Cramer–Rao bounds for intensity interferometry measurements

Abstract

The question of signal-to-noise ratio (SNR) in intensity interferometry has been revisited in recent years, as researchers have realized that various innovations can offer significant improvements in SNR. These innovations include improved signal processing. Two such innovations, the use of positivity and the use of knowledge of the general shape of the object, have been proposed. This paper investigates the potential gains offered by these two approaches using Cramer-Rao lower bounds (CRLBs). The CRLB on the variance of the positivity-constrained maximum likelihood (ML) estimate is at best 1/4 of the variance of the unconstrained estimator. This is compared to the positivity-constrained ML estimator, which delivers a best-case variance reduction of only (1-1/π)/2=34.1%. The gains offered by prior knowledge depend on the quality of such information, as might be expected from optimal weighting of such data with the measured data. Furthermore, biases are induced by the application of constraints, and these biases can eliminate some or all of the advantage of lower variances, as found when considering the total root-mean-square error. A form of CRLB for variance is presented that properly incorporates prior information.

Cramér-Rao bounds for variance of Fourier magnitude measurements

“…If the estimator T is equal to x between 0 and θ max , and constrained to 0 for negative values and to θ max for values greater than θ max , then this quantity ∂ θ Ψ Τ (θ) can be computed for a Gaussian distribution. Assuming a Gaussian distribution of mean θ, so that θ is the CRB parameter one obtains [7]:…”

“…The expression for the associated "prior-averaged" CRB, averaged over a prior uniform distribution of θ, is given by [7]:…”

“…These bounds have been applied to prior information about the range of the estimate of |O(k)| 2 , where k is the spatial frequency, and |O(k)| 2 is the corresponding square of the Fourier magnitude from intensity interferometer measurements [7]. Expressions were developed based on the prior knowledge that |O(k)| 2 is greater than zero, and less than |O(k=0)| 2 .…”

“…A well-known tool for answering such questions is the CRB, and extensions [21][22]. The biased CRB is applied to analyze the impact of constraints, and a new extension is used to address the impact of prior information on the variance bound [7]. No other extension has been found in the literature which addresses this issue of bounds that includes random prior information on the variance of biased estimators.…”

Stellar intensity interferometry over kilometer baselines: laboratory simulation of observations with the Cherenkov Telescope Array

“…Methods specifically for intensity interferometry were worked out by Holmes et al [32][33][34] for one and two dimensions, respectively. Once a sufficient coverage of the Fourier plane is available, phase recovery and imaging indeed become possible.…”