# Differences

This shows you the differences between two versions of the page.

 definitions:absolute_frequency_units [2016/12/14 10:37]fmerino definitions:absolute_frequency_units [2018/06/20 13:12] (current) Both sides previous revision Previous revision 2016/12/14 10:46 fmerino 2016/12/14 10:37 fmerino 2016/12/12 17:07 fmerino [Definition] 2016/12/12 16:56 fmerino 2016/12/12 16:51 fmerino 2016/12/12 14:30 fmerino created Next revision Previous revision 2016/12/14 10:46 fmerino 2016/12/14 10:37 fmerino 2016/12/12 17:07 fmerino [Definition] 2016/12/12 16:56 fmerino 2016/12/12 16:51 fmerino 2016/12/12 14:30 fmerino created Line 1: Line 1: + ~~NOTOC~~ + ===== Absolute frequency ===== ===== Absolute frequency ===== Line 4: Line 6: 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. 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. $\frac{1}{\text{\AA}}$) and absolute frequencies $f_a$: + A simple relation exists between spatial frequencies $f_s$ (e.g. ${Å}^{-1}$) and absolute frequencies $f_a$: - For an image with pixel size $p$ by: + For an image with pixel size $p$:  f_s=\frac{f_a}{p} f_s=\frac{f_a}{p} Line 13: Line 15:  - For an n-pixels image sampled, the k'th Fourier pixel ($0<=k<=n/2$) is related to frequency by: + For an n-pixels image, the k'th Fourier pixel (with $0 \leq k \leq 0.5n$) is related to frequency by:  - f_s=k/n/p}}} + f_s=\frac{k}{n \times ​p} \\ \\ - ​. ​     {{{f_a=k/n}}} + f_a=\frac{k}{n}  - Resolution r [Å] (defined as inverse of spatial frequency) is: - ​. ​     {{{r=1/​f_s=p**n/​k`}}} + The resolution $r$ is defined as the inverse of the spatial frequency: + $$+ r=\frac{1}{f_s}=p\frac{n}{k} +$$ ==== Within the code ==== ==== Within the code ====