The performance of the proposed metrics is highly dependent on what
transformation 
that is used. In this case, we used a a spatial gray
level dependency (SGLD) matrix of the texture when testing the
-metric. The matrix is
formed with the position operator 'one pixel to the left'. For more
information on the forming of the SGLD matrix, see for instance [6].
When testing the 
-metric we used a somewhat more
sophisticated underlying metric. First two SGLD-matrices, 
and 
, were calculated for each texture
, one with the position operator 'one pixel to the left'
and one with the position operator 'one pixel up'. If the texture is
mainly horizontal, will be rather centered arround its diagonal,
while will not, and vice versa. A measurement of a matrix
'diagonality' is the moment
This was calculated for both and . 
yields low
values for matrices that are centered around their diagonals, e.g see
[7]. Hence, it is possible to calculate the main angle of the texture by
And the underlying metric is defined as
To be able to test the metrics, textures with known distances were
generated, see Image . The angle of the texture is known
since it is generated by a known Markov random field process. The
direction 
of the texture is so that 
. is measured in degrees. Given two textures with
direction 
and 
, the reference distance is calculated
according to:
Since a metric is a relative measurement it is not very interesting
to compare the numerical values of the distances between two textures
from different metrics. However, the relative order among
textures within a metric is the same regardless of the metric
used. That is, if 
in one metric,
for all other metrics.