98
Biomedical Signal and Image Processing
As can be seen, soft thresholding can be thought of as an extension of hard thresholding
that avoids creating discontinuities. Soft computing is often considered as a relatively
more successful algorithm for automated denoising. The main factor that affects the performance of both algorithms is the selection of a suitable threshold value ξ. As discussed
later, MATLAB toolbox for WT provides several techniques for this selection; however, often the manual selection of the threshold value seems to be the most appropriate
method for a given image or signal. As a result, we leave exploring this selection process
to the readers as an exercise in the Problems section.
5.6.2 COMPRESSION
In many biomedical applications, long signals and large images are created. The storage of all these signals and images creates a serious issue. The main objective here is
to design techniques to reduce the size of a signal or image without compromising the
information contained in the signal or image.
The process of compression using the DWT is very similar to denoising: first,
the signal or image is decomposed to its DWT coefficients, and then the coefficients
that are less than a threshold value are eliminated. Since there are often many coefficients that are too small to make a sensible difference in the signal, when a signal
is reconstructed using only the surviving coefficients, a good approximation of the
signal is obtained.
The second rule for compression states that since noise often corrupts the highfrequency components, by eliminating too-high-frequency components, the useful
information in a signal would not change considerably.
The third rule for compression states that too-low-frequency components may not
contain vital information, and their corresponding coefficients can be eliminated or
reduced in the compression process. This rule can be better understood if we think of
some undesirable trends in signals such as electrocardiogram that contain no relevant
information, and, therefore, it makes sense to eliminate them before saving the signal.
In a typical electrocardiogram, the low-frequency variations caused by the patient’s
respiration constitute the trend that is often eliminated by discarding the high-scale
(low-frequency) coefficients in the DWT domain.
The compression performance of the DWT will be further illustrated in the
following example.
Example 5.6
As previously mentioned, in decomposing of the image given in Figure 5.13,
the low-pass component requires only 25% of the storage space to provide an
almost identical version of the original image. The second-level decomposition
of the image is shown in Figure 5.14. As shown, the low-pass component, while
having only 6.25% of the original image, still provides an acceptable approximation of the original image. These observations demonstrate the capabilities
of the DWT in compression applications. This is why the newer versions of
image compression technologies such as some JPEG standards apply the DWT
for compression.
Biomedical Signal and Image Processing
As can be seen, soft thresholding can be thought of as an extension of hard thresholding
that avoids creating discontinuities. Soft computing is often considered as a relatively
more successful algorithm for automated denoising. The main factor that affects the performance of both algorithms is the selection of a suitable threshold value ξ. As discussed
later, MATLAB toolbox for WT provides several techniques for this selection; however, often the manual selection of the threshold value seems to be the most appropriate
method for a given image or signal. As a result, we leave exploring this selection process
to the readers as an exercise in the Problems section.
5.6.2 COMPRESSION
In many biomedical applications, long signals and large images are created. The storage of all these signals and images creates a serious issue. The main objective here is
to design techniques to reduce the size of a signal or image without compromising the
information contained in the signal or image.
The process of compression using the DWT is very similar to denoising: first,
the signal or image is decomposed to its DWT coefficients, and then the coefficients
that are less than a threshold value are eliminated. Since there are often many coefficients that are too small to make a sensible difference in the signal, when a signal
is reconstructed using only the surviving coefficients, a good approximation of the
signal is obtained.
The second rule for compression states that since noise often corrupts the highfrequency components, by eliminating too-high-frequency components, the useful
information in a signal would not change considerably.
The third rule for compression states that too-low-frequency components may not
contain vital information, and their corresponding coefficients can be eliminated or
reduced in the compression process. This rule can be better understood if we think of
some undesirable trends in signals such as electrocardiogram that contain no relevant
information, and, therefore, it makes sense to eliminate them before saving the signal.
In a typical electrocardiogram, the low-frequency variations caused by the patient’s
respiration constitute the trend that is often eliminated by discarding the high-scale
(low-frequency) coefficients in the DWT domain.
The compression performance of the DWT will be further illustrated in the
following example.
Example 5.6
As previously mentioned, in decomposing of the image given in Figure 5.13,
the low-pass component requires only 25% of the storage space to provide an
almost identical version of the original image. The second-level decomposition
of the image is shown in Figure 5.14. As shown, the low-pass component, while
having only 6.25% of the original image, still provides an acceptable approximation of the original image. These observations demonstrate the capabilities
of the DWT in compression applications. This is why the newer versions of
image compression technologies such as some JPEG standards apply the DWT
for compression.