The image parameter determination needs to know the relevant PSF or LSF. A pre-requisite to determine the image shape is knowing the source colour.
First, quick and simple image parameters are determined in AF using a Tukey’s bi-weight method. The resulting positions
and fluxes serve two purposes. They are used as starting points for the final
image parameter determination, and they are also used to propagate the image
location from the AF field to the BP and RP field in order to obtain reliable
colours. This process is explained in more detail in Section 3.3 of Fabricius et al. (2016).
Note, however, that for Gaia DR2, the colour dependence of the image shapes has not
The final image parameters, viz. transit time, flux, and for 2D windows also
the AC position, were determined with a maximum-likelihood method described
in Section 2.4.8. For converting the fluxes from digital
units to , gain factors determined before launch were used.
The resulting parameters are stored as intermediate data for later use in
the astrometric and photometric core processes.
A general Maximum-Likelihood algorithm for CCD modelling
The general principle for Maximum-Likelihood (ML) fitting of arbitrary models
in the presence of Poissonian noise is simple and can be
formulated in a general framework which is independent of the precise
model. In this way, it is possible to use the same fitting procedure
for 1D and 2D profile fitting to CCD sample data, as well as for more complex
fitting (e.g., for estimating the parameters of the LSF model). Here, we outline
the basic model for this framework.
Model of sample data
The basic input for the estimation procedure consists of data and
a parametrised model. The estimation procedure adjusts the model
parameters until the predicted data agrees as well as possible with observed
data. At the same time, it provides an estimate of the covariance matrix
of the estimated parameters and a measure of the goodness-of-fit. The ML
criterion is used for the fit, which in principle requires that the probability
distribution of the data is known as a function of the model parameters. In practice,
a simplified noise model is used, which is believed to be accurate enough, which leads to simple and efficient algorithms.
Let be the sample data, the model
parameters, and the sample values predicted by the
model for given parameters. Thus, if the model is correct and
are the true model parameters, we have for each :
Using a noise model, we have in addition:
where is the standard deviation of the readout noise. More precisely,
the adopted continuous probability density function (pdf) for the random variable
is given by:
valid for any real value .
It is assumed that , , and are all expressed in electrons
per sample (not in arbitrary ADU units, voltages, or similar). In particular,
is the sample value after correction for bias and gain, but including
dark signal and background. The readout noise is assumed to be known;
it is never one of the parameters to be estimated by the methods described
in this document.
The functions are in principle defined by the various
source, attitude, and calibration models, including the LSF, PSF, and CTI
models. The set of parameters included in the vector varies
depending on the application. For example, in the 1D image centroiding
algorithm, may consist of just two parameters representing
the intensity and location of the image; in the LSF calibration process,
will contain the parameters (e.g., spline coefficients)
defining the LSF for a particular class of stars; and so on. The intensity
model is left completely open here; the only
thing we need to know about it is the number of free parameters,
Given a set of sample data , the ML estimation of the parameter
vector is done by maximizing the likelihood function:
where is the pdf of the sample value from the adopted
noise model (Equation 2.30). Mathematically equivalent,
but more convenient in practice, is to maximize the log-likelihood function:
Using the modified Poissonian model, Equation 2.30, we have:
where the additive constant absorbs all terms that do not depend on
. (Remember that is never one of the free model
parameters.) The maximum of Equation 2.33 is obtained by solving the
simultaneous likelihood equations