Frequency-domain filtering was introduced in topic 3. One particularly important filter is the Wiener filter which finds its main application in image restoration (Section 3.2.3). Let us recall the general equation of the Wiener filter in the frequency domain:
Here, H(u,v) is the Fourier transform of the point-spread function of the degradation process h(x,y) and H*(u,v) is its complex conjugate. Se(u,v) is the power spectrum (i.e., the squared magnitude of the Fourier transform) of the noise, and S/u,v) is the power spectrum of the ideal image f (x,y). A special case exists where blurring with the point-spread function is negligible [i.e., H(u,v)« 1] and Equation (5.19) simplifies to
Conversely, if the noise component is negligible, the filter in Equation (5.19) simply becomes the reciprocal of the point-spread function, W(u,v) = 1/H(u,v). However, dividing the degraded image by the degradation function in the frequency domain is impractical because the degradation function H(u,v) generally has lowpass characteristics (i.e., the magnitude drops off toward higher frequencies). Therefore, its reciprocal, W(u,v), assumes very large magnitudes at higher frequencies. This may lead to high frequencies of the degraded image, G(u,v), being divided by very small numbers or even by zero. Since noise has a broadband spectrum, the contribution of Se(u,v) to the denominator of Equations (5.19) and (5.20) prevents W(u,v) from assuming very large values at high frequencies. In fact, a higher noise component reduces the high-frequency magnitude of W(u,v), therefore giving the filter a more lowpass character. Conversely, if the noise component is small in Equation (5.19), the blurring term H(u,v) dominates and the filter assumes more highpass characteristics. In most practical cases, the power spectrum Sf (u,v) is unknown, and the spectrum of the noise component Se(u,v) is also unknown. Most frequently, the power spectrum ratio Se(u,v)/Sf(u,v) is therefore replaced by a constant k and the Wiener filter simplifies to
The constant k is determined experimentally for each case. A large value of k, necessary when a large noise component is present, configures the filter to act more as a lowpass, while a small value of k, allowable in images with low noise component, configures the filter as a highpass (see Figure 3.11). Two adaptive variations of this filter are possible, a frequency-adaptive form where k becomes a function of frequency, K(u,v), and a spatially adaptive form where k becomes locally dependent as k(x,y).
If the statistical properties of the noise are spatially variable, Se(u,v) in Equation (5.20) becomes a function of the spatial coordinate (x,y). In the simplified form [Equation (5.21)], the constant k would become spatially dependent and the noise-reducing Wiener filter can be formulated as
where
is the local variance of the noise component. The application of this spatially adaptive Wiener filter is computationally extremely expensive, because the inverse Fourier transform has to be performed for each pixel. It can be shown, however, that the adaptive minimum mean-square filter in Equation (5.3) is a very efficient spatial-domain implementation of the adaptive Wiener filter in Equation (5.22).40
This example explains the main challenge for the design of adaptive frequency-domain filters. Since spatial information is lost in the frequency domain, local properties can only be considered with a separate inverse transform for each region of the image where the statistical properties differ. Such a filter is computationally inefficient. However, there are alternative approaches. Abramatic and Silverman1 proposed a parametric Wiener filter,
where 7 is a function to be optimized according to specific criteria. A nonadaptive version of this filter is obtained by minimization of the difference between the idealized, noise-free image and the restored image (Backus-Gilbert approach3). It can be seen that with 7 ^ 0, the filter becomes the identity filter (a desired behavior near edges), whereas for 7 ^ 1 the filter approaches the conventional Wiener filter in Equation (5.22). The Backus-Gilbert approach still requires 7 to be a function of location, 7 (x,y), with the associated inverse Fourier transform necessary for each pixel x,y. Abramatic and Silverman propose to use a function for 7 (x,y) that falls monotoni-cally from 1 to 0 as the local image gradient increases from 0 to to. Furthermore, it is possible to consider directionality of the gradient (e.g., the four directions of the compass operator) to formulate an anisotropic function 7 (x,y). The effect of such a filter would be related to the adaptive bilateral filter in Equation (5.6). The main reason to pursue a frequency-domain approach with a Wiener filter is the fact that the Wiener filter is the optimum filter for known image and noise variance. If either is unknown, an empirical approach such as the minimum mean-squared-error filter or the adaptive bilateral filter will lead to satisfactory results with less computational effort.
Using the same basic idea, Guy17 recently proposed a Fourier-based approach to minimize Poisson noise in scintigraphic images. Poisson noise is a special case of multiplicative, intensity-dependent noise, where the standard deviation is approximately equal to the mean intensity value. The algorithm, termed Fourier block noise
reduction, uses a moving-window Fourier filter. For each pixel, a square local window is specified in which the local variance ct^xj) is computed in the spatial domain [Equation (5.4)]. The local noise component, oN(x,y) can be estimated from the Poisson model to be approximately equal to the local mean intensity. Now, the local window is subjected to a Fourier-domain lowpass with variable cutoff frequency, and the cutoff frequency is increased in repeated steps. After each step, the residual noise of the filtered block, oR(x,y), is computed by using Equation (5.4), and the algorithm halted when oR(x,y) < oN(x,y) in the local neighborhood selected. With variable step size for the cutoff frequency, an improvement in computational efficiency can be achieved. One of the advantages of this filter is the application of the Fourier transform to a local neighborhood, which increases overall computational efficiency over the computation of the Fourier transform of the entire image for each pixel as proposed in the adaptive Wiener filter implementations. The other advantage is the optimized removal of the Poisson noise component without additional filter parameters, which makes it suitable for unsupervised image processing. An example of the filter applied to a SPECT image of the brain is given in Figure 5.9.
FIGURE 5.9 Fourier block noise reduction filter to remove Poisson noise. Shown is a dopamine transporter imaging scan, that is, a SPECT scan with specific radiopharmaceuticals, in one slice of the brain (A). Since the radiopharmaceutical itself is the source of the events recorded to form an image, the signal is collected from random decay events. Therefore, a higher signal is always associated with a larger noise component (Poisson noise). Gaussian smoothing (B) blurs the image to an extent where details vanish and contrast becomes low. The Fourier block noise reduction filter (C), applied with 4 x 4 blocks on a 128 x 128 pixel image, smoothes noise while preserving details such as the ventricles.
