Image Processing Reference
In-Depth Information
14
Group Direction and
N
-Folded Symmetry
Direction of a line or a set of lines displaced in a parallel fashion were discussed
in Chap. 10. Such images, characterized by linearly symmetric isocurves, contain a
single direction that is the common direction of the parallel lines. The linearly sym-
metric image model was helpful to explain many local image phenomena, including
edges, lines, simple textures, and a rich variety of corners. In its various extensions,
the single direction model could even be useful for describing motion, simple texture,
and even geometry. Here, we wish to go one step further and increase the number of
directions that can be allowed to be contained in an image. We do this mainly because
the linear model, which assumes a single direction in some coordinate system, is not
sufficiently powerful to explain certain image phenomena. Texture patterns consist-
ing of repetitive line constallations having multiple directions, even in the local scale,
constitute an important example.
14.1 Group Direction of Repeating Line Patterns
We describe the concept of
group direction
by illustrating it through line patterns.
Two linearly symmetric images that differ with
π
4
or
5
π
4
in their directions are shown
in the first column of Fig. 14.1. The relative rotation between the two images can be
obtained from the relative directions between the constituent line directions. Using
the double-angle representation, the direction of the lines can be made unique,
θ
=2
ϕ
(14.1)
where
ϕ
is the normal of a line direction. Accordingly,
θ
for the images on the top
and on the bottom would be
θ
=
π
and
θ
=
3
π
2
, respectively. The angle
θ
of the
content can thus serve as the direction of the image itself, provided that it is known
that the image is linearly symmetric. The relative rotation between the two images
becomes
Δθ
=
θ
− θ
=
π
2
π
4
or
5
π
4
.
In the second column of Fig. 14.1, the rotated versions of an image containing
two sets of parallel lines are shown. The line sets differ directionwise maximally
, which needs to be divided by 2 to yield
Δϕ
as
