Star position reconstruction
Summary
The star position reconstruction on the r
axis (see here), and consequently x
and y
, is not accurate.
The current method takes the average of pixels r
positions weighted by the image intensity (~observed PSF). However, considering that the probability distribution (PDF) along r
is asymmetric, the average is actually distinct from the distribution center.
For the Laplace-asymmetric distribution we can calculate the average position (dark mode broken for the image):
Which does not correspond to the center (0
in this example).
What is the expected correct behavior?
From the calculation of the weighted average corresponding to the calculation above, the distribution center can therefore be obtained by shifting the result by the following offset:
-Sr * (1/K2 - K2) / (K + 1/K)
with K2 := K*K
.
Note: I did not derive any estimate for the error introduced by using the average as reconstructed value as it depends on non-trivial parameters. The error is probably not very high but deserved to be characterized.