## Noise Filtering

The reproducible resolution determined by using one of the methods described in section 3.2 of this chapter is a guide for the decision what part of the Fourier transform contains significant, signal-related information to be kept, and what part contains largely noise to be discarded. Since some signal contributions are found even beyond the nominal resolution boundary, low-pass filtration (or any nonlinear noise filtering techniques) should normally not be applied until the very end of a project, when a final 2D average or 3D reconstruction has been obtained. The reason is that low-pass filtration at any intermediate step would remove the opportunity to utilize the more spurious parts of the signal, and nonlinear filtering would prohibit the use of any linear Fourier-based processing afterwards.

Figure 3.25 Demonstration of low-pass filtration and other means of noise reduction. (a) Test image (256 x 256); (b) test image with additive ''white'' noise (SNR = 0.8); (c) Gaussian filter; (d) median filter (5 x 5 stencil); (e) adaptive Wiener filter; (f) edge enhancing diffusion; (g) coherence enhancing diffusion; (h) hybrid diffusion. From Frangakis and Hegerl (2001), reproduced with permission of Elsevier.

Figure 3.25 Demonstration of low-pass filtration and other means of noise reduction. (a) Test image (256 x 256); (b) test image with additive ''white'' noise (SNR = 0.8); (c) Gaussian filter; (d) median filter (5 x 5 stencil); (e) adaptive Wiener filter; (f) edge enhancing diffusion; (g) coherence enhancing diffusion; (h) hybrid diffusion. From Frangakis and Hegerl (2001), reproduced with permission of Elsevier.

In the following, the profiles of three linear filters for low-pass filtration are given. Other, nonlinear filters are discussed by Frangakis and Hegerl (2001). A demonstration of both types of filters, reproduced from that work, is given in figure 3.25.

Low-pass filters must be designed such that transitions in Fourier space are smooth, to avoid ''ringing'' or "overshoot" artifacts (also known as Heaviside phenomenon) that appear in real space along edges of objects. In those terms, a filter with a top-head profile would have the worst performance. A Gaussian filter, defined as

is ideal in this regard, but it has the disadvantage of attenuating, as a function of spatial frequency, too soon while coming to a complete blocking too late relative to the desired spatial frequency radius.

Two other Fourier filters are more frequently used as they offer a choice of width for the transition zone: the Fermi filter (Frank et al., 1985) and the Butterworth filter (Gonzales and Woods, 1993).

The Fermi filter follows the distribution of particles following the Fermi statistics at a given temperature; hence, the width of the transition zone is specified by the ''temperature'' parameter T: