4.1 Introduction
4.2 Notation and mathematical theorems used
4.3 Analytic image formation in 2D PET
4.4 Filtered BackProjection in 3D and 3DRP
After all corrections (e.g. for scatter, randoms and the effects of attenuation) have been applied to data acquired in a PET camera, the number of counts assigned to an LOR joining a pair of crystals is proportional to a line integral of the activity along that LOR. Parallel sets of such line integrals are known as projections. Reconstruction of images from projections is a problem to which much attention has been paid over the last 30 years, and many analytical and iterative reconstruction schema exist. For 2D reconstruction, the most commonly used algorithm is the analytical method called Filtered Backprojection (FBP). FBP is straightforward to implement but does have the property of amplifying noise in the signal (see section 4.3). Recently, considerable interest has been shown in iterative reconstruction schema, such as the Ordered Subsets  Expectation Maximisation (OSEM) algorithm (Hudson and Larkin 1994), which possess different noise properties to FBP. For 3D reconstruction, the Reprojection and Filtered Backprojection (3DRP) method of Kinahan and Rogers (1990) has been the most popular, in part because of the significant computational burden of newer 3D iterative reconstruction methods. 3DRP itself is computationally expensive, and this has led to the development of approximate 3D reconstruction algorithms. Of these, Fourier Rebinning (Defrise et al 1997), which reduces the 3D problem to a series of 2D problems without significantly distorting the image and results in a significant reduction in the computational burden, is stimulating particular interest.
In this section FBP in 2 and 3 dimensions will be
briefly summarised and the basic elements of 3DRP will be
described.
4.2 Notation and mathematical theorems used
In this work the following notation will be
adopted:







































Fourier transform 
inverse Fourier transform 

Backprojection 
Convolution operator 
The spatial coordinates for a fullring PET camera are shown in figure 11.
Figure 11. 3D coordinate system for a fullring PET camera
Several mathematical theorems will be used without proof. These include the following:
Fourier's theorem:
(5)
Fourier addition theorem:
(6)
Convolution theorem:
(7a)
and
(7b)
If a function is sampled with a sampling distance d then it may be fully recovered from its samples (apart from harmonic components which are zerovalued at the sample points) if its Fourier transform has no nonzero components at frequencies beyond a particular value k_{c }where k_{c} = (1/2d).
The critical frequency k_{c} =
(1/2d) is known as the Nyquist frequency. A more detailed
treatment of this material may be found in, for example,Bracewell
(1986).
4.3 Analytic image formation in 2D PET
In 2D PET, data from LORs are arranged into sets
of 1dimensional parallel projections. For a point source
distribution, this gives rise to a series of intensity profiles as
shown in figure
12.
Figure 12. Projections generated from a single central point source (3 projections shown).
An estimate of the original source distribution may be obtained by a process known as backprojection. In this process, the magnitude of each value in a projection is added to every point in image space corresponding to the relevant line of integration in object space. Backprojections for a single pointsource are shown in figure 13. When a small number of projections are used, the resultant image contains starshaped artefacts. In the limit of an infinite number of projection angles, this process is the equivalent of convolving the original source distribution with the function f(r,q ) = r^{1}.
An extended source distribution can be considered as being made up of a series of points of varying intensity, each of which, after backprojection, is convolved with r^{1}. Since convolution is distributive over addition, the relationship between the density function and the back projection b(r,q ) may be written:
(8)
where r (r,q ) is the extended source distribution. Application of the convolution theorem gives:
(9)
Now the Fourier transform of r^{1} is just r_{k}^{1}, so that
(1.10)
and we have recovered the extended source distribution function. In practice the source distribution is sampled with a finite samplewidth, so to enable application of the Sampling theorem it is necessary cut off the Fourier transform of the backprojection at the Nyquist frequency. This is achieved by multiplying it with a gate function of the appropriate width. In real space this is equivalent to convolution with the sinc function. This can result in "undershoot" artefacts in the image (where image values are underestimated or even negative), particularly in regions close to sharp edges in the source distribution.
Figure 13. Backprojections of a point source. With finite numbers of
backprojection angles, "star" artefacts are seen.
Filtered Backprojection is a mathematically equivalent process to that described above. Fourier transforms of the projections are first calculated and multiplied by the cutoff 1D version of r_{k} (the ramp filter). The inverse Fourier transform of the result is then calculated and backprojected to create the image.
Use of the ramp filter amplifies highfrequency components in the backprojection. Unfortunately statistical noise in the data manifests in Fourier space as highfrequency components. So the process of FBP amplifies noise in the image. In order to reduce this effect, a range of modifications to the ramp filter can be used. Perhaps the most commonly used of these is the Hanning filter (figure 14). Such filters are equivalent to some form of smoothing in image space.
Figure 14. The Ramp and Hanning filters
A more detailed mathematical treatment of analytic 2D
tomographic reconstruction is given by Brooks
and Di Chiro (1976).
4.4 Filtered BackProjection
in 3D and 3DRP
In three dimensions, the data from the LORs may be
arranged into 2D sets of parallel projections (figure
15(b)). FBP generalises to 3D directly if
the projections can be obtained over all j as well
as q .
Unfortunately in real cameras projections cannot easily be obtained
over the full range of j , and a different filter must be found for the deconvolution step
(Colsher,1980).
Figure 15(a). Parallel projections in 2D. Note that the LORs
become closer together towards the edge of the FOV. To correct for
this, the data must be resampled (arc corrected) prior to
reconstruction.
Figure 15(b). Parallel projections in 3D
Another problem is the fact that the limited axial extent of the camera causes some of the 2D projections to be truncated, and the degree of truncation depends on position (figure 16). As a result there is no single filter which is appropriate for every projection set. The 3DRP algorithm circumvents this problem by performing a 2D reconstruction on a subset of the 3D projections first. The resulting image volume is then reprojected in order to obtain estimates of the truncated projections and a 3D FBP is then performed on the combined real and synthesised data.
Figure 16. Axial cutaway diagram of a PET camera operating in
3D mode, showing the extent of the projection sets as a function of
angle j . As
j increases, the measurable extent of the
projection set decreases, requiring the reconstruction filter to
change with position. To avoid this, an initial 2D reconstruction is
performed on the j = 0 projection set, and
estimates of the missing parts of the truncated projections obtained
by reprojecting through the image volume.
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Revision date:
12 Jan 1999