Synonyms
Computed tomography; Computer-assisted tomography; Computerized axial tomography; x-ray tomography
Definition
Transmission x-ray tomography is a method to image the internal structure of an opaque body or object by combining measurements of the intensity attenuation of many x-ray beams that have passed through the object.
Overview
The possibility of x-ray CT was first foreseen by Allan M. Cormack [1, 2]. The first practical implementation was done by Godfrey M. Hounsfield. Both Cormack and Hounsfield shared the 1979 Nobel Prize in Physiology or Medicine for their discovery. See http://www.nobelprize.org/nobel_prizes/medicine/1979
The detected intensity I D of an x-ray beam after passing through the object is related to the original intensity I0 by
where L denotes the line of the ray and f is the linear attenuation coefficient. Therefore, the line integral \(\int _{L}f(x)dx\)can be determined from the measurement of the...
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Cormack, A.: Representation of a function by its line integrals, with some radiological applications. J. Appl. Phys. 34(9), 2722–2727 (1963)
Cormack, A.: Representation of a function by its line integrals, with some radiological applications II. J. Appl. Phys. 35(10), 2908–2913 (1964)
Fanelli, D., Öktem, O.: Electron tomography: a short overview with an emphasis on the absorption potential model for the forward problem. Inverse Probl. 24(1), 013001(51) (2008). doi:10.1088/0266-5611/24/1/013001
Faridani, A., Hass, R., Solmon, D.: Numerical and theoretical explorations in helical and fan-beam tomography. J. Phys.: Conf. Ser. 124, 012024(1–20) (2008). doi:10.1088/1742-6596/124/1/012024, http://iopscience.iop.org/1742-6596/124/1/012024
Gel’fand, I., Graev, M.: Crofton’s function and inversion formulas in real integral geometry. Funct. Anal. Appl. 25, 1–5 (1991)
Herman, G.: Fundamentals of Computerized Tomography: Image Reconstruction from Projections, 2nd edn. Springer, London (2009)
Kalender, W.: Computed Tomography: Fundamentals, System Technology, Image Quality, Applications, 3rd rev edn. Publicis Corporate Publishing, Erlangen (2011)
Katsevich, A.: An improved exact filtered backprojection inversion algorithm for spiral cone-beam CT. Adv. Appl. Math. 32, 681–697 (2004)
Natterer, F.: The Mathematics of Computerized Tomography. Wiley, Chichester (1986). Republished 2001 by SIAM
Natterer, F., Wübbeling, F.: Mathematical Methods in Image Reconstruction. SIAM, Philadelphia (2001)
Noo, F., Defrise, M., Clackdoyle, R., Kudo, H.: Image reconstruction from fan-beam projections on less than a short scan. Phys. Med. Biol. 47, 2525–2546 (2002). doi:10.1088/0031-9155/47/14/311
Radon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber Verh Sächs Akad Wiss Leipzig Math-phys Kl 69, 262–277 (1917)
Smith, K., Keinert, F.: Mathematical foundations of computed tomography. Appl. Opt. 24(23), 3950–3957 (1985)
Zou, Y., Pan, X.: Exact image reconstruction on PI-lines from minimum data in helical cone-beam CT. Phys. Med. Biol. 49, 941–959 (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this entry
Cite this entry
Finch, D.V., Faridani, A. (2015). X-Ray Transmission Tomography. In: Engquist, B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-70529-1_10
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70528-4
Online ISBN: 978-3-540-70529-1
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering