Absolute frequency


Inside SPHIRE, spatial frequencies are handled in absolute units. This means that the spatial frequencies are expressed in term of pixels, or absolute frequency $f_a$, instead of units of inverse distance. The Nyquist frequency, the maximum spatial frequency contained in an image, corresponds to $f_a=0.5$. In inverse distance units, it will correspond to $f_N=\frac{1}{2p}$, where $p$ is the pixel size, usually in Å.

A simple relation exists between spatial frequencies $f_s$ (e.g. ${Å}^{-1}$) and absolute frequencies $f_a$:

For an image with pixel size $p$: $$ f_s=\frac{f_a}{p} \\ f_a=f_s \times p $$

For an n-pixels image, the k'th Fourier pixel (with $0 \leq k \leq 0.5n$) is related to frequency by:

$$ f_s=\frac{k}{n \times p} \\ f_a=\frac{k}{n} $$

The resolution $r$ is defined as the inverse of the spatial frequency:

$$ r=\frac{1}{f_s}=p\frac{n}{k} $$

Within the code

For examples of code in which Fourier pixels are handled check sparx/filter.cpp.