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where <math>E</math> is the identity matrix in 2D because the two eigenvectors are always orthogonal (and sum to unity). The first term in the last expression of the decomposition, <math>(\lambda_1 -\lambda_2)e_1e_1^\text{T}</math>, represents the linear symmetry component of the structure tensor containing all directional information (as a rank-1 matrix), whereas the second term represents the balanced body component of the tensor, which lacks any directional information (containing an identity matrix <math>E</math>). To know how much directional information there is in <math>I</math> is then the same as checking how large <math>\lambda_1-\lambda_2 </math>is compared to <math>\lambda_2</math>.
Evidently, <math>\kappa_{20}</math> is the complex equivalent of the first term in the tensor decomposition, whereas <math display="block">(|\kappa_{20}|-
:<math>
\begin{array}{c}
\kappa_{20} =(\lambda_1-\lambda_2)\exp(i2\phi)&=&w*(h*I)^2\\
\kappa_{11} =\lambda_1+\lambda_2&=&w*|h*I|^2\\
\end{array}
</math>
where <math>h(x,y)=(x+iy)\exp(-(x^2+y^2)/(2\sigma^2)) </math> is the (complex) gradient filter, and <math>*</math> is convolution, constitute a complex representation of the 2D Structure Tensor. As discussed here and elsewhere <math>w</math> defines the local image which is usually a Gaussian (with a certain variance defining the outer scale), and <math>\sigma </math> is the (inner scale) parameter determining the effective frequency (determining) range in which the orientation <math>2\phi</math> is to be estimated.
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=IEEE-PAMI|volume=13|number=8|pages=775--790|year=1991 |doi=10.1109/34.85668}} </ref>
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.
==The 3D structure tensor==
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