Maximum likelihood estimation in the context of an optical measurement

Abstract

The method of maximum likelihood estimation (MLE) is a widely used statistical approach for estimating the values of one or more unknown parameters of a probabilistic model based on observed data. In this tutorial, I briefly review the mathematical foundations of MLE, then reformulate the problem for the measurement of a spatially-varying optical intensity distribution. In this context, the detection of each individual photon is treated as a random event, the outcome being the photon's location. A typical measurement consists of many detected photons, which accumulate to form a spatial intensity profile. Here, I show a straightforward derivation for the likelihood function and Fisher information matrix (FIM) associated with a measurement of multiple photons incident on a detector comprised of a discrete array of pixels. An estimate for the parameter(s) of interest may then be obtained by maximizing the likelihood function, while the FIM determines the uncertainty of the estimate. To illustrate these concepts, several simple examples are presented for the one-and twoparameter cases, revealing many interesting properties of the MLE formalism, as well as some practical considerations for optical experiments. Throughout these examples, connections are also drawn to optical applications of quantum weak measurements, including off-null ellipsometry and scatterometry.

Birefringent Fourier filtering for single molecule coordinate and height super-resolution imaging with dithering and orientation

“…In order to estimate the sensitivity of CHIDO, we use Cramér–Rao (CR) lower bounds 36 , 37 on the uncertainties of the six parameters being measured ( x , y , z , ξ , θ , Ω). These bounds were deduced from a numerical calculation of the inverse of the Fisher Information matrix, in this case of dimension 6 × 6.…”

Geometric descriptions for the polarization of nonparaxial light: a tutorial

“…Techniques that allow characterizing simultaneously the position, orientation and wobble of multiple fluorophores are therefore a current thrust of research in microscopy [35][36][37][38][39][40][41][42][43][44][45]. The performance of a given technique is often evaluated in terms of the lower bounds for the accuracy in the estimation of these parameters in the presence of Poisson noise, according to what is known as a Cramer-Rao bound [151,152]. As mentioned in this article, the wobbling properties of a fluorophore translate into the polarization properties of the light it emits and, therefore, a meaningful measure of estimation must take into account the intrinsic geometric/topologic properties of the polarization matrix.…”

SNR enhancement in brillouin microspectroscopy using spectrum reconstruction

“…Finally we note that although both considered denoising algorithms produce better estimates of the Brillouin shift and linewidth, there remains further scope for improvement since neither achieve the CRLB, although the difference between the theoretical lower bound and the simulation results may be partially accounted for by pixelation effects mentioned above. Statistical methods such as the method of Maximum Likelihood Estimation (MLE) [55] can routinely achieve the CRLB but have yet to be applied to Brillouin spectroscopy. This remains an interesting area for future study.…”