# (p.327) Appendix 2 Profiles, Point-Spread Functions, and Effects of Commonly Used Low-Pass Filters

# (p.327) Appendix 2 Profiles, Point-Spread Functions, and Effects of Commonly Used Low-Pass Filters

The effect of three commonly used filter functions is demonstrated in the following, using the image of Lena (figure A2.1) as an example. In each case, the following information is provided for three choices of a characteristic parameter: (i) filter function, in the range of Nyquist frequencies 0… 0.5; (ii) point-spread
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function (PSF); and (iii) the resulting images. Among the artifacts produced by application of the filter, the *ringing* artifact is the most noticeable.

# 1. Gaussian Low-Pass Filter

Gaussian low-pass filters (LPFs) are two-dimensional (2D) Gaussian functions in the Fourier domain. One-dimensional profiles are shown below, at several different values for the filter radius. The radius specifies the 2-σ level of the Gaussian. The corresponding PSFs are in real (or image) space. The wider the PSF, the more blurring. It should be noted that, in contrast with the Fermi and Butterworth filters below, even the smallest radius leads to no ringing artifacts (figure A2.2).

# 2. Fermi Low-Pass Filter

The Fermi LPF is defined as 1/(1 +exp[(F–radius)/*T*]), which negotiates between “top hat” and Gaussian characteristics, depending on the value of the temperature factor, *T*. The temperature defines the reciprocal distance (in terms of frequency units), within which the filter falls off (*T* = 0 gives a sharp cutoff). The radius in all three filters below has been set to 0.15. The sharp cutoff (*T* = 0) causes “ringing” artifacts in the filtered image. The PSFs of all three filters are similar, but the filters that decay more slowly reduce the undulations in the PSF (figure A2.3).

# 3. Butterworth Low-Pass Filter

The Butterworth LPF can be roughly thought of as a smoothed trapezoid. The corners of the trapezoid are controlled by the passband and the stopband. Frequencies lower than the passband are unaltered, frequencies beyond the stopband are set to zero, while those in between these limits are attenuated.

The first filter (passband = stopband = 0.15) is nearly a sharp cutoff filter (top hat), causing “ringing” artifacts in the filtered image. This “ringing” is also seen as undulations in the outer lobes of the PSF.

The third filter (passband = 0.02, stopband = 0.75) passes more high frequencies, so that details are sharper. However, attenuating the low frequencies results in loss of contrast in some areas (e.g., the face). Note that, although the stopband can be specified beyond 0.5 and still determines the filter shape, the filter is cut off at the 0.5 (Nyquist) frequency (figure A2.4).