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--- definitions:absolute_frequency_units [2016/12/12 16:51]
fmerino
+++ definitions:absolute_frequency_units [2016/12/12 17:07]
fmerino [Definition]
@@ Line 1: / Line 1: @@
 ===== Absolute frequency =====
-==== Definition ====
+==== Definitions ====
-Inside SPHIRE, we define the spatial frequencies are handled in absolute units. This means that the spatial frequencies are expressed in term of pixels units, or absolute frequency $f_a$, instead of units of inverse distance. In this way $f_a=0$ corresponds to the zero Fourier term (zero frequency), while $f_a=0.5$  corresponds to Nyquist frequency $f_N$.
+Inside SPHIRE, we define the spatial frequencies are handled in absolute units. This means that the spatial frequencies are expressed in term of pixels units, or absolute frequency $f_a$, instead of units of inverse distance. In this way $f_a=0$ corresponds to the zero Fourier term (zero frequency), while $f_a=0.5$  corresponds to Nyquist frequency $f_N$. The Nyquist frequency corresponds the maximum spatial frequency contained in an image. In absolute frequencies this always has a value of 0.5 while in inverse distance it will correspond to $f_N=\frac{1}{2p}$, where $p$ is the pixel size usually in Å.
- . '''''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]`}}}.
+A simple relation exists between spatial frequencies $f_s$ and absolute frequencies $f_a$:
-For the pixel size p [Å], spatial frequency {{{`f_s`}}} [1/Å] is related to absolute frequency {{{`f_a`}}} by:
+For an image with pixel size $p$ [Å] by:
+$$
+f_s=\frac{f_a}{p}
+\\
+f_a=f_s \times p
+$$
- .      {{{`f_s=f_a/p`}}}
+For an n-pixels image sampled, the k'th Fourier pixel ($0<=k<=n/2$) is related to frequency by:
- .      {{{`f_a=f_s**p`}}}
-For an image size n pixels sampled at p Å, k'th Fourier pixel (0<=k<=n/2) is related to frequency by:
- .      {{{`f_s=k/n/p`}}}
+$$
+f_s=k/n/p`}}}
+\\
  .      {{{`f_a=k/n`}}}
+$$
 Resolution r [Å] (defined as inverse of spatial frequency) is:

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