Generalized structure tensor: Difference between revisions

Revision as of 06:15, 23 August 2021

Revision as of 20:20, 18 October 2022

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The curvilinear coordinates of GST can explain physical processes applied to images. A well known pair of processes consist in rotation, and zooming. These are related to the coordinate transformation <math>\xi=\log(\sqrt{x^2+y^2})</math> and <math>\eta=\tan^{-1}(x,y)</math>.

If an image <math>f</math> consists in iso-curves that can be explained by only $<math>\xi$ </math> 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 differentiable 1D function <math>g</math> then, the image <math>f</math> is invariant to scaling (w.r.t. the origin).