Geometric tomography is a mathematical field that focuses on problems of reconstructing homogeneous (often
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
) objects from tomographic data (this might be X-rays, projections, sections, brightness functions, or covariograms). More precisely, according to R.J. Gardner (who introduced the term), "Geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on planes or cross-sections by planes."
[
Gardner, R.J., Geometric Tomography, Cambridge University Press, Cambridge, UK, 2nd ed., 2006]
Theory
A key theorem in this area states that any convex body in
can be determined by parallel, coplanar
X-rays in a set of four directions whose slopes have a transcendental
cross ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, ...
.
Examples
*
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 ...
*
Funk transform
In the mathematical field of integral geometry, the Funk transform (also known as Minkowski–Funk transform, Funk–Radon transform or spherical Radon transform) is an integral transform defined by integrating a function on great circles of the s ...
(a.k.a. spherical Radon transform)
See also
*
Tomography
Tomography is imaging by sections or sectioning that uses any kind of penetrating wave. The method is used in radiology, archaeology, biology, atmospheric science, geophysics, oceanography, plasma physics, materials science, cosmochemistry, ...
*
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 projection (linear algebra), projections. The mathematical basis for tomographic imag ...
*
Discrete tomography
Discrete tomography
Herman, G. T. and Kuba, A., Discrete Tomography: Foundations, Algorithms, and Applications, Birkhäuser Boston, 1999
Herman, G. T. and Kuba, A., Advances in Discrete Tomography and Its Applications, Birkhäuser Boston, 2007
fo ...
*
Generalized conic In mathematics, a generalized conic is a geometrical object defined by a property which is a generalization of sums defining property of the classical conic. For example, in elementary geometry, an ellipse can be defined as the locus of a point ...
References
{{reflist
External links
Website summarizing geometric tomography– Describes its history, theory, relation to computerized and discrete tomography, and includes interactive demonstrations of reconstruction algorithms.
Tomography
Projective geometry