Image Processing Techniques in Emission Tomography - Correction Techniques in Emission Tomography - page 142

compartments. The solution of this system gives an estimation of the true
signal for each compartment without the influence of the spill-over effect.
A disadvantage of this approach lies in the need for an exact delineation
of the individual compartments. In PET imaging the selection of functional
regions is usually made using anatomical data taken from registered CT or
MRI data [71]. Similar to the case of the RC, the uniform tracer uptake is a
limiting assumption.

Deconvolution
Deconvolution tries to reverse the convolution of a clean image I u with the
PSF P (see Equation (7.24)). In the absence of additive noise N and given
a known P, I = P ? I u can be deconvolved using the convolution theorem
(Section 7.2.2). In real PET systems, however, noise is always present and the
PSF may not be known exactly. This complicates the deconvolution process
enormously. During direct deconvolution, noise is amplified strongly. We clar-
ify this relationship with the help of the model of image degradation: Fourier
transform of Equation (7.24) and application of the convolution theorem leads
to

F(I u ) = F(I)

F(P) F(N) F(P) ;

(7.28)

where F denotes the Fourier transform. Noise is often located in the higher
frequencies. Since P is a Gaussian function, the values of F(P) are close to
zero at high frequencies. For this reason the noise F(N) gets amplified due to
the division by F(P).
Figure 7.13 shows the results of different deconvolution techniques for an
artificial example in the absence of noise. Iterative deconvolution algorithms
try to avoid the amplification of additive noise, which is hard to put into
practice. In the following we describe suitable approaches for partial volume
correction based on deconvolution techniques.

Van-Cittert deconvolution
The van-Cittert iteration can be used to perform PVC if the point spread
function is known approximately. The iteration scheme is given by

I 0 = I (7.29)

I j+1 = I j + (I 0 P ? I j ) ; (7.30)

where I is the given PET image and 2R + a relaxation parameter. The
criterion for ending the iteration scheme is crucial since the amount of noise
increases with every iteration step [64].

Richardson{Lucy deconvolution
The Richardson{Lucy (RL) deconvolution uses a statistical approach. The
PSF is assumed to be known approximately. The basic idea of the RL de-
convolution is to correct the observed image I towards a maximum likelihood