Probability Theory In Conditional-Averaging Ghost Imaging With Thermal Light

In ghost imaging with thermal light (GITL), the bucket signals are analog signals playing the role of weights in the algorithms of image reconstruction. However, diverse bucket signals seemed to be completely ignored in conditional-averaging GITL, which was recently proposed by two groups [Appl. Phys. Lett. 100, 131114 (2012); Chin. Phys. Lett. 29, 074216 (2012)]. Even so, positive and negative ghost images were obtained by averaging just partial reference signals. To understand this effect, one needs to determine the statistical relation between random bucket signals and reference signals. Here, we apply probability theory to GITL by regarding thermal light intensities as stochastic variables, and then deduce the joint probability density function between the bucket and reference signals. Positive and negative images are formed in ensemble averaging of the product of the reference signal and two logic quantities of the bucket signals. The visibility and contrast-to-noise ratio of these images in conditional-averaging GITL are analyzed in detail in both theory and experiment. As for applications, we perform an experiment of remote conditional-averaging GITL and verify that the image quality is maintained when the bucket signals are extremely attenuated. Our method can be applied to any ghost imaging scenarios in which the specific statistics of the optical source is given.