Biomedical Engineering Reference
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lines are attributed to measurement values on transaxial planes. The main
diculty is that to implement the Fourier transforms involved, Pf must be
available everywhere. Therefore, missing data are computed from the mea-
surements for = 0. For details of the complex implementation, see [4].
The dominating term with respect to complexity is a 3D Fourier transform
with interpolation, which makes Fourier rebinning not too much slower than
SSRB.
3.2.4.2
3D filtered backprojection
This is simply an implementation of the 3D X-ray backprojection formula.
Note that to implement the corresponding algorithm, we need all lines pass-
ing through our object, but only a small number will actually be available.
To get these, usually a low{quality preliminary reconstruction like SSRB is
performed, and the missing data is taken from projections of the resulting
image. The algorithm itself is on the order of N
5
.
3.2.5 Limitations
The main advantages of the analytical methods are their speed and the
fact that they can be fully analyzed mathematically, so exact bounds on reso-
lution can be given and optimal choice of parameters is guaranteed. However,
they are inflexible with respect to changes to the physical model and to the
incorporation of non-Gaussian noise.
3.3 Discrete algorithms
Assume that the activity function f can be written (or at least approxi-
mated) as a linear combination of a nite number of ansatz{functions
k
such
that f =
P
k=1
k
k
with unknown coecients
k
. Typically,
k
are trans-
lates of each other, and could be Gaussians or voxels/pixels (characteristic
functions of cubes or rectangles). For simplicity, we assume that the latter is
the case.
Our continuous problem is thus reduced to the discrete problem of finding
N numbers, satisfying the measurements. Assuming that data gl
l
are available
for lines of response L
l
, l = 1 :::M, our problem reads:
Find pixel values
k
, such that the line integral over Ll
l
of the image is gl:
l
:
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