Structure tensor: Difference between revisions

Revision as of 02:12, 10 May 2019

Revision as of 18:52, 3 June 2019

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</math>

where <math>i=\sqrt{-1}</math> and <math>\phi</math> is the direction angle of the most significant eigenvector of the structure tensor <math>\phi=\angle{e_1}</math> whereas <math>\lambda_1</math> and <math>\lambda_2</math> are the most and the least significant eigenvalues. From, this it follows that <math>\kappa_{20} </math> contains both a certainty <math>|\kappa_{20}|=\lambda_1-\lambda_2</math> and the optimal direction in double angle representation since it is a complex number consisting of two real numbers. It follows also that if the gradient is represented as a complex number, and is remapped by squaring (i.e. the argument angleangles of the complex gradient is doubled), then averaging acts as an optimizer in the mapped ___domain, since it directly delivers both the optimal direction (in double angle representation) and the associated certainty. The complex number represents thus how much linear structure (linear symmetry) there is in image <math>I</math>, and the complex number is obtained directly by averaging the gradient in its (complex) double angle representation without computing the eigenvalues and the eigenvectors explicitly.

Likewise the following second order complex moment of the power spectrum of <math>I</math>, which happens to be always real because <math>I</math> is real,