Geometry of cone-beam reconstruction

Geometry is the synthetic tool we use to unify all existing analytical cone-beam reconstruction methods. These reconstructions are based on formulae derived by Tuy [Tuy, 1983], Smith [Smith, 1985] and Grangeat [Grangeat, 1991] which explicitly link the cone-beam data to some intermediate functions in the Radon transform domain. However, the essential step towards final reconstruction, that is, differential-backprojection, has not yet achieved desired efficiency. A new inversion formula is obtained directly from the 3D Radon inverse [Radon, 1917, Helgason, 1999]. It incorporates the cone-beam scanning geometry and allows the theoretical work mentioned above to be reduced to exact and frugal implementations. Extensions can be easily carried out to 2D fan-beam reconstruction as well as other scanning modalities such as parallel scans by allowing more abstract geometric description on the embedding subspace of the Radon manifold. The new approach provides a canonical inverse procedure for computerized tomography in general, with applications ranging from diagnostic medical imaging to industrial testing, such as X-ray CT, Emission CT, Ultrasound CT, etc. It also suggests a principled frame for approaching other 3D reconstruction problems related to the Radon transform. The idea is simple: as was spelled out by Helgason on the opening page of his book, The Radon Transform [Helgason, 1999] - a remarkable duality characterizes the Radon transform and its inverse. Our study shows that the dual space, the so-called Radon space, can be geometrically decomposed according to the specified scanning modality.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2002

Author dc:creator

• Yang, Xiaochun, 1971-

Advisor dc:contributor.advisor

• Daniel J. Kleitman.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/8338

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/8338