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In mathematics, the Radon transform in two dimensions, named after the Austrian mathematician Johann Radon, is the integral transform consisting of the integral of a function over straight lines. The transform was introduced in 1917 by 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. It was later generalised to higher-dimensional Euclidean spaces, and more broadly in the context of integral geometry. The complex analog 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 ''ƒ'' 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 image reconstruction. The Radon transform data is often called a sinogram because the Radon transform of a Dirac delta function is a distribution supported on the graph of a sine wave. 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 ''ƒ''(x) = ''ƒ''(''x'',''y'') be a compactly supported continuous function on R2. The Radon transform, ''Rƒ'', is a function defined on the space of straight lines ''L'' in R2 by the line integral along each such line: : Concretely, the parametrization of any straight line ''L'' with respect to arc length ''z'' can always be written : where ''s'' is the distance of ''L'' from the origin and is the angle the normal vector to ''L'' makes with the ''x'' axis. It follows that the quantities (α,''s'') can be considered as coordinates on the space of all lines in R2, and the Radon transform can be expressed in these coordinates by : More generally, in the ''n''-dimensional Euclidean space R''n'', the Radon transform of a compactly supported continuous function ''ƒ'' is a function ''Rƒ'' on the space Σ''n'' of all hyperplanes in R''n''. It is defined by : for ξ ∈Σ''n'', where the integral is taken with respect to the natural hypersurface measure, ''d''σ (generalizing the |''d''x| term from the 2-dimensional case). Observe that any element of Σ''n'' is characterized as the solution locus of an equation : where α ∈ ''S''''n''−1 is a unit vector and ''s'' ∈ R. Thus the ''n''-dimensional Radon transform may be rewritten as a function on ''S''''n''−1×R via : It is also possible to generalize the Radon transform still further by integrating instead over ''k''-dimensional affine subspaces of R''n''. The X-ray transform is the most widely used special case of this construction, and is obtained by integrating over straight lines. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「radon transform」の詳細全文を読む スポンサード リンク
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