Structure tensor: Difference between revisions

Revision as of 10:10, 27 February 2020

Revision as of 20:59, 19 May 2020

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Namely, if <math>\lambda_1 > \lambda_2</math>, then <math>e_1</math> (or <math>-e_1</math>) is the direction that is maximally aligned with the gradient within the window.

In particular, if <math>\lambda_1 > 0, \lambda_2 = 0</math> then the gradient is always a multiple of <math>e_1</math> (positive, negative or zero); this is the case if and only if <math>I</math> within the window varies along the direction <math>e_1</math> but is constant along <math>e_2</math>. This condition of eigenvalues is also called linear symmetry condition because then the iso-curves of <math>I</math> consist in parallel lines, i.e there exists a one dimensional function function <math>g </math> which can generate the two dimensional function <math>I</math> as <math>I(x,y)=g(d^\text{T} p)</math> for some constant vector <math>d=(d_x,d_y)^T </math> and the coordinates <math>p=(x,y)^T </math>.

If <math>\lambda_1 = \lambda_2</math>, on the other hand, the gradient in the window has no predominant direction; which happens, for instance, when the image has [[rotational symmetry]] within that window. This condition of eigenvalues is also called balanced body, or directional equilibrium condition because it holds when all gradient directions in the window are equally frequent/probable.