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 formula_1 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 formula_2be a function that satisfies the three regularity conditions:
The Radon transform, formula_7, is a function defined on the space of straight lines formula_8 by the line integral along each such line as:formula_9Concretely, the parametrization of any straight line "formula_10" with respect to arc length formula_11 can always be written:formula_12where formula_13 is the distance of formula_14 from the origin and formula_15 is the angle the normal vector to "formula_10" makes with the formula_17-axis. It follows that the quantities formula_18 can be considered as coordinates on the space of all lines in formula_19, and the Radon transform can be expressed in these coordinates by: formula_20More generally, in the formula_21-dimensional Euclidean space formula_22, the Radon transform of a function formula_1 satisfying the regularity conditions is a function "formula_7" on the space formula_25 of all hyperplanes in formula_22. It is defined by:
formula_27where the integral is taken with respect to the natural hypersurface measure, formula_28 (generalizing the formula_29 term from the formula_30-dimensional case). Observe that any element of formula_25 is characterized as the solution locus of an equation formula_32, where formula_33 is a unit vector and formula_34. Thus the formula_21-dimensional Radon transform may be rewritten as a function on formula_36 via: formula_37It is also possible to generalize the Radon transform still further by integrating instead over formula_38-dimensional affine subspaces of formula_22. 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: formula_40 For a function of a formula_30-vector formula_42, the univariate Fourier transform is: formula_43 For convenience, denote formula_44. The Fourier slice theorem then states: formula_45 where formula_46
Thus the two-dimensional Fourier transform of the initial function along a line at the inclination angle formula_15 is the one variable Fourier transform of the Radon transform (acquired at angle formula_15) 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: formula_49
Dual transform.
The dual Radon transform is a kind of adjoint to the Radon transform. Beginning with a function "g" on the space formula_25, the dual Radon transform is the function formula_51 on R"n" defined by: formula_52The integral here is taken over the set of all hyperplanes incident with the point formula_53, and the measure formula_54 is the unique probability measure on the set formula_55 invariant under rotations about the point formula_56.
Concretely, for the two-dimensional Radon transform, the dual transform is given by: formula_57 In the context of image processing, the dual transform is commonly called "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 formula_58 denote the Laplacian on formula_22 defined by:formula_60This is a natural rotationally invariant second-order differential operator. On formula_25, the "radial" second derivative formula_62 is also rotationally invariant. The Radon transform and its dual are intertwining operators for these two differential operators in the sense that: formula_63In analysing 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 (or function formula_1 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 formula_1 from its Radon transform is the "Filtered Back-projection Formula" or "Radon Inversion Formula": formula_66where formula_67 is such that formula_68. The convolution kernel formula_67 is referred to as Ramp filter in some literature.
Ill-posedness.
Intuitively, in the "filtered back-projection" formula, by analogy with differentiation, for which formula_70, 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:formula_71where formula_72 is the previously defined adjoint to the Radon Transform. Thus for formula_73, we have: formula_74The complex exponential formula_75 is thus an eigenfunction of formula_76 with eigenvalue formula_77. Thus the singular values of formula_78 are formula_79. Since these singular values tend to formula_80, formula_81 is unbounded.
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 formula_21 dimensions can be inverted by the formula: formula_83where formula_84, and the power of the Laplacian formula_85 is defined as a pseudo-differential operator if necessary by the Fourier transform: formula_86For computational purposes, the power of the Laplacian is commuted with the dual transform formula_87 to give: formula_88where formula_89 is the Hilbert transform with respect to the "s" variable. In two dimensions, the operator formula_90 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 formula_91 of two variables: formula_92Thus in an image processing context the original image formula_91 can be recovered from the 'sinogram' data formula_94 by applying a ramp filter (in the formula_13 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: formula_96The dual transform can also be inverted by an analogous formula: formula_97
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 formula_99 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.