Radon and Abel Transforms
DOI link for Radon and Abel Transforms
Radon and Abel Transforms book
The Austrian mathematician Johann Radon (1887-1956) wrote a classic paper in 1917, ‘‘Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten’’ (on the determination of functions from their integrals along certain manifolds) (Radon, 1917). This work forms the foundation for what we now call the Radon transform. English translations are available in the monograph by Deans (1983, 1993) and the translation by Parks (1986). The problem of determining a function f(x, y) from knowledge of its line integrals (the twodimensional (2D) case), or a function f(x, y, z) from integrals over planes the (three-dimensional [3D] case) arises in widely diverse ﬁelds. These include medical imaging, astronomy,
crystallography, electron microscopy, geophysics, optics, and material science. In these applications the central aim is to obtain certain information about the internal structure of an object either by passing some probe (such as x-rays) through the object or by using information from the source itself when it is self-emitting, such as an organ in the body that contains a radioactive isotope, or perhaps the interior of the Earth when motions occur. Comprehensive reviews of these and other applications are contained in Brooks and Di Chiro (1976), Scudder (1978), Barrett (1984), Chapman (1987), and Deans (1983, 1993).