Radon transform

It was later generalized to higher-dimensional Euclidean spaces and more broadly in the context of integral geometry.

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.

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.Let

, is a function defined on the space of straight lines

where the integral is taken with respect to the natural hypersurface measure,

The X-ray transform is the most widely used special case of this construction, and is obtained by integrating over straight lines.

Thus the two-dimensional Fourier transform of the initial function along a line at the inclination angle

In the context of image processing, the dual transform is commonly called back-projection[4] 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.

This is a natural rotationally invariant second-order differential operator.

In analysing the solutions to the wave equation in multiple spatial dimensions, the intertwining property leads to the translational representation of Lax and Philips.

[6] In imaging[7] and numerical analysis[8] this is exploited to reduce multi-dimensional problems into single-dimensional ones, as a dimensional splitting method.

In the two-dimensional case, the most commonly used analytical formula to recover

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:

is the previously defined adjoint to the Radon transform.

[9] 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.

Explicit and computationally efficient inversion formulas for the Radon transform and its dual are available.

is defined as a pseudo-differential operator if necessary by the Fourier transform:

For computational purposes, the power of the Laplacian is commuted with the dual transform

appears in image processing as a ramp filter.

[13] One can prove directly from the Fourier slice theorem and change of variables for integration that for a compactly supported continuous function

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:[4]

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.

Radon transform. Maps f on the ( x , y )-domain to Rf on the ( α , s )-domain.
Radon transform of the indicator function of two squares shown in the image below. Lighter regions indicate larger function values. Black indicates zero.
Original function is equal to one on the white region and zero on the dark region.
Horizontal projections through the shape result in an accumulated signal (middle bar). The sinogram on the right is generated by collecting many such projections as the shape rotates. Here, color is used to highlight which object is producing which part of the signal. Note how straight features, when aligned with the projection direction, result in stronger signals.
Example of reconstruction via the Radon transform using observations from different angles. The applied inversion to the projection data then reconstructs the slice image. [ 2 ]
Computing the 2-dimensional Radon transform in terms of two Fourier transforms.