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The elegance of the complex representation stems from that the two components of the structure tensor can be obtained as averages and independently. In turn, this means that <math>\kappa_{20}</math> and <math>\kappa_{11}</math> can be used in a scale space representation to describe the evidence for presence of unique orientation and the evidence for the alternative hypothesis, the presence of multiple balanced orientations, without computing the eigenvectors and eigenvalues. A functional, such as squaring the complex numbers have to this date not been shown to exist for structure tensors with dimensions higher than two. In Bigun 91, it has been put forward with due argument that this is because complex numbers are commutative algebras whereas quaternions, the possible candidate to construct such a functional by, constitute a non-commutative algebra.<ref name=bigun91>
{{cite journal|author1=J. Bigun |author2=G. Granlund |author3=J. Wiklund |lastauthoramp=yes |title=Multidimensional Orientation Estimation with Applications to Texture Analysis and Optical Flow| journal=
The complex representation of the structure tensor is frequently used in fingerprint analysis to obtain direction maps containing certainties which in turn are used to enhance them, to find the locations of the global (cores and deltas) and local (minutia) singularities, as well as automatically evaluate the quality of the fingerprints.
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{{cite conference|author1=S. Arseneau |author2=J. Cooperstock |last-author-amp=yes |title=An Improved Representation of Junctions through Asymmetric Tensor Diffusion|booktitle=International Symposium on Visual Computing|date=November 2006}}
</ref> and several other image processing problems. The structure tensor can be also applied in [[geology]] to filter [[Seismology|seismic]] data.<ref>{{Cite journal|last=Yang|first=Shuai|last2=Chen|first2=Anqing|last3=Chen|first3=Hongde|date=2017-05-25|title=Seismic data filtering using non-local means algorithm based on structure tensor|journal=Open Geoscienes|volume=9|issue=1|pages=151–160|doi=10.1515/geo-2017-0013|issn=2391-5447|bibcode=2017OGeo....9...13Y}}</ref>
===Processing spatio-temporal video data with the structure tensor===
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