Image Processing Techniques in Emission Tomography - Correction Techniques in Emission Tomography - page 124

(a) Original (b) Result

(c) Original amplitude (d) Mask amplitude (e) Filtered amplitude

FIGURE 7.2: The original noisy image (a) was filtered in Fourier domain us-
ing the frequency version of the mean filter. (b) Mean filtering result. (c) Am-
plitude of the original image jF(I)j. (d) Amplitude of the lter mask jF(M)j.
(e) Amplitude of jF(M) · F(I)j.

where M is the mean filter mask. Hence, convolution in the image domain is
equivalent to a point-wise multiplication in frequency domain.
The amplitude of the FFT of the noisy input image F(I) is shown in
Figure 7.2(c). The amplitude of the Fourier-transformed mean filter F(M)
can be seen in Figure 7.2(d). The result of F(M)

F(I) is illustrated in Fig-
ure 7.2(e). The resulting filtered image I d in Figure 7.2(b) is equal to the
result in Figure 7.1(b) (up to machine precision).
One of the advantages of filtering in the Fourier domain is the possibility
of performing deconvolution, as described in Section 7.5.2. A popular decon-
volution filter is the frequency space version of the Wiener filter. The charac-
teristics lie in the special choice of the deconvolution mask M F . For a detailed
discussion regarding this filter we refer to [21, 22].

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7.2.3 Wavelet transform domain
Similar to the Fourier transform, wavelets represent images as a linear
combination of basis functions of different frequencies. Wavelets have local
support in image space, whereas Fourier basis functions have infinite support.
Thus, a disadvantage of Fourier analysis is the total loss of local information
as compared to the wavelet transform.
Wavelet-based filters are very promising at present. There is a huge number
of publications on this topic. A brief overview is given in [45]. Wavelet filters
can be subdivided into linear and nonlinear methods. A wavelet realization
of the Wiener filter is a typical example of linear filters. The key principle of
non-linear procedures is the property that white noise in the image domain
maps to white noise in the transformed domain; thus noise can be separated
eciently from the pure signal.