Biomedical Engineering Reference
In-Depth Information
a particular pair of detectors, and each row represents the projected activity of
parallel detector pairs at a given angle relative to the detector ring. In other
words, if p represents the sinogram and p ( r ) represents the value recorded
at the ( r ) position of p , then p ( r ) represents the total number of photon
emissions occurring along a particular line joining two detectors at a distance
r from the center of the tomograph, viewed at an angle θ with respect to the
y -axis (or the x -axis, depending on how the coordinate system is chosen) of
the tomograph. However, the sinogram provides only little information about
the radiopharmaceutical distribution in the body. The projection data in the
sinogram has to be reconstructed to yield an interpretable tomographic image.
2.10 Image Reconstruction
The goal of image reconstruction is to recover the radiotracer distribution from
the sinogram. The reconstruction of images for the data acquired with the two-
dimensional mode is simple, while the reconstruction of a three-dimensional
volumetric PET data is more complicated, but the basic principles of recon-
struction are the same as those for the two-dimensional PET imaging. We focus
the discussion on the two-dimensional PET image reconstruction for simplicity.
A more thorough discussion of three-dimensional data acquisition and image
reconstruction can be found elsewhere [29].
The theory of image reconstruction from projections was developed by
Radon in 1917 [4]. In his work, Radon proved that a two-dimensional (or
three-dimensional) object can be reconstructed exactly from its full set of one-
dimensional projections (two-dimensional projections for three-dimensional ob-
ject). In general, image reconstruction algorithms can be roughly classified into
(1) Fourier-based and (2) iterative-based.
2.10.1 Fourier-Based Reconstruction
The Radon transform defines a mathematical mapping that relates a two-
dimensional object, f ( x , y ), to its one-dimensional projections, p ( r ), mea-
sured at different angles around the object [4, 30]:
p ( r ) =
f ( x , y ) dl r
(2.6)
0
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