Biomedical Engineering Reference
In-Depth Information
Straightdirection
Axialdirection
Object
Slantdirection
FIGURE 3.3: Single slice rebinning: Measurements of slant directions are
used as additional measurements in the straight reconstruction plane.
3.2.4.1
2D Rebinning
Assume that our object is constant in the axial direction. Then, up to a
constant factor, the integrals over lines that share the same projection on the
transaxial plane will give the same measurement.
In this case, the single slice rebinning algorithm SSRB will be exact (see
Figure 3.3):
1. Compute approximations to the 2D{Radon transform Rf(;s;z 0 ) of the
activity function f on the plane z = z 0 in the following way: For a
reconstruction in a plane E perpendicular to the axial direction, we
take into account the measurements g L over all lines L that have their
center point in E in the following way: Compute the projection of L
onto the plane and use g L as an additional measurement on E.
2. Reconstruct using 2D filtered backprojection. Obviously, this will not
generally be exact for functions with variations in the z coordinate.
In order to derive an exact formula, we rst observe that if f is a 3D{
function, Pf is a 4D{function. Thus, we expect the range of P to satisfy a
one-dimensional consistency condition. One formulation is easily derived
from the Fourier slice theorem for Pf: If , 0 are in S n1 , and is in
the intersection of ? and 0? , we nd that
Pf(;) = (2) 1=2 f() = Pf( 0 ;)
which basically means that it is possible to convert data measured in a given
plane to measurement data for a different plane.
In fact, [3] proves that using an appropriate consistency condition, the
measurement values for straight directions can be computed from the values
for any slant direction. This is the starting point for Fourier rebinning type
algorithms: Using approximations to the equation, measured values on slant
 
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