PET Image Reconstruction - Correction Techniques in Emission Tomography - page 38

longer exact, and can be only an approximation: since, obviously, h is bandlim-
ited, f is, too. So, our reconstruction formula can be exact, that is f = f,
only for bandlimited functions f with band limit .
Shannon's sampling theorem states that bandlimited functions can be
uniquely reconstructed from samples, provided the sampling frequency is bet-
ter than the Nyquist limit. Since for bandlimited f the Radon transform Rf
is bandlimited too (using the projection theorem), we expect that a filter fre-
quency of corresponds to optimal sampling schemes of Rf that ensure that
all {bandlimited functions can be uniquely reconstructed.
Turning that argument around: given a scanner with fixed sampling geom-
etry, there is a corresponding such that ltered backprojection is exact for
all functions f with bandlimit , which gives an easily computable optimal
choice for , depending on scanner geometry only. Note that ltered back-
projection thus has no more free parameters; everything can be chosen in an
optimal way.
With respect to complexity, the backprojection step is dominating. As-
suming that all parameters (resolution, number of data in each dimension, ...)
are on the order of N, R h gives an image of size N 2 and needs N operations
for the discretized integral, so the complexity of the overall algorithm is on
the order of N 3 . Note that using fast backprojection, this can be cut down
to O(N 2 log N) operations (see, e.g., [13] for an introduction and multiple
references). The same order can be achieved by using implementations of the
Fourier slice theorem [6].
3.2.4 Implementation and rebinning
In the measurement process, single coincident photons on lines of response
are recorded and counted. In order to satisfy the assumption that the number
of photons on an LOR is proportional to the line integral over the activity
function, we may need to aggregate lines of response into bins: in a measure-
ment system of infinite accuracy, the probability of measuring two events on
the same LOR is 0, so we would have a data vector of zeros and ones which
does not satisfy our assumption. Note that this is the starting point of list
mode algorithms.
Further, in the algorithm given above, we will be able to make use of only
2D-data, corresponding to a single ring of detectors, perpendicular to the axial
direction of the detector in investigation. In order to make use of measure-
ments in slant directions, we need to incorporate these into the reconstruction
process.
There is no pure mathematical reason for doing this; using infinite mea-
surement time, and infinite dose, the approximation of the data as an integral
over activity function becomes exact, so there is no need for data from slant
directions. However, of course, in an application we will try to make use of as
many detected events as possible to enhance the signal to noise ratio.