Information about the distribution of point sources on the sky can be derived in a number of ways. The methods can be divided in those that detect individual stars and those that use a statistical description of the distribution of sources, without the need to detect individual sources. The magnitude limit for individual detections is given by Saha (1995), and is dependent on the allowed rate of spurious detections. Zepka et al. (1994) use information from the statistics of the noise to reduce the rate of spurious detections. Analysis of the probability density function of the observed intensities is in use in radio (Condon & Dressel 1978) and X-ray (Barcons 1992) astronomy. The intensity distribution of sources can be derived to source densities of about one source per beam (Scheuer 1974). LUMINOUS falls in the same category as probability density analysis.
Detection of individual stars is an accurate method for determining the LF, as long as the stars can easily be detected and measured. When the completeness of detection becomes too small, probability density analysis yields better results.
Solutions obtained with LUMINOUS should be checked for uniqueness. Both
in generating the image from the LF and in generating the histogram
from the image, information can potentially get lost or distorted.
The correlation between the number of stars of a certain magnitude and the
intensity of the background determines the limiting magnitude.
As the correlation coefficient approaches -1, it becomes impossible to
discern between adding a number of stars or increasing the diffuse
background. Likewise, the correlation coefficient between two adjacent
magnitude bins in the LF can approach -1, which makes them
indiscernible.
As seen from the derived LF in Fig. 4 (click here), there were some large
fluctuations in the faint part of the LF. Smoothing effectively removed
the fluctuations, leaving the LF close to the input one. The effect of
smoothing on 
was negligible, indicating that the unsmoothed and
smoothed solutions were equivalent. The derived solution was not unique.
But since it is unlikely that the fluctuations are physical, the smoothed
LF is preferred.
All stars in the simulated image have the same shape, the PSF. Errors in the PSF affect the derived LF. When the PSF is measured directly from the image, there may be several faint, undetected stars present within the radius of measurement. The shape of the analytical part of the PSF may contain errors (e.g. the wings are too bright or faint). In practical cases, a perfect PSF cannot be obtained. Noise in the PSF can affect the noise in the background and change the appearance of the histogram. Care should therefore be taken when extracting the PSF from an image.
The simulations for which the PSF was extracted from the image (Fig. 5 (click here)) show that the choice of a large PSF radius was not the best. The smaller PSF gave a better derived LF, despite the fact that part of the information in the wings was discarded.
In classical CCD photometry it is customary to correct for sensitivity
variations of the detector with a flat fielding procedure. This is of
significant importance for the accuracy of the photometry. LUMINOUS
does not operate on the image, but on the histogram of the image.
To achieve best results, one must correct for
variations in the histogram bin width. An ADC
does the actual binning of electron counts into discrete ADU values. However,
this conversion
is not perfect: both non-linearity effects and variable bin width affect
the appearance of the histogram. The effect on classical photometry is
usually negligible. When trying to adjust the histogram of a simulated image to
that of an observed image, accurate modelling of the ADC effects
is of similar importance as the flat field is for photometry.
The simulation with the ADC errors (Fig. 6 (click here)) showed that large systematic errors
in the derived LF can arise, even for relatively bright stars. In the
first applications of LUMINOUS to the WFPC2 image, the
fit to the histogram of the image was not good in some bins.
This resulted in both a large and large systematic
errors in the
derived LF. It is of importance to correctly incorporate the ADC effects,
specifically for those intensity bins that contain the majority of the pixels
in the image. Suitable calibration images for characterisation of the effect
could e.g. have a flat histogram of pixel intensities, such as a linear
intensity ramp in the image.
As can be seen from the
uncertainties in the numbers of stars in the simulation (Fig. 8 (click here)), the faintest part of the
derived luminosity function had large uncertainties.
The uncertainty in the derived LF grew approximately exponentially with magnitude,
with an exponent larger than the exponent of the input LF. The actual values of
the uncertainties increased with the degree of crowding.
The uncertainties were determined from the spread of the derived LF parameters for
solutions having a lower than a fixed value.
The solutions were checked against Scheuer's (1974) limit of one source per beam. The
area of the beam was calculated as 
pixels, FWHM being the full
width at half maximum of the PSF. The size of the image was 
pixels.
With
the PSF, ADC effects, flat field, read-out noise, bias level, gain and location
and intensities of the detectable stars perfectly known, the derived LF was
composed of approximately 0.4 sources per beam for FWHM = 1.5, and about four
sources per beam for FWHM =11. The order of magnitude of detected source density
is correct, both for the case of background and read-out noise limited detection
(FWHM =1.5) and confusion limited detection (FWHM =11). The ratio of the number
of directly detectable stars versus derived number of stars was in both cases
about 1:10.
If the detected LF was truncated at the magnitude where completeness of detection dropped below 0.5, and the derived LF at the magnitude where the uncertainty in the LF exceeded the star count, LUMINOUS could measure the LF about 2 magnitudes deeper.
Systematic errors could limit this range somewhat. Errors in the bias level and read-out noise affected mainly the counts for the faintest 1.5 magnitudes of the LF. The error in the read-out noise was of importance, indicating that an accurate value for the read-out noise is desirable. However, knowledge of the ADC effects was of much greater importance. Errors in the simulation of the ADC effects caused systematic errors for the counts of stars that were bright enough to be individually detected.
The sampling of the intensities in the image had no significant effect on the accuracy. It was possible to retrieve an LF with either an increasing or decreasing slope. The shape of the LF affected the accuracy through the limit of one source per beam, causing a somewhat larger uncertainty for a steeper LF. The crowding of the image, changed by increasing the FWHM of the PSF, caused a larger uncertainty for higher crowding, as could be expected from the source density limit. However, the number of sources per beam increased to four, indicating that LUMINOUS effectively derived the LF, despite the fact that a lower limiting magnitude was reached.
In the LMC Bar exposure with the Wide Field Camera of WFPC2 it was possible
to reach about 1.5 magnitudes deeper than direct detection and obtain
reproducible results, but with systematic errors. The fit was not
optimal, judged by the value of 1.2. The cause of this error is
most likely a combination of effects, some of which have been discussed
above. Since the LFs derived with the measured and modelled PSF are so
similar, and considering the simulations with the PSF errors, it is unlikely
that the errors in the PSF are dominating the systematic errors. A
likely candidate for these systematic errors is the ADC correction. The
simulations have shown that large systematic errors can arise due to
ADC correction errors. Because of the coarse intensity sampling, only a
few histogram bins are effectively used to derive information about the
LF. A modelling error affecting one or two of these bins will have large
impact on the LF.
Another possible error source is the assumption about the shape of the
distribution of the read-out noise: was sensitive to the width
of the assumed Gaussian distribution. If the actual distribution was not
Gaussian (bias jumps can widen the distribution), this would affect the
fit and the derived LF. A better intensity
sampling would allow for a better determination of the read-out noise
characteristics.
The deviations between the derived LF from the single exposure and the detected LF from the averaged image, together with the less than optimal fit to the histogram, stress the importance of calibration of the effects that contribute to the shape of the histogram. Accurate calibrations of the ADC effects and shape of the read-out noise would allow for more accurate modelling, reducing the systematic errors due to these effects. High resolution sampling of the intensities should facilitate the determination of these effects.
A point where LUMINOUS can be improved is the measure of similarity used. The histogram of the image was chosen because it contains no information about the exact location of the stars. The information is not available from the image, and not necessary for derivation of the LF. Comparison between the observed and simulated image must be done with some measure that is insensitive to the location of the stars. The texture of the image could be such a measure of similarity.
A histogram of pixel intensities of the image loses all information about
neighbouring pixels. Instead of taking a one dimensional histogram, a second
dimension can be added, describing the surroundings of the pixel in question.
If a simple average intensity of the eight surrounding pixels is taken as a
second dimension, a new sort of histogram is obtained. For a well sampled
image, the intensities of the pixels of stars (
) are correlated
with the intensities of their surrounding pixels (
). This
means that the major part of the pixels will end up near the diagonal of
the histogram, where 
. CREs, hot pixels,
image defects, etc will generally have little correlation with their
surroundings, and do not adhere to the diagonal. Using the
two dimensional histogram of the image as a measure of similarity may
improve LUMINOUS. An extra advantage of the two dimensional
histogram is that changes
in the width of the PSF are reflected in the degree of correlation, or the
spread around the diagonal.
The derived LF fluctuated with large gradients around the input LF. An additional constraint for the derived LF could be one of smoothness, using a maximum entropy method to adjust the number of stars in the LF bins.
The simulated image should be as similar as possible to the observed image. A choice must be made if an effect is important enough to be put in the simulation, and if it is known well enough to be modelled instead of assuming some generic description. A more careful description of the various aspects responsible for the appearance of the image could prevent systematic errors in the derived LF.
The problem of finding the best LF to describe the image is one of
minimisation. Like with any such problem, it is essential to use a good
algorithm to find the minimum in space.
A better algorithm may be able to find a better minimum, possibly with a
better LF.