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Théorème de Radon

Résumé
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. 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 (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.
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Johann Radon
Johann Karl August Radon (né le à Tetschen - mort le à Vienne) est un mathématicien autrichien. Radon est né à Tetschen (royaume de Bohême, Autriche-Hongrie. Il passe son doctorat à l'université de Vienne en 1910 et séjourne le semestre d'hiver 1910/11 à l'université de Göttingen ; ensuite, il est assistant à l' (Brno), et de 1912 à 1919 à la Grande École technique de Vienne. En 1913-1914, il passe son habilitation à l'université de Vienne. thumb|left|Johann Radon en 1920 environ En 1919, il est nommé professeur extraordinaire à l'université de Hambourg récemment fondée.
Théorème de Radon
Le théorème de projection de Radon établit la possibilité de reconstituer une fonction réelle à deux variables (assimilable à une image) à l'aide de la totalité de ses projections selon des droites concourantes. L'application la plus courante de ce théorème est la reconstruction d'images médicales en tomodensitométrie, c'est-à-dire dans les scanneurs à rayon X. Il doit son nom au mathématicien Johann Radon. En pratique, il est impossible de disposer de toutes les projections d'un objet solide, seulement un échantillonnage.
Tomographic reconstruction
Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections. The mathematical basis for tomographic imaging was laid down by Johann Radon. A notable example of applications is the reconstruction of computed tomography (CT) where cross-sectional images of patients are obtained in non-invasive manner.
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