Image Processing Techniques in Emission Tomography - Correction Techniques in Emission Tomography - page 121

7.2 Denoising
A variety of intrinsic factors degrade image quality in emission tomography.
These include scatter, randoms, out-of-field counts, detector dead time, de-
tector noise, patient motion, attenuation, non-colinearity of photons, positron
range and image reconstruction artifacts. Further, a trade-off between image
quality on the one hand and examination time and radiation exposure on the
other is required. Prolonged examination time usually implies inconvenience
for the patient. In contrast, short acquisition time and low radiation dose lead
to a reduction of image quality due to a lowered statistic.
In the following, we describe techniques for noise removal. The focus is put
on denoising as a post-reconstruction process, leaving aside noise reduction
during image reconstruction. The image degradation process can be modeled
as

I = P ? I u + N ; (7.1)

where I is the measured image signal, I u is the (uncorrupted) image free of
noise, P a convolution mask and N additive noise. The discrete convolution
of an nm mask P and an image I u at pixel (x;y) is dened as

X X

(P ? I u )(x;y) := P(i;j)I u (xi;yj) : (7.2)

i=1 j=1

The focus of this section is put on the elimination of the additive noise N.
Making good estimates of P is an important part of the more general problem
of image restoration. Image restoration is discussed in connection with partial
volume correction (PVC) in Section 7.5 where P represents the point spread
function (PSF).
Compared to the underlying image I u , the additive noise N is mainly
present in high frequencies. In order to improve image quality by eliminating
the high-frequency noise component the image has to be low-pass filtered.
Image quality can be quantified by means of the signal-to-noise ratio (SNR).
The definition of the SNR is given in Section 7.7.1.
As denoising has a long history, many approaches exist. A short introduc-
tion into basic theories is given in the following. For further reading a survey
of denoising techniques is given in [45]. Publications dealing especially with
denoising in emission tomography are [1, 6, 24, 59]. Standard image processing
literature usually deals with this topic as well [21, 22, 48, 52].
The rest of this section is structured according to three main approaches
for image denoising. First, image domain{based lter methods are discussed
in Section 7.2.1. They can be understood intuitively and thus provide a good
starting point. In Section 7.2.2 filtering in the frequency domain using the
Fourier transform is presented. Finally, the actual popular filtering in the
wavelet transform domain is introduced in Section 7.2.3.