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Quantum Information and Measurement (QIM)

Biased Estimate For Superresolving Quantum Imaging

Alexander B. Mikhalychev, Ilya Karuseichyk, Anton Sakovich, Piotr Novik, Dmitri S. Mogilevtsev

Published Jan 1, 2019• 1 citation
Publisher

Abstract

We apply Fisher information analysis to the problem of superresolving quantum imaging and investigate the effect of imposing constraints on the reconstructed object. An approach for resolution quantification for unconstrained and constrained problems is proposed.
DOI
10.1364/qim.2019.t5a.62

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Advanced superresolution imaging techniques require specific approaches for accurate and consistent estimation of the achievable spatial resolution. Fisher information supplied to the Cram\'er-Rao bound (CRB) has proved to be a powerful and efficient tool for resolution analysis and optical setups optimization. However, the standard CRB is not applicable to constrained problems violating the unbiasedness condition, while such models are frequently encountered in quantum imaging of complex objects. Complementary to the existing approaches based on modifying CRB, we propose a practical algorithm for approximate construction of a modified Fisher information matrix, which takes the constraints into account and enables accurate estimation of errors for constrained problems by the standard CRB. We demonstrate the efficiency of the proposed technique by applying it to one-, two-, and multiparameter model problems in quantum imaging. The approach provides quantitative explanation of previous results with successful experimental reconstruction of objects with the spatial scale smaller than the theoretical limit predicted by the standard CRB.
While standard Fisher Information (FI) and Cramer-Rao bound (CRB) provide essential and easily obtained information about the estimating parameter sensitivity and error bounds, they fail to address the constraints. As attempts to address the constraints by modifying the CRB proved cumbersome, a practical algorithm to modify the CRB was proposed in our previous study. Here, we further investigate the effect of a imaging system's parameters on the standard and modified CRB obtained by this algorithm in the single-parameter model not limited to biphoton systems.
Quantum imaging emerged quite recently as a breakthrough technology of overcoming the diffraction limit in microscopy and enhancement of optical resolution without the necessity to use hard radiation or perform scanning in the near field [1] , [2] . Both, the quantum imaging itself and the more “classical” techniques inspired by it (for example, super-resolution optical fluctuations imaging – SOFI [3] ), rely on detection and analysis of photon (intensity) correlations. Typically, it is believed that the more correlated illuminating light is used and the higher order of the correlations is measured, the larger super-resolution can be achieved. That conclusion is based on efficient narrowing of the point-spread function, which, however, does not necessarily imply better resolution as the ability to reconstruct smaller features of the investigated object successfully.
Quantum imaging generally leads to nonlinear estimation problems intractable for a larger number of problem parameters. We connect the features of the Fisher information with the parametric locality of the problem and build the efficient iterative scheme reconstructing only a subset of the whole set of parameters in each step. We demonstrate super-resolving reconstruction of grey transmission objects with the experimental data obtained by imaging with entangled twin-photons and pseudo-thermal sources. We predict and experimentally confirm the existence of optimal photon correlations providing for the best image resolution. We also show how an estimation bias may drastically improve the resolution.