Image Processing Reference
In-Depth Information
Fig. 14.9. The P2 texture patch composition consisting of real aerial images (
left
) and its
automatic segmentation using group direction tensors for 2, 4, and 6-folded symmetries.
14.7 Further Reading
Texture as a vision problem has been covered by a large body of publications in
psychology. Texture recognition in its various forms appears in many applications,
e.g., as diagnostic tools, multi-spectral image segmentation, and tracking in image se-
quences [78,82,233]. The prevailing view is that the second order statistics of images
is the only property that humans can discriminate in textures. The study in [178] sug-
gests certain Lie operators, which are differential operators, to define textures. While
the complex moments provide for optimal solutions to the line-fitting, cross-fitting,
etc., problems in the power spectrum, they correspond precisely to Lie operators that
can identify the direction(s) along which the image is translation invariant. As has
been discussed here, these complex moments can be computed by means of symme-
try derivatives applied to subbands of the original in the spatial domain. The scheme
was possible to implement via separable filtering thanks to direct tensor sampling
and the theoretical results studied in Sect. 11.9. However, the group direction tensor
can also be obtained by spectrum sampling in analogy with Sect. 10.13. The theo-
retical details of this approach to implement the group direction tensor are studied
in [26], whereas experimental results can be found in [25]. Even if a texture contains
symmetries of high order when no subband decomposition is applied, it can still be
discriminated against another texture by use of only structure tensor features applied
at the subbands level [134], e.g., the P3 patch shown in Fig. 16.8. By contrast, some
textures cannot be discriminated by applying the structure tensor to subbands, but
need the descriptive power of higher order symmetry features, e.g., Figs. 14.9 and
16.8. Accordingly, as the number of textures involved in the discrimination increases,
and/or the complexity of the individual textures increases, the structure tensor fea-
tures need to be completed. This can be done by the group direction tensors.
