## Determining Motion Parameters from Spherical Flow Fields

(or the Advantages of Having Eyes in the Back of Your Head)

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
motion 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 motion field 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 motion field can be partitioned into 180 degree arcs
of clockwise and counterclockwise 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.
Because information is correlated over large distances, the method can
be considered a pattern recognition rather than a numerical algorithm.
The algorithm is shown to be robust and
relatively insensitive to noise and to missing data.
Both theoretical and empirical studies of the error sensitivity are
presented.
The theoretical analysis shows 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%.