Probability theory of intensity correlation in ghost imaging with thermal light

To well reveal the imaging mechanism of ghost imaging (GI) with thermal light, we propose a probability theory using more general assumptions than existing theoretical explanations that the reference patterns obey an arbitrary identical distribution and the objects are of grayscale. It is not limited to a specific correlation function, even multi-functional variants of reference patterns can be used in correlation calculation. We have proven in both theory and experiments that the reconstruction means are linear transformations of original object's gray values and the probabilities of recovered pixel values in the pixel region of the same original gray value follow a Gaussian distribution, whose variance explains the appearance of reconstruction noise. We also derive the relationship between pattern variance and contrast-to-noise ratio under noisy conditions, which will be instructive for better pattern construction. This work will deepen the understanding of GI and enrich intensity correlation forms.