  This article aims at developing a model based optimization for reduction of
temporal unwrapping and field estimation errors in multi-echo acquisition of
Gradient Echo sequence. Using the assumption that the phase is linear along the
temporal dimension, the field estimation is performed by application of unity
rank approximation to the Hankel matrix formed using the complex exponential of
the channel combined phase at each echo time. For the purpose of maintaining
consistency with the observed complex data, the linear phase evolution model is
formulated as an optimization problem with a cost function that involves a
fidelity term and a unity rank prior, implemented using alternating
minimization. Itoh s algorithm applied to the multi-echo phase estimated from
this linear phase evolution model is able to reduce the unwrapping errors as
compared to the unwrapping when directly applied to the measured phase.
Secondly, the improved accuracy of the frequency fit in comparison to
estimation using weighted least-square regression and penalized maximum
likelihood is demonstrated using numerical simulation of field perturbation due
to magnetic susceptibility effect. It is shown that the field can be estimated
with 80 percent reduction in mean absolute error in comparison to wLSR and 66
percent reduction with respect to penalized maximum likelihood. The improvement
in performance becomes more pronounced with increasing strengths of field
gradient magnitudes and echo spacing.
