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The leading difficulty in achieving the contrast necessary to directly imageexoplanets and associated structures (eg. protoplanetary disks) at wavelengthsranging from the visible to the infrared are quasi-static speckles, and theyare hard to distinguish from planets at the necessary level of precision. Thesource of the quasi-static speckles is hardware aberrations that are notcompensated by the adaptive optics system. These aberrations are callednon-common path aberrations (NCPA). In 2013, Frazin showed how, in principle,simultaneous millisecond (ms) telemetry from the wavefront sensor (WFS) and thescience camera behind a stellar coronagraph can be used as input into aregression scheme that simultaneously and self-consistently estimates the NCPAand the sought-after image of the planetary system (the exoplanet image). Thephysical principle underlying the regression method is rather simple: thewavefronts, which are measured by the WFS, modulate the speckles caused by theNCPA and therefore can be used as probes of the optical system. The mostimportant departure from realism in the author's 2013 article was theassumption that the WFS made error-free measurements. The simulations in Part Iprovide results on the joint regression on the NCPA and the exoplanet imagefrom three different methods, called the ideal, the naive, and thebias-corrected estimators. The ideal estimator is not physically realizable butis a useful as a benchmark for simulation studies, but the other two are, atleast in principle. This article provides the regression equations for allthree of these estimators as well as a supporting technical discussion.Briefly, the naive estimator simply uses the noisy WFS measurements without anyattempt to account for the errors, and the bias-corrected estimator usesstatistical knowledge of the wavefronts to treat errors in the WFSmeasurements.

Millisecond Exoplanet Imaging, II: Regression Equations and Technical Discussion