  Ghost imaging (GI) is an intriguing imaging technology which achieves the
object images through intensity correlation between reference patterns and
bucket signal. Here, we propose a probability model to explain the imaging
mechanism of this modality, by assuming that the reference patterns fulfill an
arbitrary identical distribution and that the objects are of gray-scale. We
have proven that the probability of the reconstructed pixel values in the pixel
region of the same original gray value obeys a Gaussian distribution, no matter
which functional form of the reference patterns is used in correlation
calculation. Both simulation and experiments have demonstrated that the
probability of recovered pixel values are highly consistent with their Gaussian
theoretical distribution, while their variance explains the appearance of
reconstruction noise. In addition, we have also extend this theory to other
classic correlation functions, e.g., normalized GI and differential GI. The
results have shown that there is a linear relationship between reconstruction
means in specified pixel regions and original gray values, which might provide
a unified explanation for GI with thermal light.
