Classical crowded field photometry detects and measures individual stars down to a limiting magnitude at which they can not be discerned from spurious noise peaks in the background of the image. Severe crowding limits detection of fainter stars measurable in an uncrowded field. The combined effect of the undetectable stars on the image is still measurable using statistical techniques.
Saha (1995) formulates a notion of detection significance (for
uncrowded point sources) independent of any special choice of
statistics for source detection. Completeness and spurious detections are
related and depend on the source brightness, the pixel size of the detector and
the point spread function (PSF). Also for uncrowded sources,
Zepka et al. (1994) describe a method using the histogram of the data and
the statistical properties of the background. With their method, it is
possible to detect a fraction of the objects that are well below five
, but without detecting large numbers of spurious noise peaks.
The stellar luminosity function (LF) is defined as the number of stars per cubic parsec within an absolute magnitude interval. The initial mass function and star formation history of a sample of stars are reflected in the LF, which in itself is, given the distance to the stars, an easily accessible observational quantity. For the determination of the LF of an observed field of stars the actual detection of the stars is not strictly necessary, since no individual positions and intensities for the stars are needed. Only the number of stars as a function of magnitude is needed. The incompleteness of detection can partly be eliminated through estimation of the fraction of detected stars per magnitude interval. When this fraction becomes small, the uncertainty in the estimated LF gets large, and when the fraction is zero, the limit of the detection has been reached.
Figure 1 (click here) shows an extreme case, where the stars are so faint that none can be detected without detecting a large number of spurious noise peaks as well. It is not possible to say anything about the LF using the method described above. However, if one compares this noisy image with an image containing only noise, the images will not be statistically equivalent, as if drawn from the same statistical ensemble. All deviations in intensity, due to the undetectable stars, will be positive. If the number of stars is large, the net effect of all these small contributions can be significant, causing an increase in the apparent background level and a shape change (especially an increase in the width) of the noise distribution of the apparent background level (Fig. 2 (click here)).
Figure 1: Left: image showing pure Gaussian noise. Right:
image showing pure Gaussian noise plus numerous faint stars.
The noise has a 1 amplitude of 0.8 analog to digital units (ADU).
The original image of 800 
800 pixels contains 10000
equal magnitude stars, the
sub-image shown here contains around 80 stars. The full width at half
maximum of the PSF for the
stars is 1.3 pixels, the maximum amplitude for a single
star is 0.8 ADU
Figure 2: Pixel intensity histograms of the images shown in Fig. 1 (click here). A
100 ADU bias level has been added to the images
In classical crowded field photometry this net effect will contribute to the overall background in the image. Stetson (1987) suggests that the sky background could be calculated from the observed LF and PSF. Here, the opposite is proposed: to derive the faint part of the LF using the properties of the noise and background in the image.
Comparing the appearance of the observed image with a simulated image with known LF, the LF can be adjusted to present statistically equivalent images, thus deriving an estimate for the LF. With this procedure, it is possible to derive the LF for stars that are up to two magnitudes fainter than those that can be detected as individual stars.