Generalized structure tensor: Difference between revisions

Revision as of 21:05, 4 January 2016

Revision as of 21:47, 4 January 2016

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Accordingly, such curvilinear coordinates <math>\xi,\eta</math> are locally orthogonal.

Then GST consists in

==Basic concept for its use==

: <math> GST=(\lambda_{max}-\lambda_{min})

\left[

\begin{array}{c}

\cos(\theta) \\

\sin(\theta) \\

\end{array}

\right]

[\cos(\theta), \sin(\theta)] +\lambda_{min} I </math>

where <math> 0\le \lambda_{min}\le \lambda_{max}</math> are the (infinitesimal) errors of translation in the best direction (designated by the angle <math> \theta </math>) and the worst direction (designated by <math> \theta+\pi/2</math>). The matrix <math> I </math> is the identity matrix.

Efficient detection of <math>\theta</math> in images is possible by image processing, if the pair <math>\xi</math>, <math>\eta</math> is given. Complex convolutions (or the corresponding matrix operations) and point-wise non-linear mappings are the basic computational elements of GST implementations. A total least square error estimation of <math>2\theta</math> is then obtained along with the two errors, <math>\lambda_{max}</math> and <math>\lambda_{min}</math>, in analogy with the Cartesian [[Structure tensor]]. The estimated <math>2\theta</math> can be used as a shape feature whereas <math>\lambda_{max}-\lambda_{min}</math> alone or in combination with

Thus the Cartesian [[Structure tensor]] is a special case of the GST where <math> \xi=x</math>, and <math> \eta=y</math>.

==Basic concept for its use in image processing and computer vision ==

Efficient detection of <math>\theta</math> in images is possible by image processing, iffor thea pair <math>\xi</math>, <math>\eta</math> is given. Complex convolutions (or the corresponding matrix operations) and point-wise non-linear mappings are the basic computational elements of GST implementations. A total least square error estimation of <math>2\theta</math> is then obtained along with the two errors, <math>\lambda_{max}</math> and <math>\lambda_{min}</math>, in analogy with the Cartesian [[Structure tensor]]. The estimated <math>2\theta</math> can be used as a shape feature whereas <math>\lambda_{max}-\lambda_{min}</math> alone or in combination with

<math>\lambda_{max}+\lambda_{min}</math> can be used as a quality (confidence, certainty) measure for the estimation.

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==Physical and mathemical interpretation==

The curvilinear coordinates of GST can explain physical processes applied to images. TwoA ofwell theknown mostpair knownof such processes consist in rotation, and zooming. TheThese first process isare 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 suchonly a$\xi$ transformation, 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).

Likewise the second process, zoomingZooming (comprising unzooming) operation is modeled explainedsimilarly. byIf the image has iso-curves that look like a "star" or bicycle spokes, i.e. <math>f(\xi,\eta)=g(\eta)</math> withfor some differentable 1D function <math>\eta=\tan^{-1}(x,y)g</math>. Suchthen, a functionthe image <math>f</math> is invariant to scaling, i.e. zooming/dezooming (w.r.t. the origin).

In combination,

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Analogously, the Cartesian [[structure tensor]] is a representation of a translation too. Here the physical process consists in an ordinary translation of a certain amount along <math>x</math> combined with translation along <math>y</math>,

: <math> \cos(\theta) x+\sin(\theta) y= \text{constant} </math>