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In this study we compare four different denoising methods Tree-Adapted wavelet shrinkage (TAWS), SureShrink, Median Adaptive and Median (see Table 4.1)) via three error measures (Sup, RMS and SNR) (Table 4.1). Secondly we design two different wavelet transform schemes the DWT and MODWT (see Chapter 2) with four different wavelet bases (Haar, Daub 2, Daub 4, Ladaub 8) (Table 4.2) and test them at various window sizes for denoising performance (Table 4.3). The aim is to study the suitability of different wavelet bases and also of different window sizes for denoising brain image data discussed in detail in Chapter 5. Thirdly in Section 4.3.3 we compare wavelet denoising versus independent component analysis (ICA) denoising (using P28 brain image data).
4.3.1 Comparison of the four different denoising methods
We examined four different denoising methods by comparing error estimates obtained via FAWAVE (version 2.0, Walker, 2008). The first method is the Tree- Adapted Wavelet Shrinkage (TAWS) technique implemented by the TAWS algorithmas described in Walker (1999; 2000; 2002; 2003) and in Walker & Chen (2000). In
Figure 4.11 Denoised images using several denoising methods for P28.
the TAWS procedure, shrinkage, as well as averaging, is implemented which reduces the computation time required.
The second routine is the famous SureShrink method (Donoho & Johnson, 1995b, 1994; Gencay, et al.,2002; Ogden, 1997).
SureShrink uses a hybrid of the universal threshold (Donoho & Johnson, 1994, 1995b) and the Stein's Unbiased Risk Estimator (SURE) threshold (Donoho & Johnson, 1994, 1995b; Gencay et al., 2002). We now describe our remaining non-linear methods (Median and Adapted median, see Table 4.1).
With non-linear filters in spatial filtering, noise is removed without any attempt to explicitly identify it. Spatial filters employ a low pass filtering on groups of pixels with the assumption that noise occupies the higher region of the spectrum. In recent years, a variety of non-linear median type filters have been developed to overcome the blurring of images (Ben Hamza et al., 1999; Hardie & Barner, 1994; Yang et al., 1995). The third method we use is non-linear filtering which is a 3 by 3 median denoising (Fitch et al., 1984). This is used to denoise images that are corrupted by impulse noise. The fourth method denoises an image by a 3 by 3 adaptive median filter (Wang & R.A, 1995; Xing & Wang, 2001). In this procedure the pixel value centred on a 3 by 3 region is only replaced by the median, if its values differ from the median by more than %. It usually results in a more sharply resolved denoising than does median denoising. Denoised images (P28) via the four methods are given in Figure 4.11. It is difficult to distinguish between the original image and the four denoised images. We then use three ways of measuring the amount of error between the original image and the resultant noisy image. All of these measures aim to provide quantitative evidence for the effectiveness of the given noise removal method (as given in Table 4.1) (Sup, RMS and SNR).
The first error measure is the supreme error (Sup), which is taken to be the largest magnitude error between the original image and denoised image values at all pixels (Walker, 1999). The second measure is the root mean square error (RMS), which is the classical error between the original image and the denoised image (Walker, 2000). A third measure, commonly used in image processing, is the signal-to-noise ratio (SNR) (Walker, 1999, 2000). The results of applying these three measures to P28 are summarized in Table 4.1. When we compare the Sup error and RMS, across denoising methods, SureShrink gives the smallest error. SureShrink appears to be superior among the four denoising methods. For the SureShrink denoising method the RMS error is 0.246, which is less than the RMS error of 0.565 for TAWS denoising. Both the Median and
the Adaptive Median denoising methods are not much different from each other across all error estimate types. Unlike other measures, an increase in SNR represents a decrease in error. The SureShrink denoising method appears to be superior to the other three denoising methods according to SNR.
The measures of error discussed above have all been used for many years. Their deficiencies in relation to accurate quantification of the perceptions of our visual systems are well known. It is generally recognized that they have remained in use, despite their deficiencies, mainly because they fit well into the type of mathematics used in image processing, making theoretical predictions concerning their values relatively easy to obtain (Walker, 1999, 2000).
Table 4.1 Error measurements for the SPECT 2D images.
Figure 4.12 Original brain image and denoised images up to 5 levels for P28 using the soft
threshold (via the Haar wavelet and DWT).
4.3.2 Wavelet domain for image denoising
Now we consider only the wavelet domain for image denoising. By choosing a threshold that is a sufficiently large multiple of the standard deviation of the random
noise, it is possible to remove most of the noise by thresholding the wavelet transform values. We examine how this procedure performs on the P28's brain image using the soft thresholding technique. The soft thresholding method will be compared with hard thresholding.
Our first example of image denoising is examined by using the Haar wavelet on the noisy version of P28's brain image. The result is shown in Figure 4.12. The original image is transformed to various window sizes using the DWT and MODWT.
The decomposition process can be iterated, with successive approximations being decomposed in turn, so that one signal is broken down into many lower resolution components as illustrated by Figure 4.12. In Figure 4.12, the levels 3, 4, and 5 seem to be oversmoothed and the soft threshold seems to be too aggressive. Nevertheless, the histogram of the details is quite good for level 3 since it is close to a Gaussian distribution (see Figure 4.13).
Figure 4.13 Histogram of 5 level details for P28 via the DWT.
Table 4.2 Wavelet soft threshold values used to generate the denoised image of P28
using four different wavelets.
Table 4.3 The standard deviation of the residuals of the denoised image for Patient 28 (P28).
Given the Gaussianity of level 3 details (Figure 4.13) we now deal only with level 3 wavelets.
Next we examine to denoised images using first the DWT and then MODWT using the same choice as before of wavelet bases. We use threshold values for the soft threshold, as given, in Table 4.2.
When we compare Figure 4.14 with Figure 4.15, whilst it is difficult to distinguish in the printed versions between the original image and the four denoised images, Figure 4.15 is more clear with the MODWT denoised images appear to be slightly superior. The residuals between the original image and denoised image
Original image Haar Daub2 Daub 4 LaDaub8
Figure 4.14 Denoised images using the DWT with varying wavelet types (P28).
Original image Haar Daub2 Daub 4 Ladaub8
Figure 4.15 Denoised images using the MODWT with varying wavelet types (P28).
confirm this subjective statement (see the standard deviation of the residuals given in Table 4.3).
Note that if the filter cleans up the given image effectively, the residual image is essentially just noise. Therefore small standard deviation (SD) gives a better denoised image (Table 4.3). Comparison with the performance of the wavelet transform was made. Results show that the MODWT produce better results, although the standard deviation of the residuals is often higher than the DWT error.
When comparing the four different wavelet bases (see Table 4.3), LaDaub8 gives a superior denoised image. We can conclude from our experiment that choice of the wavelet basis is crucial. The experiments were done using a window size of 256x256, 256x512, 256x1024, 512x256, 512x512, and 512x10204 (Table 4.3). The extended images 256x1024 and 512x10204 are good choices. It is noteworthy that the small
SD values of residuals do not always correspond to good visual quality.
4.3.3 Comparison between Wavelet and ICA denoising
In Section 4.3.1 we compared the different denoising methods on the basis of three well-defined criteria: the supreme error (Sup), the root mean square error (RMS), and the signal to noise ratio (SNR). Every criterion measures a different aspect of the denoising method. It is easy to show that only one criterion is not sufficient to judge the image, and so one expects a good solution to have a high performance under the three criteria.
In this section we compare two source signal extraction algorithms, namely the Wavelet Denoising (WD) (by soft thresholding) method (using the Haar wavelet) and Independent Component Analysis (ICA) on a Patient 28's brain image. Patient 28-2D data denotes the30th slice generated in 3D (79x95x68) among 68 slices of the patient's brain image (P28).
A common definition of SNR is the ratio of mean to standard deviaation of a signal in statistics (Säckinger, 2005; Schroeder, 2000)
where ï is the signal mean and Ïƒ is the standard deviation of the noise. The higher the ratio, the less obtrusive the background noise is. Recall that the discrete wavelet transform (DWT) is a time-scale representation technique of a signal with a mother wavelet function. Wavelet transforms can thus be used to reduce the noise in a signal by a method called wavelet shrinkage proposed by Donoho (1995) that is mentioned in Chapter 3. DWT localizes information of the deterministic signal into a limited number of the wavelet coefficients (see Equations (2.1)-(2.5) and Figures (2.5)- (2.9) in Chapter 2).
Figure 4.16 Top- Noisy signal for P28-30th slice, the middle is the denoised signal using ICA and the bottom is the denoised signal using Haar wavelet thresholds.
The use of ICA denoising is motivated by the fact that the linear model consists of a set of independent signals additively combined (see Chapters 1 and 5). Since ICA is capable of identifying the components
of a mixture, the demixing becomes effectively a de facto denoising (Jung et al., 2000; Makeig et al., 1999; Ylipaavalniemi et al., 2006). To perform ICA, we need two conditions to be satisfied. First, the components should be statistically independent and secondly the mixing should be linear. FastICA (Hyvärinen, 2001a; Hyvärinen et al., 2001b) does not distinguish between signal and noise, but simply separates the components (Bobin et al., 2007; Hyvärinen et al., 2001b; Üzümcü et al., 2003), in contrast to the WD technique. The original image name becomes the denoised version of the desired signal by using the Haar wavelet. An application of the FastICA algorithm that yields the estimated signal waveform is given in Figure 4.17, showing only parts of the signal since the whole series is too big. To quantify the ability of ICA in recovering the desired signal we use two different performance measures: the signal-to-noise (SNR) ratio and the correlation coefficient. The SNR values are computed before and after ICA denoising (see Table 4.4). The correlation values measure how much the original signal resembles its corrupted version. Therefore the correlation coefficient yields a measure of the recovery performance in the denoising process. Values of SNR and the correlation coefficients for P28 between the noisy and denoised signal are given in Table 4.4.
Table 4.4 SNR and Correlation coefficients between the noisy and denoised signal: P28.
From Table 4.4, we conclude that the performance of ICA remains near optimal while WD's degrades substantially. This indicates that ICA is a robust denoising technique, in that its performance is not affected by the severity of the mixing conditions. Similarly, the ICA correlation coefficient has a value of 1, which corresponds to near perfect waveform matching, even when the input signal is highly corrupted. On the other hand, wavelet denoising has a correction coefficient of 0.8581 that shows poor performance than ICA. Therefore we conclude that, in the presence of non-Gaussian noise contamination, as in the case of brain image slice signals, WD fails to recover the original source signal. In contrast, ICA denoising has a robust and near optimal performance independent of the severity of mixing and of the statistical distribution of the mixing components. Furthermore ICA does not necessitate any outlier elimination (Emir et al., 2003; Matthews et al., 2008; Vigneron et al., 2003).
An application ICA to brain image denoising is described in detail in Chapter 5.