Maximum-likelihood estimation of parameterized wavefronts from multifocal data

Abstract

A method for determining the pupil phase distribution of an optical system is demonstrated. Coefficients in a wavefront expansion were estimated using likelihood methods, where the data consisted of multiple irradiance patterns near focus. Proof-of-principle results were obtained in both simulation and experiment. Large-aberration wavefronts were handled in the numerical study. Experimentally, we discuss the handling of nuisance parameters. Fisher information matrices, Cramér-Rao bounds, and likelihood surfaces are examined. ML estimates were obtained by simulated annealing to deal with numerous local extrema in the likelihood function. Rapid processing techniques were employed to reduce the computational time.

Phase‐retrieved pupil function and coherent transfer function in confocal microscopy

“…The accompanying error bars depict the maximum and minimum of the retrieved coefficients from all realizations. It should be noted that due to the nonlinear dependence of the retrieved phase on the input intensity, a bias can be introduced into the retrieved coefficients in the presence of noise (Sakamoto & Barrett, 2012), which can also be seen in Figure 3. Similar Monte Carlo simulations were also performed for SNRs ranging from 10 −1 to 10 5 .…”

“…In other scenarios, such as the calibration of preexisting industrial set-ups, avoiding the need of additional optics and system modification is preferable. Here, use of phase retrieval algorithms (Gureyev et al, 1995;Almoro et al, 2006;Maiden & Rodenburg, 2009;Sakamoto & Barrett, 2012), which use wave propagation laws to extract phase information from intensity only images, present a further noninterferometric technique. In the context of optical imaging, phase retrieval algorithms have successfully been applied in wide-field and fluorescence microscopy for retrieval of the complex pupil function and APSF (Hanser et al, 2004).…”

Three-Dimensional Single-Molecule Localization Microscopy in Whole-Cell and Tissue Specimens

“…Additionally, GS-based PR algorithms do not take into account the photon statistics of the recorded PSFs; therefore, this leads to inaccurate results when retrieving pupil functions from PSFs with low signal-to-noise ratios. Maximum likelihood estimator (MLE)-based phase retrieval algorithms (101,102) were developed to address these difficulties. MLE phase retrieval methods estimate a set of Zernike polynomial amplitudes by maximizing a likelihood function that represents the similarity between the PR-PSFs and the experimental PSFs.…”

Synthetic phase-shifting for optical testing: Point-diffraction interferometry without null optics or phase shifters

“…The advantages of the statistical approach in imaging are well understood [6]. ML estimation in wavefront sensing has been rigorously discussed by Barrett et al [7] and ML estimation of parameterized wavefronts from multifocal data has been developed using simulated annealing, a global optimization algorithm [8]. Despite its advantages, the limitation of search algorithms for ML estimation in a high-dimensional parameter space, such as ZF coefficient space, is the computational time.…”

“…A fast conjugate-gradient method can be applied if the likelihood is monotonic even in high-dimensional parameter space. However, if the likelihood surface has many local minima and strong coupling between parameters, time-consuming global optimization is often the only choice [8,9]. To overcome this challenge, the innovative iterative search algorithm, called the synthetic phase-shifting (SPS) algorithm is developed.…”