Generalized structure tensor: Difference between revisions

Revision as of 15:00, 10 February 2019

Revision as of 13:19, 14 April 2019

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If an image <math>f</math> consists in iso-curves that can be explained by only $\xi$ i.e. its iso-curves consist in circles <math>f(\xi,\eta)=g(\xi)</math>, where <math>g </math> is any real valued differentiable function defined on 1D, the image is invariant to rotations (around the origin).

Zooming (comprising unzooming) operation is modeled similarly. If the image has iso-curves that look like a "star" or bicycle spokes, i.e. <math>f(\xi,\eta)=g(\eta)</math> for some differentabledifferentiable 1D function <math>g</math> then, the image <math>f</math> is invariant to scaling (w.r.t. the origin).

In combination,

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where the amount is specified by the parameter <math>\theta</math>. Evidently <math>\theta</math> here represents the direction of the line.

Generally, the estimated <math>\theta</math> represents the direction (in <math>\xi,\eta</math> coordinates) along which infinitisemalinfinitesimal translations leave the image invariant, in practice least variant.

With every curvilinear coordinate basis pair, there is thus a pair of infinitesimal translators, a linear combination of which is a [[Differential operator]]. The latter are related to [[Lie algebra]].