Group Direction and N-Folded Symmetry - Vision with Direction - page 306

Fig. 14.3. The gradient directions ( left ) in the local images of FMTEST and the corresponding
group directions ( right ). The two gradient direction types that can occur in local images, e.g.,
the colored region , are drawn at the same point, to avoid clutter

image. This is illustrated in Fig. 14.3, where the gradient directions and the group
directions are overlayed on the original pattern. The gradient and the group direc-
tion vectors are drawn for neighborhoods at the same distance from the center and
represent the various types of directions that can occur in the respective neighbor-
hoods. Whereas the gradient directions are periodic with 2 π , the group directions are
periodic with the period π . Per neighborhood, there are two gradient vector types,
differing in their directions with π , whereas there is a unique group direction. Be-
cause the partial derivatives D x and D y are linear, the normal vector at a point, the
gradient, is the vectorial summation of the constituent normal vectors. Consequently
there are four and six types of gradient vector directions that can locally occur in
FMTEST2 and FMTEST3, respectively, but the local group directions are unique for
all local images in both test charts. We illustrate this only for the latter, in Fig. 14.4,
where we note that the group directions are unique and periodic with the period

π
3

,
whereas there are six gradient direction types that can occur in a neighborhood, e.g.,
the green region. As one walks around a circle, the local images rotate continuously,
and the group directions reflect this change by rotating continuously.

14.3 Group Direction Tensor by Complex Moments

In Chapter 10 we discussed a technique that allows detection of patterns having one
direction, to be precise, linearly symmetric images, while it could quantitate the di-
rection. Studying the FT of images that are linearly symmetric reveals that the power
spectrum is concentrated to a line passing through the spectral origin. Here we will
be interested in the same except that the number of directions that are jointly present