Geometric tomography explained

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."^[1]

Theory

A key theorem in this area states that any convex body in

E ^{n} can be determined by parallel, coplanarX-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

External links

• Website summarizing geometric tomography – Describes its history, theory, relation to computerized and discrete tomography, and includes interactive demonstrations of reconstruction algorithms.
• Geometric tomography applet I
• Geometric tomography applet II

Notes and References

1. Gardner, R.J., Geometric Tomography, Cambridge University Press, Cambridge, UK, 2nd ed., 2006