Using the variance ensures that the NMSE will not be affected by adding a bias to . The measure NMSE is similarly defined. The SNR improvement due to processing is defined by

SNR improvement	(6.2.25)

A human observing two images affected by the same type of degradation will generally judge the one with the smaller NMSE to be closer to the original. A very small NMSE generally can be taken to mean that the image is very close to the original. It is important to note, however, that the NMSE is just one of many possible objective measure and can be misleading. When images with different types of degradation are compared, the one with the smallest NMSE will not necessarily seem closest to the original. As a result, the NMSE and SNR improvements are stated for reference only and should not be used in literally comparing the performance of one algorithm with another. Figure 6.5 illustrates the performance of a Wiener filter for image restoration. Figure 6.5(a) shows an original image of 512 X 512 pixels, and Figure 6.5(b) shows the image degraded by zero-mean white Gaussian noise at an SNR of 7dB. The SNR is defined as

	(6.2.26)

Figure 6.5(c) shows the result of the Wiener filter applied to the degraded image. In the Wiener filter, was assumed given and was estimated by averaging for ten different images. For white noise degradation, is constant independent of . The processed image has an SNR improvement of 7.4dB. As Figure 6.5 shows. Wiener filtering clearly reduces the background noise. This is also evidenced by the SNR improvement of. However, it also blurs the image significantly. Many variations of Wiener filtering have been proposed to improve its performance. Some of these variations will be discussed in the next section.

Variations of Wiener Filtering

The Wiener filter discussed was derived by minimizing the mean square error between the original and processed signals. The mean square error is not, however, the criterion used by a human observer in judging how close a processed image is to the original. Since the objective criterion consistent with human judgment is not known, many adhoc variations have been proposed. One variation is power spectrum filtering. In this method, the filter used has the frequency response given by.

	(6.2.27)

The function in (6.2.27) is the square root of the frequency response of the Wiene filter. If and are samples of stationary random processes linearly independent of each other, the output of the filter will have the same power spectrum as the original signal power spectrum . The method is thus known as power spectrum filtering. To show this,

	(6.2.28)

From (6.2.27) and (6.2.28)

	(6.2.29)

Several variations of Wiener filtering that have been proposed for image restoration can be expressed by the following :

	(6.2.30)

where and are some constants. When and reduces to Wiener filtering. When and reduces to power spectrum filtering. When is a parameter and , the result is called a parametric Wiene filter, all the comments made in section 6.21 apply to this class of filters. Specifically, they are zero-phase filters, and tend to preserve high SNR frequency components. The power spectra and are assumed known, and such filters are typically implemented by using the DFT and inverse DFT. In addition, these filters are typically lowpass. They reduce noise but blur the image significantly. The performance of the power spectrum filtering is shown in Figure 6.6. The original and degraded images used are those shown in figure 6.5. The SNR improvement is 6.6 dB.