Tomographic reconstruction

The mathematical basis for tomographic imaging was laid down by Johann Radon. It is applied in Computed Tomography to obtain cross-sectional images of patients. This article applies in general to tomographic reconstruction for all kinds of tomography, but some of the terms and physical descriptions refer directly to X-ray computed tomography.

The projection of an object at a given angle heta is made up of a set of "line integrals". In X-ray CT, the line integral represents the total attenuation of the beam of x-rays as ittravels in a straight line through the object. As mentioned above, the resulting image is a 2D (or 3D) model of the attenuation coefficient. That is, we wish to find the image mu(x,y). The simplest and easiest way to visualise method of scanning is the system of parallel projection, as used in the first scanners. For this discussion we consider the data to be collected as a series of parallel rays, at position r, across a projection at angle heta. This is repeated for various angles. Attenuation occurs exponentially in tissue:

:I = I_0expleft({-intmu(x,y),ds} ight)

where mu(x) is the attenuation coefficient at position x along the ray path. Therefore generally the total attenuation p of a ray at position r, on the projection at angle heta, is given by the line integral:

:p(r, heta) = ln (I/I_0) = -intmu(x,y),ds

Using the coordinate system of Figure 1, the value of r onto which the point (x,y) will be projected at angle heta is given by:

:xcos heta + ysin heta = r

So the equation above can be rewritten as

:p(r, heta)=int^infty_{-infty}int^infty_{-infty}f(x,y)delta(xcos heta+ysin heta-r),dx,dy

where f(x,y) represents mu(x,y). This function is known as the Radon transform (or "sinogram") of the 2D object. The projection-slice theorem tells us that if we had an infinite number of one-dimensional projections of an object taken at an infinite number of angles, we could perfectly reconstruct the original object, f(x,y). So to get f(x,y) back, from the above equation means finding the inverse Radon transform. It is possible to find an explicit formula for the inverse Radon transform. However, the inverse Radon transform proves to be extremely unstable with respect to noisy data. In practice, a stabilized and discretized version of the inverse Radon transform is used, known as the filtered back projection algorithm.

Further reading

*Avinash Kak & Malcolm Slaney (1988), Principles of Computerized Tomographic Imaging, IEEE Press, ISBN 0-87942-198-3.

External links


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Reconstruction algorithm — In tomography, a variety of practical reconstruction algorithms have been developed to implement the process of reconstruction of a 3 dimensional object from its projections. These algorithms are designed largely based on the mathematics of the… …   Wikipedia

  • Iterative reconstruction — is a method or group of algorithms used to reconstruct 2D and 3D images from the projections of an object. The technique differs greatly from the more ubiquitous filtered back projection (FBP) method. In X ray computed tomography, this method is… …   Wikipedia

  • Software tools for molecular microscopy — There are a large number of software tools or software applications that have been specifically developed for the field sometimes referred to as molecular microscopy or cryo electron microscopy or cryoEM. Several special issues of the Journal of… …   Wikipedia

  • Computed tomography — tomos (slice) and graphein (to write).Computed tomography was originally known as the EMI scan as it was developed at a research branch of EMI, a company best known today for its music and recording business. It was later known as computed axial… …   Wikipedia

  • Tomography — Basic principle of tomography: superposition free tomographic cross sections S1 and S2 compared with the projected image P Tomography refers to imaging by sections or sectioning, through the use of any kind of penetrating wave. A device used in… …   Wikipedia

  • X-ray computed tomography — For non medical computed tomography, see Industrial CT Scanning. catSCAN redirects here. For the Transformers character, see Transformers: Universe. X ray computed tomography Intervention A patient is receiving a CT scan for cancer. Outsid …   Wikipedia

  • Single photon emission computed tomography — (SPECT, or less commonly, SPET) is a nuclear medicine tomographic imaging technique using gamma rays. It is very similar to conventional nuclear medicine planar imaging using a gamma camera. However, it is able to provide true 3D information.… …   Wikipedia

  • William H. Oldendorf — William Henry Oldendorf (1925 December 14, 1992) was an American neurologist, physician, researcher, medical pioneer, founding member of the American Society for Neuroimaging (ASN), and originator of the technique of Computed Tomography. Early… …   Wikipedia

  • radiation — radiational, adj. /ray dee ay sheuhn/, n. 1. Physics. a. the process in which energy is emitted as particles or waves. b. the complete process in which energy is emitted by one body, transmitted through an intervening medium or space, and… …   Universalium

  • Radon transform — In mathematics, the Radon transform in two dimensions, named after the Austrian mathmematician Johann Radon, is the integral transform consisting of the integral of a function over straight lines. The inverse of the Radon transform is used to… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.