Information in a photon: Relating entropy and maximum-likelihood range estimation using single-photon counting detectors

Range estimation at low light-levels is accomplished using pulsed illumination of the target and time-of-flight measurement of backscattered light using single-photon detectors. Photon arrival statistics for this problem are time-inhomogeneous Poisson point processes where the rate function is determined by the illumination waveform. Given the flexibility to choose from different illumination waveforms, an important design question is - how does the range estimation performance depend on the pulse shape? The maximum-likelihood (ML) range estimation problem is nonlinear and thus it is difficult to analytically compare the estimation performance from different illumination waveforms. In this paper, we present an information-theoretic framework for evaluating ML range estimation performance. We derive relationships between the entropy of the photon arrival observations and the Cramér-Rao lower bound (CRLB) on the range estimate by extending De Brujin´s identity and isoperimetric properties for non-Gaussian distributions.