  One of the major difficulties limiting ground-based direct imaging of
exoplanets with adaptive optics is quasi-static speckles in the science camera
(SC) that obscure the planetary image. These speckles are caused by
aberrations, called non-common path aberrations (NCPA), that are not corrected
in the adaptive optics loop, and all attempts to subtract them in
post-processing have been problematic. The method of Frazin (2013) (F13) uses
simultaneous millisecond telemetry from wavefront sensor (WFS) and the SC to
estimate the both the NCPA and the exoplanet image in a self-consistent manner.
Rodack et al. (2018) proposed correcting for the NCPA in real-time while on-sky
using the F13 estimation method, and called this procedure the "Real-Time
Frazin Algorithm." The original regression model underlying the F13 method did
not account for uncertainty in the WFS measurements, and this cannot be done
with standard statistical methodology since these uncertainties manifest
themselves in the independent variables (i.e., they cannot be treated as
another source of noise in the SC data). Further, simulations show that simply
using the noisy wavefront measurements without accounting for their
uncertainties leads to estimates of the NCPA with unacceptably large bias.
Here, the source of this bias is explained in terms of an "errors in variables"
statistical model. Then, the method of F13 is generalized to account for WFS
measurement error using a new sequential estimation technique that treats the
nonlinear coupling between NCPA, WFS measurements and the error covariance of
the WFS measurements. This new technique keeps a running estimate of the NCPA,
the exoplanet image and their joint covariance matrix. The sequential
implementation of the method should make it computationally efficient enough to
be suitable for on-sky correction of the NCPA as well as off-line analysis.
