Tomographic reconstruction

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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.

Figure 1: Parallel beam geometry. Each projection is made up of the set of line integrals through the object.

The projection of an object at a given angle $θ$ 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 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 $μ(x, y)$ . 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 $θ$ . This is repeated for various angles. Attenuation occurs exponentially in tissue:

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

where $μ(x)$ 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 $θ$ , is given by the line integral:

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

Projected X-rays are clearly visible on this slice taken with a CT-scan

Using the coordinate system of Figure 1, the value of $r$ onto which the point $(x, y)$ will be projected at angle $θ$ 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 $μ(x, y)$ . 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.