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Pixel sensitivity
The basic assumption for this analysis is that the camera has an
unintensified CCDdetector giving a linear response to both photon
intensity and total number of photons. From this assumption it
follows that the detected intensity depends linearly on both the total
number of photons and the intensity. Thus, provided that no
nonlinearities are added (Brändström private communication), the
output from the pixel with image coordinates is:

(5.18) 
where is the intensity in pixel ,
is the sensitivity of the pixel,
is the number of
photons that hit the pixel, are the zero level, and
the dark current of the pixel.
To calibrate the camera the first step is to make a large series of
images with zero time of exposure to get as good an estimate as
possible of . The second step is to take a large number of
images without any light to estimate the dark current
;
this has been done with sufficient accuracy for the ALIS cameras.
When the zero level and dark current is determined it is possible
to rewrite equation (5.18):

(5.19) 
where
.
The third step is to determine the pixel sensitivity matrix .
This can be done by taking set of images with different exposure
times, which by simple algebra gives the estimate of the
sensitivity matrix

(5.20) 
where
is the number of photons per second that hit
the pixel . The number of photons that hit the pixel has some
random variation due to the Poisson characteristics of the emitting
source. In addition to the Poisson noise the measured images
and
contain the internal noise of
the camera. Even with an ideal detector, without internal noise, there
will be a stochastic variance which is at least as large as the square
root of the measured signal
. From equation (5.20) it follows that the
variance of the sensitivity
estimate is

(5.21) 
If the requirement on the accuracy of the sensitivity estimate
is a relative error of , it is perfectly straightforward
to derive the necessary condition on the total measured intensity
needed:

(5.22) 
This infers that if the accuracy requirements on the sensitivity is
the needed total intensities in the pixel,
and is
counts. With images with
counts it is necessary to take the average of some 50
images to get the necessary accuracy. For ALIS, however, there is as yet
no estimate of the sensitivity matrix with the required
accuracy. However, there exists relative sensitivity estimates that
give the relative average sensitivity of the different cameras.
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Up: The Forward model 
Previous: Point spread function
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copyright Björn Gustavsson 20001024
