Self-shadowing and cast shadows are a particular difficulty in measuring the membership likelihoods, however we have found the following approach sufficient to compensate for shadowing. First, we observe that if a pixel is significantly brighter (has a larger Y component) than the class statistics say it should, then we do not need to consider the possibility of shadowing. It is only in the case that the pixel is darker that there is a potential problem.
When the pixel is darker than the class statistics indicate, we
therefore normalize the chrominance information by the brightness,
This normalization removes the effect of changes in the overall
amount of illumination. For the common illuminants found in an
office environment this step has been found to produce a stable
chrominance measure despite shadowing.
The log likelihood computation then becomes
where 
is 
for the image pixel at
location (x,y), 
is the mean
of class k and 
is the corresponding
covariance.