Low Pass and High Pass Filtration

For a noncrystalline object, which we are exclusively dealing with, the signal and noise components of the Fourier transform are superimposed and normally inseparable. However, the two components frequently have different behaviors as a function of spatial frequency radius R: while the signal component falls off at the resolution limit (see section 4), the noise component has significant contributions beyond that limit. Hence, multiplication of the Fourier transform of such an image with a cutoff function:

0 elsewhere will eliminate part of the noise, thus enhancing the signal-to-noise ratio. Application of such a function, or variants of it having a smooth transition instead of a sharp cutoff, is termed low-pass filtration, as it passes only Fourier components with low spatial frequencies.

Functions with smooth radial transition are usually preferred, since application of equation (A1.18) would cause an artificial enhancement of image features whose size corresponds to the spatial frequencies at the cutoff (e.g., see Frank et al., 1985). The most ''gentle'' function used for Fourier filtration is one with a Gaussian profile (e.g., Frank et al., 1981b):

Its falloff behavior is controlled by the parameter R0: at a spatial frequency radius of R = Vu2 + v2 = R0, the filter function reduces the Fourier amplitude to 1/e of its original value.

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