ALMA Optical Pointing System Peak Finding Algorithm

TIP Last Update: JeffMangum - 11 June 2009

The peak finding algorithm used for all optical pointing measurements is as follows:

X_c = \frac{\sum_i x_i I_i}{\sum_i I_i}

...where x_i and I_i are the spatial coordinates and intensity distribution, respectively, of the star image. This algorithm appears to the the "standard" for peak finding applications in industry (c.f. Ares and Arines (2004), Alexander and Ng (1991)). The theoretical accuracy of this algorithm is given by (Cao and Yu (1994); note that a typo in the original Cao and Yu formulation of the following equation is corrected in the form listed below):

\sigma^2_{X_c} = \frac{\sigma^2_\eta L}{\langle I_T \rangle^2}\left(\frac{L^2 - 1}{12}\right)

  • \sigma_\eta is the noise in the image (in pixels)
  • \langle I_T \rangle is the mean of the total intensity of the star
  • L is the length of the "detector window" (subimage in our case) in pixels

Note that this noise estimate is likely to be on the conservative side, as the effects of thresholding (which we do) have not been accounted for. See the references listed above for details. I was not able to figure out how to apply a thresholding correction to the equation above, so left that out for now.

In the following I post three typical nighttime star images of a 6.5 magnitude star acquired with pOPT2 on DV03 using a 5 second integration time:
  • Raw star image (before dark frame subtraction)
  • Dark frame image
  • Star image after dark frame subtraction

Image 5s 20090609 091826.png

Dark 05s 20090609 092246.png

ImageDarkSub 5s 20090609 091826.png

The mean intensity of the star image is about 1000, while the RMS of the star image (excluding the region around the star) is 9.8, which is a mean signal-to-noise of about 100, which is typical for the ALMA pOPT measurements. Noting that
  • The peak finding algorithm is applied to a sub-image which is 19x19 pixels in size, so L = 19
  • \frac{\sigma_\eta}{\langle I_T \rangle} = \frac{1}{100}
...then using the equation for \sigma^2_{X_c} above yields:

\sigma^2_{X_c} = \frac{1}{10000}\left(\frac{6840}{12}\right) = 0.057~pixels^2

The measured plate scale for pOPT2 on DV03 is 0.88 arcsec/mm, which implies that:

\sigma_{X_c} = \sqrt{0.057}*0.88 = 0.21~arcsec

Now, using a parallel argument to that applied to the noise contribution due to seeing during offset pointing measurements, this noise contribution actually hits our offset pointing measurement twice (once for the collimation determination, then again with the residual measurement relative to the derived collimation). Therefore, the total noise contribution which needs to be subtracted from our offset pointing measurements due to peak finding algorithm uncertainty is:

2\sigma^2_{X_c} = 2*\left(0.21\right)^2 arcsec^2

-- JeffMangum - 2009-06-10
Topic revision: r4 - 2009-06-12, JeffMangum
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