A new method is proposed for determining the faint end of the luminosity function: LUMINOUS (LUminosityfunction Modelling of Image Noise to Observe Undetectable Stars). The basic principle is that if two images look "similar" enough, the luminosity functions are similar.
A simulated image is created, using information derived from the observed image. This includes noise characteristics and sensitivity of the detector, analog to digital converter (ADC) effects, PSF, diffuse sky brightness, detected stars, and the location of cosmic ray effects (CREs) and defective pixels. Iteratively modifying the number of detected stars for certain magnitude intervals, the simulated image is made to look more like the observed image. Any additional stars needed to improve the appearance of the simulated image are located at random.
To determine the similarity between the images, the histograms of pixel intensities in the images are used. The histogram has the advantage of independence of the location of the stars in the image. A pixel-to-pixel comparison of the simulated and real image would be sensitive to the location of the (undetectable) faint stars.
The histogram of the simulated image is compared with that of the original
image. The goodness of fit of the two histograms is expressed with the
parameter, defined as
where i0 is the first significant intensity bin in the histograms, N+1 is
the number of histogram bins over which is determined, and Ri and
Si are the counts in the 
bin of the two histograms.
This measure covers the interesting range of intensities, where
the contribution from bright stars is small, but the largest effect from
the undetectable and faint stars is located.
Statistically equivalent histograms should have a of unity.
Since the aim of LUMINOUS is to detect stars that are of the same intensity as fluctuations in the noise of the image, this noise should be modelled as accurately as possible. In the original raw image, all intensities are in the form of integer values. Subsequent image processing will change these intensities to real values, more closely representing the true intensity of the light falling on the CCD pixels. This processing includes e.g. bias subtraction and flat fielding.
With real values it is not obvious how to define a unique histogram. The properties of the noise are changed in a complex way. In the raw image the noise is easily defined as the quadratic sum of the read-out noise and the Poisson (shot) noise. As an example, consider a part of the image with a low value for the flat field. The noise in this part will be amplified during flat fielding of the image. If the flat field has a small systematic error, this will affect the intensities and give them a larger uncertainty. If, on the other hand, the true intensity of the image is modelled, and then multiplied with the flat field, the systematic error will affect the intensities as well. The Poisson noise in the intensities will add to the effect of this error, but since the value of the flat field was low, and the error small, the effect of the Poisson noise will dominate, and can easily be calculated.
For these reasons the raw image is modelled. Instead of correcting for effects of the detector, they are modelled. The method chosen here is to model the physical processes taking place. An LF dictates a certain distribution of stars projected on the detector. Spatial sensitivity variations of the detector are modelled through the flat field. The photons are converted to electrons, which are read out through electronics with a certain gain factor, linearity effects, dynamic range, read-out noise etc. Raw data is the final product. The following steps are taken when converting an input LF to a simulated histogram: add dark image, add known stars (from photometry), add unknown stars (from input LF), apply flat-field effects, add Poisson noise, add read-out noise, convert from electrons to analog to digital unit (ADU), add bias level, add bias image, apply ADC nonlinearity effects, multiply with CRE mask, create histogram.
Careful mapping of the noise characteristics of the individual pixels of the image during processing could be done as an alternative, and should give the same results in the end, but the effects of the noise on the histogram would complicate matters and make comparison of two images through the histogram impractical.
The following assumptions have been made for the method described here:
Each object present in the image is defined through only three parameters: two spatial coordinates and an intensity. This means that extended objects like galaxies are not modelled. Nor are CREs modelled.
LUMINOUS is designed to work in crowded stellar fields. In all reasonably practical cases (e.g. globular clusters, nearby galaxies), the stars will dominate. Resolved galaxies can be identified, and excluded from the histogram. Unresolved galaxies are treated as single stars, but since their number is low, they will not affect the LF.
Clustering of stars is assumed to be purely random. Brighter stars are detected by conventional means, and can be modelled with appropriate position and intensity.
For sufficiently small fields of view, the assumption is valid. For globular clusters, a larger field of view may require modelling of the distribution of stars, e.g. adopting the over-all intensity profile of the cluster.
Depending on e.g. the optics of the telescope, this assumption may not be true. Deviations from a constant PSF are usually largest at the edge of the field, so using a smaller part of the detector may reduce the problem.
This assumption is easy to check by inspecting the image.
Not all of these assumptions are critical. If any of the assumptions is invalid, the corresponding effect should be included when creating the simulated image.
A number of properties of the image should be known before LUMINOUS can be applied. These include:
For classical PSF fitting photometry the PSF has to be determined. If the PSF is defined to a sufficiently large radius it can be used as-is. If not, some form of model PSF could be used.
Since LUMINOUS works with raw image data, the sensitivity of the CCD should be included in the model.
The transformation from detected number of electrons to registered ADU is not completely linear. Non-linearity effects of the ADC are modelled.
Dark current and warm or hot pixels are modelled. The dark time is the time from the last clearing of the CCD to the time at read-out.
The signal of CREs can affect the histogram. The affected pixels can be excluded from the histogram of the image.
Much of this information is needed in standard image processing, and is usually known for the instrument used. The accuracy with which some of the effects are known is not always sufficient for LUMINOUS. For classical photometry the read-out noise is usually only used to determine the signal to noise ratio, and a low accuracy value suffices. For LUMINOUS it should be checked if the given value is correct and stable, since the statistical properties of the read-out noise are very important.
Except for the effects discussed, the CCD is considered perfect. This includes absence of effects like geometric distortion, deferred charge, charge transfer problems or trailing of the images, video noise, sub-pixel scale sensitivity variations, residual image effects, etc.