## Determining Motion Parameters from Spherical Flow Fields

Randal C. Nelson and John Aloimonos

Department of Computer Science

University of Rochester, University of Maryland

Rochester, NY 14627

**Abstract:**
A theory is developed for determining the motion of an observer given the
flow field over a full 360 degree image sphere.
The method is based on the fact that for an observer translating without
rotation, the projected circular flow about any equator can be divided
into disjoint semicircles of clockwise and counterclockwise flow,
and on the observation that the effects of rotation decouple
around the three equators defining the three principal axes of rotation.
Since the effect of rotation is geometrical,
the three rotational parameters can be determined independently by
searching, in each case, for a rotational value for which the derotated
equatorial flow field can be partitioned into 180 degree arcs
of opposite flow.
The direction of translation is also obtained from this analysis.
This search is two dimensional in the motion parameters,
and can be performed relatively efficiently.
The algorithm is shown to be robust and
relatively insensitive to noise and to missing data.
Results of theoretical analysis show that for white noise of bounded
magnitude M, the expected error is at worst linearly proportional to M.
Empirical tests demonstrate negligible error for perturbations of up to 20%
in the input, and errors of less than 20% for perturbations of up to 200%.