From (6.2.15) and (6.2.18), the noncausal Wiener filter is given by

	(6.2.19)

Application of the noncausal Wiener filter in (6.2.10) to the signal estimation problem is shown in Figure 6.1. The noncausal Wiener filter has been the basis of many systems developed in such applications as speech enhancement and image restoration. These results can also be applied to solving the linear mean square error estimation problem when and the means of the signal and noise processes, are nonzero. The solution in this case involves subtracting from the noisy observation applying the noncausal Wiener filter developed for zero-mean processes, and adding to the resulting signal. This is illustrated in Figure 6.2.

If we assume that and given by equ.(6.2.7), are samples of zero-mean stationary random processes that are linearly independent of each other and that their power spectra and are known, the optimal linear minimum mean square error estimate of is obtained by filtering with a Wiener filter whose frequency response is given by

	(6.2.20)

If we impose the additional constraint that and are samples of Gaussian random processes then the Wiener filter in (6.2.20) is the optimal minimum mean square error estimator of the signal among both linear and nonlinear estimators. The Wiener filter was first considered for image restoration in the early 1960s. It has influenced the development of many other image restoration systems. The Wiener filter in (6.2.20) was derived under the assumption that and are samples of zero-mean processes. If has a mean of and has a mean of , then and are first subtracted from the degraded . The resulting signal is next filtered by the Wiener filter. The signal mean is then added to the filtered signal. This is shown in Figure 6.3.

Figure 6.3 Noncasual Wiener filter for linear minimummean square error estimation of from =+

The treatment of nonzero means, as shown in Figure 6.3 minimizes the mean square error between and for Gaussian random processes and . It also ensures that will be an unbiased estimate of . If is zero , will be identical to the mean . In this case, can be estimated from .

The Wiener filter in (6.2.20) is a zero-phase filter. Since the power spectra and are real and nonnegative, is also real and nonnegative. Therefore, the Wiener filter preserves the high SNR frequency components while attenuating the low SNR frequency components. If we let approach will approach 1, indicating that the filter tends to preserve the high SNR frequency components. If we let approach will approach 0, indicating that the filter tends to attenuate the low SNR frequency components.

The Wiener filter is based on the assumption that the power spectra and are known or can be estimated. In typical problems, estimating the noise power spectrum is relatively easy Estimating the image power spectrum is not a simple matter. One method used is to average for many different images . This is analogous to the periodogram averaging method for spectral estimation. Another method is to model ,with a simple function such as

	(6.2.21a)

	(6.2.21b)

For some constant . The parameter is estimated from the degraded image .

The Wiener filter is generally implemented in the frequency domain by

	(6.2.22)

The sequences represent the discrete Fourier transforms (DFTs) of and . In (6.2.22), the size of the DFT and inverse DFT is at least (N+M-1) when the image size is N X N and the filter size is M X M, If the DFT size is less that (N+M-1) (N+M-1), then the IDFT will not be identical to near the boundaries of the processed image because of aliasing effects. In most cases, however, the effective size of is small, and adequate results can be obtained with a DFT and inverse DFT of size N N. One way to obtain is to sample the Wiener filter frequency response by

	evaluated at		(6.2.23)

where the size of the DFT and inverse DFT is L X L.

The Wiener filter is typically a lowpass filter. Typically, the energy of a typical image is concentrated in low-frequency regions. Since random background noise is generally wideband, the Wiener filter is lowpass in character. This is illustrated in Figure 6.4.

Figure 6.4(a) shows an example of a which decreases in amplitude as and increases. Figure 6.4(b) shows an example of a that is constant independent of and . The resultant Wiener filter given by (6.2.20) is lowpass in character, as shown in figure 6.4(c)

When the information is available, we will use the normalised mean square error (NMSE) between and and between and degraded image to illustrate the performance of image restoration algorithms. This is defined as

	(6.2.24)

where Var[.] is the variance.