Structure tensor: Difference between revisions

Revision as of 13:06, 20 July 2018

Revision as of 13:09, 20 July 2018

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</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>