So far in this book we have assumed that time-resolved measurements of the wave field are available. This assumption is natural for applications in seismology and in acoustics, where the sampling rate is larger than the operating frequencies. It is not the case in optics where only intensity measurements are available. This is because the measurements are local time averages of the square of the wave field. The goal of this chapter is to show through the analysis of a particular imaging problem that correlation-based imaging is feasible with intensity-only measurements, when the illumination is provided by noise sources.

In this chapter we analyze an imaging modality called ghost imaging that can produce an image of an object by correlating the intensities measured by two detectors: one that does not view the object and another that does view the object. In ghost imaging a high-resolution detector measures the intensity of a field generated by a partially coherent source which has not interacted with the object to be imaged. A bucket (or single-pixel) detector collects the total intensity of the field generated by the same source that has interacted with the object. The correlation of the intensity of the field measured at the high-resolution detector with the intensity measured by the bucket detector gives an image of the object.

In Section 11.2 we express the correlation of the measured intensities in terms of the Green's function and the source covariance function and we define the ghost imaging function. In Section 11.3 we analyze the resolution properties of the ghost imaging function. We emphasize the importance of the partial coherence of the source and study how scattering affects the resolution properties in the paraxial regime. The overall conclusion is that the image resolution improves as the source coherence decreases, and deteriorates as the scattering in the medium increases.

The ghost imaging setup

The experimental setup proposed in Valencia et al. (2005); Cheng (2009); Li et al. (2010); Shapiro and Boyd (2012) is shown in Figure 11.1. The waves are generated by a partially coherent source.