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==GST in 2D and locally orthogonal bases==
Let the term image represent a function
<math>f(\xi(x,y),\eta(x,y))</math>
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Accordingly, such curvilinear coordinates <math>\xi,\eta</math> are locally orthogonal.
==Basic concept for its use==
Efficient detection of <math>\theta</math> in images is possible by image processing, if the pair <math>\xi</math>, <math>\eta</math> is given. Logarithmic spirals, including circles, can for instance be detected by (complex) convolutions.<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.▼
Efficient detection of <math>\theta</math> in images is possible by image processing, if the pair <math>\xi</math>, <math>\eta</math> is given. 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 <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 ().
▲
Generalized structure tensor can be used as an alternative to [[Hough transform]] in [[image processing]] and [[computer vision]] to detect patterns whose local orientations can be modelled, for example junction points. The main differences comprise:
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*With one template multiple patterns belonging to the same family can be detected;
*Image binarization is not required.
==Physical interpretation==
The curvilinear coordinates of GST can explain physical processes applied to images. Two of the most known such processes consist in rotation, and zooming. The first process is related to the transformation <math>\xi=\log(\sqrt{x^2+y^2})</math>. If an image <math>f</math> consists in iso-curves that can be explained by such a transformation, i.e. its iso-curves consist in circles <math>f(\xi,\eta)=g(\xi)</math>, where <math>g </math> is any real valued 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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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 infinitisemal translations leave the image invariant, in practice least variant.
This is connected to [[Lie operators]].
==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.
== See also ==
*[[Lie operators]]
*[[Structure tensor]]
*[[Hough transform]]
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