Tomographic reconstruction

Tomographic imaging 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 mathematical basis for tomographic imaging was laid down by Johann Radon.

Description

File:Tomographic fig1.png

Figure 1: Parallel beam geometry utilized in tomography and tomographic reconstruction. Each projection, resulting from tomography under a specific angle, is made up of the set of line integrals through the object.

File:Ct skull.jpg

Resulting tomographic image from a plastic skull phantom. Projected X-rays are clearly visible on this slice taken with a CT-scan as image artifacts, due to limited amount of projection slices over angles.

The projection of an object, resulting from the tomographic measurement process at a given angle $\theta$, is made up of a set of line integrals (see Fig. 1). As set of many such projections under different angles organized in 2D is called sinogram (see Fig. 3). In X-ray CT, the line integral represents the total attenuation of the beam of x-rays as it travels 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 the 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 $\theta$. This is repeated for various angles. Attenuation occurs exponentially in tissue:

$I = I_0\exp\left({-\int\mu(x,y)\,ds}\right)$

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 $\theta$, is given by the line integral:

$p(r,\theta) = \ln (I/I_0) = -\int\mu(x,y)\,ds$

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

$x\cos\theta + y\sin\theta = r\$

So the equation above can be rewritten as

$p(r,\theta)=\int^\infty_{-\infty}\int^\infty_{-\infty}f(x,y)\delta(x\cos\theta+y\sin\theta-r)\,dx\,dy$

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.

Usage and reconstruction algorithms

In practice of tomographic image reconstruction, often a stabilized and discretized version of the inverse Radon transform is used, known as the filtered back projection algorithm. Recent developments have seen the Radon transform and its inverse used for tasks related to realistic object insertion required for testing and evaluating Computed Tomography use in Airport Security. An alternative family of tomographic reconstruction algorithms are the Algebraic Reconstruction Technique ones.

Gallery

Shown in the gallery is the complete process for a simple object tomography and the following tomographic reconstruction based on ART.

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  Fig. 2: Phantom object, two kitty-corner squares.

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  Fig. 3: Sinogram of the phantom object (Fig.2) resulting from tomography. 50 projection slices were taken over 180 degree angel, equidistantly sampled (only by coincidence the x-axis marks displacement at -50/50 units).

• Algebraic Reconstruction Technique - animated.gif

  Fig.4: ART based tomographic reconstruction of the sinogram of Fig.3, presented as animation over the iterative reconstruction process. The original object could be approximatively reconstructed, as the resulting image has some visual artifacts.

Further reading

• Avinash Kak & Malcolm Slaney (1988), Principles of Computerized Tomographic Imaging, IEEE Press, ISBN 0-87942-198-3.
• Bruyant, P.P. "Analytic and iterative reconstruction algorithms in SPECT" Journal Of Nuclear Medicine 43(10):1343-1358, 2002

External links

• http://www.slaney.org/pct/
• http://www.itk.org/ Insight ToolKit; open source tomographic support software
• http://tomopy.readthedocs.org
• http://www.visielab.ua.ac.be/software ASTRA (All Scales Tomographic Reconstruction Antwerp) toolbox; very flexible, fast and open source software for computed tomographic reconstruction
• http://niftyrec.scienceontheweb.net/ NiftyRec; comprehensive open source tomographic reconstruction software; Matlab and Python scriptable