In addition to local curvature, there are many kinds of symmetries that can be detected in an orientation field. One class of symmetries which lends itself to a particularly elegant implementation are the class of rotational symmetries, for which the orientation field is polar separable with an angular component given by
where k is an integer. Some examples of such fields are shown in Figure 5 for positive and negative values of k. (In these figures, note that the angle is halved for visualisation purposes.)
Figure 5: Examples of orientation maps with rotational symmetry for
different values of 
.
If we regard 
(when evaluated over a finite
area) as an orientation template, then we can think of
as a matched filter for various types of
rotational symmetry. When k=2, exhibits
circular symmetry, which, depending on the value of 
, can
take on an appearance which is either radially pointing outward,
uniformly circular, or pinwheel shaped (left or right handed). When
k=1, takes on an approximately
parabolic shape. In this case, varying has the fortuitous
effect of globally rotating the pattern formed by the collection
of vectors which comprise . To emphasize, by simply
applying a constant phase factor 
locally to each
vector in the parabolic symmetry field, a global rotation of the
pattern is achieved. Thus, should a parabolic symmetry exist anywhere
and at any angle in a given orientation field, a single correlation
with the conjugate of 
will suffice to detect it.
Specifically, the magnitude of the correlation indicates the
certainty of the detected symmetry while the phase indicates the
rotation angle.
As discussed in [GK95], one drawback of this simple approach is that an operator which is designed to detect a certain kind of symmetry will also have a nonzero response to other kinds of symmetries. For example, the operators for k=0 and k=1 exhibit a large degree of overlap onto one another. This problem can be dealt with by means of a projection-subtraction method described in [GK95] and [HGK82] which the authors have termed a consistency operation. Please refer to the cited references for details on this method.