Group Direction and N-Folded Symmetry - Vision with Direction - page 309

where k n
min

,k n
max

are complex representations of the extremal group directions,

θ min ,θ max ,

I n, 0
I n
2

k n
min

k n
max

k n
min

= =exp( iθ min ) , and = − . (14.8)

n
2

The quantities I n
2

,I n, 0 represent a tensor because they measure a physical prop-
erty, unbiased by the observation coordinate system. For odd n , the complex mo-
ments I n, 0 vanish when f is real. This is because the power spectrum of real images
is even. Accordingly, to be useful, n must be even when f is real.

, n
2

14.4 Group Direction and the Power Spectrum

In the theorem discussed above, F was the spectrum of an image without being spe-
cific about whether it is local or global, as it holds for both interpretations. However,
the case when f is a local image is more interesting for texture analysis applications,
therefore we assume this henceforth. The local image can be readily obtained by
multiplying the original image with a window concentrated to a small support. Ac-
cordingly, |F | 2 will be the local power spectrum. This yields a useful interpretation

in texture analysis, because the local power spectrum measurements, e.g., I n, 0 , will
then reflect the properties of the local texture, which should not change as long as
one moves the window within the same texture.
The view supported by psychology experiments suggests that [131] the gray
value of a local image in comparison with that of another point in the same neigh-
borhood is more significant than the absolute gray values for human texture discrim-
ination. This is sometimes called the second-order statistics because the gray image
correlations of two points are measured. Some studies limit the definition of texture to
images consisting of regions having the same second-order statistics, which depends
on the distance and the direction between two points [81, 96]. However, more gener-
ally a texture can be defined as an image consisting of translation-invariant local im-
age properties. The local power spectrum, which is translation-invariant, is therefore
rich in texture features. Within homogeneous regions, local phase information which
allows us to discriminate between textures consisting of differently shaped lines and
edges is neglected, e.g. [57]. In other words, only the directional information of the
geometrical structures are taken into account, irrespective of whether, for example,
the component lines are caused by crests or valleys. Therefore, when there is a tex-
ture with one distinct orientation (linear or 2-folded symmetry) around an inspected
point, the power spectrum will be concentrated to a line. When there is a texture with
two mutually orthogonal directions (rectangular or 4-folded symmetry) the power
spectrum will be concentrated to a cross. It is similar for hexagonal/triangular struc-
tures (6-folded symmetry) and octagonal ones (8-folded symmetry). The arguments
of the complex moments give the orientation of the estimated n -folded symmetry,
whereas the magnitudes give measures of the estimation quality, that is, certainties.