Circularly symmetric lowpass filters can be designed by assuming an idealized frequency response of the form
is minimum, where W(f) is weighting function.
Figure shows the desired and the fitted spectrum obtained using a 21 term zero-phase filter. The filter in the space domain is a symmetrical filter as shown in Figure . The filter coefficients correspond to hn in Figure . The McClellan transformation filter is designed using the 5x5 filter shown in the Appendix. This filter corresponds to the G filter shown in Figure . Once the filter coefficients hn are computed they are incorporated into the recursive structure shown in Figure . The implementation is done in both space and wavenumber domains. The wavenumber domain response of the filter is shown in Figure . A spike in x-y space was used to test the space-domain implementation. The output of the lowpass filter is a smoothed spike as shown in Figure
Figure 4 Twenty-one coefficient symmetric 1-D filter in the space domain corresponding to Fig 3. These are the filter coefficients hn used in the Chebyshev filter structure shown in Figure 2.
Figure 6 Output of Lowpass Filter in the x-y domain. The filter implementation is in the x-y domain. The modified McClellan filter in space-domain is used to do the transformation.