Biomedical Engineering Reference
In-Depth Information
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(a) Original
(b) Wavelet filtered
(c) Wavelet function
(d) Scaling function
FIGURE 7.3: The original noisy image (a) was filtered using Daubechies
wavelets (D4). (b) Wavelet filtering result. (c) D4 wavelet function. (d) D4
scaling function.
A wavelet transform is performed for a finite number of levels. On each
level the image is processed using a wavelet function which is in principle a
band-pass filter (for high frequencies). The wavelet function is scaled on each
level, i.e., its bandwidth is halved compared to the previous level. The filter
results of the differently scaled wavelet functions as a whole are called details.
To cover the whole frequency spectrum a scaling function, in effect a low-
pass filter, preserves low frequencies. The filter result of the scaling function
is called approximation. The original image can be recovered completely from
the wavelet representation of details and approximation.
In general, non-linear wavelet denoising consists of three steps:
1. Transformation of the original image into the wavelet transform domain.
2. Thresholding of the details (on each level).
3. Transformation back into the image space.
Apart from choosing suitable basis functions, the main problem is to determine
an adequate threshold value. Adaptive as well as non-adaptive methods for
threshold determination have been proposed [45].
An illustrative example can be found in Figure 7.3. The noisy image in
Figure 7.3(a) was filtered with D4 Daubechies wavelets on the basis of the
three steps described above with a manually chosen threshold. The D4 basis
functions are given in Figure 7.3(c) and 7.3(d).
According to the three steps of wavelet denoising, some intermediate re-
sults of the 3-level D4 wavelet transformation are shown in Figure 7.4. The
details are shown as examples in Figures 7.4(a) and 7.4(b) for level 1 and 3,
respectively. After applying thresholding to the details, reduced versions result
which are shown in Figures 7.4(c) and 7.4(d). The next step is the transforma-
tion back into the image domain yielding the image shown in Figure 7.3(b).
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