# Differences

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definitions:absolute_frequency_units [2016/12/14 10:37] fmerino |
definitions:absolute_frequency_units [2018/06/20 13:12] (current) |
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===== Absolute frequency ===== | ===== Absolute frequency ===== | ||

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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} | ||

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$$ | $$ | ||

- | 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 ==== |