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 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. 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, 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 = ƒ be a function that satisfies the three regularity conditions:
  1. ƒ is continuous
  2. the double integral, extending over the whole plane, converges
  3. for any arbitrary point on the plane it holds that
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 as
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 function ƒ satisfying the regularity conditions 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.

Relationship with the Fourier transform

The Radon transform is closely related to the Fourier transform. We define the univariate Fourier transform here as
and for a function of a 2-vector,
For convenience, denote. The Fourier slice theorem then states
where
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 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 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.

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
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.

Reconstruction approaches

The process of reconstruction produces the image 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.

Ill-posedness

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:
We have
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 could provide metal artifact 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 n dimensions can be inverted by the formula
where
and the power of the Laplacian /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:

Radon transform in algebraic geometry

In algebraic geometry, a Radon transform is constructed as follows.
Write
for the universal hyperplane, i.e., H consists of pairs where x is a point in d-dimensional projective space and h is a point in the dual projective 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.