  We use the Millennium Galaxy Catalogue (MGC) to study the effect of compact
galaxies on the local field galaxy luminosity function. Here we observationally
define as `compact' galaxies that are too small to be reliably distinguished
from stars using a standard star-galaxy separation technique. In particular, we
estimate the fraction of galaxies that are misclassified as stars due to their
compactness. We have spectroscopically identified all objects to B_MGC = 20 mag
in a 1.14 deg^2 sub-region of the MGC, regardless of morphology. From these
data we develop a model of the high surface brightness incompleteness and
estimate that ~1 per cent of galaxies with B_MGC < 20 mag are misclassified as
stars, with an upper limit of 2.3 per cent at 95 per cent confidence. However,
since the missing galaxies are preferentially sub-L* their effect on the faint
end of the luminosity function is substantially amplified: we find that they
contribute ~6 per cent to the total luminosity function in the range -17 < M_B
< -14 mag, which raises the faint end slope alpha by 0.03^{+0.02}_{-0.01}.
Their contribution to the total B-band luminosity density is ~2 per cent.
Roughly half of the missing galaxies have already been recovered through
spectroscopy of morphologically stellar targets selected mainly by colour. We
find that the missing galaxies mostly consist of intrinsically small, blue,
star-forming, sub-L* objects. In combination with the recent results of Driver
et al. (2005) we have now demonstrated that the MGC is free from both high and
low surface brightness selection bias for giant galaxies (M_B <~ -17 mag).
Dwarf galaxies, on the other hand, are significantly affected by these
selection effects. To gain a complete view of the dwarf population will require
both deeper and higher resolution surveys.
