Millisecond exoplanet imaging: II. regression equations and technical discussion

The leading difficulty in achieving the contrast necessary to directly image exoplanets and associated structures (e.g., protoplanetary disks) at wavelengths ranging from the visible to the infrared is quasi-static speckles (QSSs). QSSs are hard to distinguish from planets at the necessary level of precision to achieve high contrast. QSSs are the result of hardware aberrations that are not compensated for by the adaptive optics (AO) system; these aberrations are called non-common path aberrations (NCPAs). In 2013, Frazin showed how simultaneous millisecond telemetry from the wavefront sensor (WFS) and a science camera behind a stellar coronagraph can be used as input into a regression scheme that simultaneously and self-consistently estimates NCPAs and the sought-after image of the planetary system (exoplanet image). When run in a closed-loop configuration, the WFS measures the corrected wavefront, called the AO residual (AOR) wavefront. The physical principle underlying the regression method is rather simple: when an image is formed at the science camera, the AOR modules both the speckles arising from NCPAs as well as the planetary image. Therefore, the AOR can be used as a probe to estimate NCPA and the exoplanet image via regression techniques. The regression approach is made more difficult by the fact that the AOR is not exactly known since it can be estimated only from the WFS telemetry. The simulations in the Part I paper provide results on the joint regression on NCPAs and the exoplanet image from three different methods, called ideal, naïve, and bias-corrected estimators. The ideal estimator is not physically realizable (it is useful as a benchmark for simulation studies), but the other two are. The ideal estimator uses true AOR values (available in simulation studies), but it treats the noise in focal plane images via standard linearized regression. Naïve regression uses the same regression equations as the ideal estimator, except that it substitutes the estimated values of the AOR for true AOR values in the regression formulas, which can result in problematic biases (however, Part I provides an example in which the naïve estimate makes a useful estimate of NCPAs). The bias-corrected estimator treats the errors in AOR estimates, but it requires the probability distribution that governs the errors in AOR estimates. This paper provides the regression equations for ideal, naïve, and bias-corrected estimators, as well as a supporting technical discussion.