Radon transform

Radon transform. In the field of integral geometry, this operation maps a function defined on a plane to the set of its line integrals. Introduced in a seminal 1917 paper by the Austrian mathematician Johann Radon, it provides the mathematical foundation for reconstructing images from their projections. Its most profound application is in computed tomography (CT) scanning, where it enables the creation of cross-sectional images from X-ray data. The transform and its inverse are central to problems in image processing, seismology, and astrophysics.

Definition and mathematical formulation

The transform is defined for a function \( f \) on the Euclidean plane \(\mathbb{R}^2\). A line \( L \) can be parameterized by its distance \( s \) from the origin and its angle \( \theta \) relative to a fixed axis. The operation yields a new function \( Rf(s, \theta) \), which is the integral of \( f \) over the line \( L(s, \theta) \). Formally, it is expressed as a line integral: \( Rf(s, \theta) = \int_{L(s, \theta)} f(x) \, dl \). Using the Dirac delta function \(\delta\), this can be written in Cartesian coordinates as \( Rf(s, \theta) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) \, \delta(x \cos \theta + y \sin \theta - s) \, dx \, dy \). This formulation elegantly captures the essence of projecting a two-dimensional density onto a one-dimensional sensor array, a principle directly utilized in the design of CT scanner systems developed by companies like General Electric and Siemens.

Properties

The transform possesses several key analytical properties. It is a linear operator, meaning \( R(af + bg) = aRf + bRg \) for functions \( f, g \) and constants \( a, b \). A crucial property is its relation to the Fourier transform, encapsulated in the Fourier slice theorem. This theorem states that the one-dimensional Fourier transform of a projection at a fixed angle \( \theta \) equals a slice through the two-dimensional Fourier transform of the original function. Another important property is its behavior under translation and rotation; a translation of the function induces a shift in the sinogram, while a rotation corresponds to a simple shift in the angular variable. The transform of a radially symmetric function simplifies to an Abel transform, connecting it to problems in astrophysics like determining stellar density profiles.

Inversion

Reconstructing the original function \( f \) from its projections \( Rf \) is the central inversion problem. The existence of an explicit inversion formula was proven by Johann Radon himself. A common method utilizes the Fourier slice theorem: take the one-dimensional Fourier transform of each projection, interpolate these slices into the two-dimensional frequency domain, and then apply an inverse two-dimensional Fourier transform. A more computationally efficient and stable approach is filtered backprojection, which involves applying a ramp filter to the projections in the frequency domain before backprojecting them. The mathematical rigor of these methods was solidified through the work of researchers like Allan MacLeod Cormack and Godfrey Hounsfield, who shared the Nobel Prize in Physiology or Medicine for their contributions to CT scanning.

Applications

The primary application is in medical and industrial computed tomography, forming the core algorithm for devices manufactured by Toshiba and Philips. In seismology, it is used in seismic tomography to image the Earth's interior by analyzing the travel times of waves from events like earthquakes. In astronomy, it aids in reconstructing images in radio astronomy from interferometric data collected by instruments like the Very Large Array. Other uses include non-destructive testing in aerospace engineering, for inspecting components from companies like Boeing or Airbus, and in electron microscopy for determining three-dimensional molecular structures, a technique advanced at institutions like the MRC Laboratory of Molecular Biology.

Generalizations and related transforms

The concept extends naturally to higher dimensions; the transform in \(\mathbb{R}^n\) integrates a function over hyperplanes. The John transform deals with integrals over lines in three dimensions, relevant in SPECT imaging. A closely related operation is the X-ray transform, which integrates over lines but retains the direction, unlike the transform which uses oriented lines. The Funk–Radon transform integrates functions on the sphere over great circles, with applications in integral geometry and the study of the cosmic microwave background. These generalizations share deep connections with the theory of partial differential equations and were further explored by mathematicians such as Fritz John and Sigurdur Helgason. Category:Integral transforms Category:Tomography Category:Mathematical analysis