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year = 2004}}</ref> It is mainly used to detect and to represent the "direction" parameters of curves, just as the Cartesian structure tensor detects and represents the direction in Cartesian coordinates. Curve families generated by pairs of locally orthogonal functions have been the best studied.
It is a widely known method in
title=Local features for enhancement and minutiae extraction in fingerprints|
author = H. Fronthaler and K. Kollreider and J. Bigun|
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\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.
Thereby, Cartesian [[Structure tensor]] is a special case of GST where <math> \xi=x</math>, and <math> \eta=y</math>.
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Efficient detection of <math>\theta</math> in images is possible by image processing for a pair <math>\xi</math>, <math>\eta</math>. 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 angle is in double angle representation, i.e. <math>2\theta</math> is delivered by computations, and 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 angle estimation.
Logarithmic spirals, including circles, can for instance be detected by (complex) convolutions and non-linear mappings.<ref name=bigun04pami3 /> The spirals can be in gray (valued) images or in a binary image, i.e. locations of edge elements of the concerned patterns, such as contours of circles or spirals, must not be known or marked otherwise.
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==Physical and mathemical interpretation==
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 $\xi$ 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).
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is invariant to a certain amount of rotation combined with scaling, where the amount is precised by the parameter <math>\theta</math>.
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>,
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Generally, the estimated <math>\theta</math> represents the direction (in <math>\xi,\eta</math> coordinates) along which infinitisemal 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]].
==Miscelenous==
Image in the context of the GST means both an ordinary image and an image neighborhood therein (local image), the context determining. For example, a photograph as well as any neighborhood of it are images.
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