

${\displaystyle \{\xi |\mathbf {x} \in \xi \))$

${\displaystyle \mathbf {x} }$

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

$${\displaystyle {\mathcal {R))^{*}g(\mathbf {x} )={\frac {1}{2\pi ))\int _{\alpha =0}^{2\pi }g(\alpha ,\mathbf {n} (\alpha )\cdot \mathbf {x} )\,d\alpha .}$$

Intertwining property

Let ${\displaystyle \Delta }$ denote the Laplacian on ${\displaystyle \mathbb {R} ^{n))$ defined by:

$${\displaystyle \Delta ={\frac {\partial ^{2)){\partial x_{1}^{2))}+\cdots +{\frac {\partial ^{2)){\partial x_{n}^{2))))$$

${\displaystyle \Sigma _{n))$

${\displaystyle Lf(\alpha ,s)\equiv {\frac {\partial ^{2)){\partial s^{2))}f(\alpha ,s)}$

$${\displaystyle {\mathcal {R))(\Delta f)=L({\mathcal {R))f),\quad {\mathcal {R))^{*}(Lg)=\Delta ({\mathcal {R))^{*}g).}$$

Reconstruction approaches

The process of reconstruction produces the image (or function ${\displaystyle f}$ 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 ${\displaystyle f}$ from its Radon transform is the Filtered Back-projection Formula or Radon Inversion Formula^[9]:

$${\displaystyle f(\mathbf {x} )=\int _{0}^{\pi }({\mathcal {R))f(\cdot ,\theta )*h)(\left\langle \mathbf {x} ,\mathbf {n} _{\theta }\right\rangle )\,d\theta }$$

${\displaystyle h}$

${\displaystyle {\hat {h))(k)=|k|}$

${\displaystyle h}$

Ill-posedness

Intuitively, in the filtered back-projection formula, by analogy with differentiation, for which ${\textstyle \left({\widehat ((\frac {d}{dx))f))\right)\!(k)=ik{\widehat {f))(k)}$, 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:

$${\displaystyle {\widehat ((\mathcal {R))^{*}{\mathcal {R))[g]))(\mathbf {k} )={\frac {1}{\|\mathbf {k} \|)){\hat {g))(\mathbf {k} )}$$

${\displaystyle {\mathcal {R))^{*))$

${\displaystyle g(\mathbf {x} )=e^{i\left\langle \mathbf {k} _{0},\mathbf {x} \right\rangle ))$

$${\displaystyle {\mathcal {R))^{*}{\mathcal {R))[g](\mathbf {x} )={\frac {1}{\|\mathbf {k_{0)) \|))e^{i\left\langle \mathbf {k} _{0},\mathbf {x} \right\rangle ))$$

${\displaystyle e^{i\left\langle \mathbf {k} _{0},\mathbf {x} \right\rangle ))$

${\displaystyle {\mathcal {R))^{*}{\mathcal {R))}$

${\textstyle {\frac {1}{\|\mathbf {k} _{0}\|))}$

${\displaystyle {\mathcal {R))}$

${\textstyle {\frac {1}{\sqrt {\|\mathbf {k} \|))))$

${\displaystyle 0}$

${\displaystyle {\mathcal {R))^{-1))$

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^[10]) 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 ${\displaystyle n}$ dimensions can be inverted by the formula:^[11]

$${\displaystyle c_{n}f=(-\Delta )^{(n-1)/2}R^{*}Rf\,}$$

${\displaystyle c_{n}=(4\pi )^{(n-1)/2}{\frac {\Gamma (n/2)}{\Gamma (1/2)))}$

${\displaystyle (-\Delta )^{(n-1)/2))$

$${\displaystyle \left[{\mathcal {F))(-\Delta )^{(n-1)/2}\varphi \right](\xi )=|2\pi \xi |^{n-1}({\mathcal {F))\varphi )(\xi ).}$$

${\displaystyle R^{*))$

$${\displaystyle c_{n}f={\begin{cases}R^{*}{\frac {d^{n-1)){ds^{n-1))}Rf&n{\text{ odd))\\R^{*}{\mathcal {H))_{s}{\frac {d^{n-1)){ds^{n-1))}Rf&n{\text{ even))\end{cases))}$$

${\displaystyle {\mathcal {H))_{s))$

${\displaystyle {\mathcal {H))_{s}{\frac {d}{ds))}$

${\displaystyle f}$

$${\displaystyle f={\frac {1}{2))R^{*}H_{s}{\frac {d}{ds))Rf.}$$

${\displaystyle f}$

${\displaystyle Rf}$

${\displaystyle s}$

Explicitly, the inversion formula obtained by the latter method is:^[4]

$${\displaystyle f(x)={\begin{cases}\displaystyle -\imath 2\pi (2\pi )^{-n}(-1)^{n/2}\int _{S^{n-1)){\frac {\partial ^{n-1)){2\partial s^{n-1))}Rf(\alpha ,\alpha \cdot x)\,d\alpha &n{\text{ odd))\\\displaystyle (2\pi )^{-n}(-1)^{n/2}\iint _{\mathbb {R} \times S^{n-1)){\frac {\partial ^{n-1)){q\partial s^{n-1))}Rf(\alpha ,\alpha \cdot x+q)\,d\alpha \,dq&n{\text{ even))\\\end{cases))}$$

$${\displaystyle c_{n}g=(-L)^{(n-1)/2}R(R^{*}g).\,}$$

Radon transform in algebraic geometry

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

Write

${\displaystyle \mathbf {P} ^{d}\,{\stackrel {p_{1)){\gets ))\,H\,{\stackrel {p_{2)){\to ))\,\mathbf {P} ^{\vee ,d))$

for the universal hyperplane, i.e., H consists of pairs (x, h) where x is a point in d-dimensional projective space ${\displaystyle \mathbf {P} ^{d))$ 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

${\displaystyle \operatorname {Rad} :=Rp_{2,*}p_{1}^{*}:D(\mathbf {P} ^{d})\to D(\mathbf {P} ^{\vee ,d}).}$

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.^[14]