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CD "D

Frequency (cm-1)

Figure 9. Plot of filter amplitude versus frequency for Metz filters with orders 1 (equal to the CDRF), infinity (equal to the CDRF1), 4, 10, and 20. In all cases a Gaussian CDRF with a FWHM of 1.5 cm was used.

matched filter; as n the Metz filter becomes the inverse filter. Thus, for low n, the low pass term in square brackets dominates, while for higher orders the filter behaves more like an inverse filter. This is illustrated in Figure 9. King et a/.10'11 have applied the Metz filter to nuclear medicine images with a count-dependent, non-integral order.

An alternative filter that has been applied in the context of SPECT imaging is the Wiener filter.12'13 The Wiener filter is the filter that minimizes the mean squared error between the restored and true projections. It has the form:

CDRF(n)2

where N( v) is the noise power spectrum and \O(v)\ is the object power spectrum. Again, the term in square brackets is a low pass filter, in this case depending on both the noise power spectrum and the object power spectrum (which, paradoxically, is what we are trying to recover). Note that for frequencies where the noise power spectrum is much smaller than the object power spectrum, Eq. 12 becomes the inverse filter. However, at frequencies where the noise dominates, the Wiener filter approaches zero. One of the difficulties with the Wiener filter is that it requires estimates of the noise and object power spectra. One solution to this problem is to assume that noise is stationary. In this case, the noise power spectrum for the projections, assuming Poisson noise statistics, is flat with a value equal to the mean number of counts. For reconstructed images, approximate expressions for the noise power spectrum can be used.14'15 The object power spectrum can be approximated in the region where the noise power is high by extrapolating a fit of the image power spectrum minus the approximate noise power spectrum.12

One important issue in applying either a Metz or Wiener filter is the selection of the CDRF. Since the true CDRF is spatially varying, an approximation must be made. Among several approaches are using the CDRF for a distance from the face of the collimator equal to the center of rotation or half the distance from the center of rotation. King et a/.10,11 have fit a parameterized CDRF to minimize the mean square error of the measured images. Finally, the spatial variance of the CDRF can be improved by taking the geometric mean of opposing views.

One limitation of these restoration filters is that, since they are spatially invariant, they will tend to overcompensate in some areas and undercom-pensate in others. Also, they do not improve the asymmetry of the reconstructed point response (except perhaps by degrading it).

The second class of analytic methods explicitly includes the distance dependence of the CDRF. The basic idea is to write expressions for the projections of a function that include the CDRF and attempt to invert them, i.e. to solve for the object in terms of the projections. Since attenuation has an important effect on the measured projections, the attenuation distribution is assumed to be zero or uniform in the development of the expressions for the projections; analytic CDRF compensation methods for nonuniform attenuators have not yet been developed. To date, a closed form exact solution has not been found. However, several approximate methods have been derived.

An important basis for a number of these methods is the frequency-distance relation (FDR).16 This approximation relates the position of a point source to the FT of the sinogram of the source. Note that since the angular coordinate in the sinogram is periodic, a Fourier series transform can be used in this direction. The FDR predicts that the energy in this Fourier transform will be concentrated along a line in Fourier space through the origin and with a slope linearly related to the distance from the point source to the center of rotation (COR). For a distributed source, this implies that the activity from all sources at a specific distance from the COR lies along a single line in Fourier space. For a circular orbit, the activity at a fixed distance from the COR is also at a fixed distance from the collimator face. Thus, one could perform inverse filtering along these lines using the appropriate CDRF. However, it should be noted that the derivation of the FDR requires invoking an approximation, the stationary phase approximation. Thus, methods based on the FDR will necessarily be approximate.

Lewitt et a/.1718 were the first to use the FDR to compensate for the spatially varying CDRF. Glick et a/.19 exploited the FDR in combination with restoration filtering using the Wiener filter. This method has the advantage that it explicitly includes noise control. It was shown to provide improved symmetry in the reconstructed point spread function. However, one limitation of all these methods is that they tend to result in correlated noise with a texture that appears to have unfavourable properties compared to the noise correlations introduced by iterative methods.20

A second method for deriving analytical filters is to assume a form for both the shape and distance dependence of the CDRF that allows analytic inversion. Appledorn21 has analytically inverted the attenuated Radon transform equation assuming a CDRF shaped like a Cauchy function:

where w is width parameter. It should be noted that the Cauchy function having the same FWHM as a Gaussian will have longer tails and be somewhat more sharply peaked in the center of the response function. Soares et al.22 have implemented this method and evaluated it with and without combined analytic attenuation compensation. While the method did reduce the spatial variance of the response, it seemed to do so largely by degrading the resolution in the tangential direction.

A second class of methods approximates the CDRF as a Gaussian, not an unreasonable approximation as demonstrated above. However, in order to invert the equations, an approximate form for the distance dependence is assumed. van Elmbt et al.23 used the approximation that the square of the width of the CDRF is proportional to the distance from the center-of-rotation. The result is a modification of Bellini's method24 for uniform attenuators that includes approximate compensation for the spatially variant CDRF.

Pan et al.25 have developed a method that improves on this approximation, assuming that the change in width of the CDRF over the object was less than the width at the center of the object. This assumption leads to an expression for the CDRF of the form:

where s is the width parameter for the Gaussian CDRF, D, is the distance to the face of the collimator, r is the distance from the collimator face to the center-of-rotation, s0 is the width of the CDRF at the center of rotation, and s1 is the slope of the assumed linear relationship between D and the width of the Gaussian CDRF. This method has been evaluated by Wang et al}6'21 It was found that, even for noise-free data, the assumptions about the shape and distance-dependence of the Gaussian FWHM resulted in high-frequency artefacts unless a low-pass filter was applied. In the presence of noise, low pass filtering was even more essential. Pan's method25 resulted in the greatest improvement in resolution near the center of rotation, where the approximation in Eq. 14 is most accurate. The radial resolution improved but the tangential resolution degraded with distance away from the center of rotation. These effects are likely due to the approximate nature of the model for the detector response.

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