'''Geometric tomography''' is a mathematical field that focuses on problems of reconstructing homogeneous (often convex) 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."<ref name="ref1"> Gardner, R.J., Geometric Tomography, Cambridge University Press, Cambridge, UK, 2nd ed., 2006</ref>
==Theory== A key theorem in this area states that any convex body in <math> E^n</math> can be determined by parallel, coplanar X-rays in a set of four directions whose slopes have a transcendental cross ratio.
== Examples == * Radon transform * Funk transform (a.k.a. spherical Radon transform)
==See also== *Tomography *Tomographic reconstruction *Discrete tomography *Generalized conic
==References== {{reflist}}
== External links == *[https://geometrictomography.com Website summarizing geometric tomography] – Describes its history, theory, relation to computerized and discrete tomography, and includes interactive demonstrations of reconstruction algorithms. *[https://cgm.cs.mcgill.ca/~godfried/research/tomography.html Geometric tomography applet I] *[https://web.archive.org/web/20120325121819/http://faculty.wwu.edu/gardner/GeometricTomography.html Geometric tomography applet II]
Category:Tomography Category:Projective geometry
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