  Visual estimation of heading in the human brain is widely believed to be
based on instantaneous optic flow, the velocity of retinal image motion.
However, we previously found that humans are unable to use instantaneous optic
flow to accurately estimate heading and require time-varying optic flow
(Burlingham and Heeger, 2020). We proposed the hypothesis that heading
perception is computed from optic acceleration, the temporal derivative of
optic flow, based on the observation that heading is aligned perfectly with a
point on the retina with zero optic acceleration. However, this result was
derived for a specific scenario used in our experiments, when retinal heading
and rotational velocity are constant over time. We previously speculated that
as the change over time in heading or rotation increases, the bias of the
estimator would increase proportionally, based on the idea that our derived
case would approximate what happens in a small interval of time (when heading
and rotation are nearly constant). In this technical report, we characterize
the properties of the optic acceleration field and derive the bias of this
estimator for the more common case of a fixating observer, i.e., one that moves
while counter-rotating their eyes to stabilize an object on the fovea. For a
fixating observer tracking a point on a fronto-parallel plane, there are in
fact two singularities of optic acceleration: one that is always at fixation
(due to image stabilization) and a second whose bias scales inversely with
heading, inconsistent with human behavior. For movement parallel to a ground
plane, there is only one singularity of optic acceleration (at the fixation
point), which is uninformative about heading. We conclude that the singularity
of optic acceleration is not an accurate estimator of heading under natural
conditions.
