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--- definitions:absolute_frequency_units [2016/12/12 14:30]
fmerino created
+++ definitions:absolute_frequency_units [2016/12/14 10:46]
fmerino
@@ Line 1: / Line 1: @@
+~~NOTOC~~
 ===== Absolute frequency =====
- . {{{`f_a=0`}}} corresponds to the zero Fourier term (zero frequency), while {{{`f_a=0.5`}}}  corresponds to Nyquist frequency {{{`f_N`}}}.
+==== Definitions ====
- . '''''Nyquist frequency''''' is the maximum spatial frequency in an image scanned with a pixel size p [Å] and in pixels {{{`f_N=1/2`}}}, while in Å {{{`f_N=1/(2p) [1/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.
-For the pixel size p [Å], spatial frequency {{{`f_s`}}} [1/Å] is related to absolute frequency {{{`f_a`}}} by:
- .      {{{`f_s=f_a/p`}}}
+A simple relation exists between spatial frequencies $f_s$ (e.g. ${Å}^{-1}$) and absolute frequencies $f_a$:
- .      {{{`f_a=f_s**p`}}}
+For an image with pixel size $p$:
+$$
+f_s=\frac{f_a}{p}
+\\
+f_a=f_s \times p
+$$
-For an image size n pixels sampled at p Å, 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`}}}
+$$
+f_s=\frac{k}{n \times p}
+\\
+f_a=\frac{k}{n}
+$$
- .      {{{`f_a=k/n`}}}
+The resolution $r$ is defined as the inverse of the spatial frequency:
-Resolution r [Å] (defined as inverse of spatial frequency) is:
+$$
+r=\frac{1}{f_s}=p\frac{n}{k}
+$$
- .      {{{`r=1/f_s=p**n/k`}}}
+==== Within the code ====
 For examples of code in which Fourier pixels are handled check sparx/filter.cpp.

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