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Let the term image represent a function $f(\xi(x,y),\eta(x,y))$ where $\xi,\eta,x,y $, and $f$, are real valued.
Generalized Structure Tensor is an extension of the Cartesian [[Structure Tensor]] to [[Curvilinear coordinates]]. It finds the direction along which an image can undergo a translation with minimal error, measured in [[L2 norm]] amounting to [[total least squares]] sense, where the translation is along the curvilinear coordinates (instead of Cartesian).▼
▲Generalized Structure Tensor, GST, is an extension of the Cartesian [[Structure Tensor]] to the [[Curvilinear coordinates]], $\xi,\eta$. It
1. The "lines" are ordinary lines in the curvilinear coordinate basis
$$ \cos(\theta) \xi(x,y)+\sin(\theta) \eta(x,y)= Constant$$
which are curves in Cartesian coordinates as depicted by the equation above. The error is measured in the $L^2$ norm and the minimality of the error refers to [[$L^2$ norm]].
2. The functions $\xi(x,y), \eta(x,y)$ constitute a harmonic pair, i.e. they fulfill Cauchy-Riemann conditions. Thus, the curvilinear coordinates, of the Generalized Structure Tensor are locally orthogonal coordinates.
The curvilinear coordinates of GST are thereby invariants of physical processes, i.e. the latter transform the coordinates.
One of the most known such processes are in-plane rotations and zooming/dezoooming.
For the first process, it is related to the transformation $\xi=log(\sqrt{x^2+y^2})$. If any image $f$ consists in iso-curves that can be represented by circles, i.e. $f(\xi,\eta)=g(\xi)$, where $g $ is any real valued function defined on 1D, it is invariant to rotations around the origin. Likewise $f(\xi,\eta)=g(\eta)$ with $\eta=atan^{-1}(x,y)$ are invariant to scaling, i.e. zooming/dezooming with respect to the origin. Besides, $f(\xi,\eta)=g( \cos(\theta) \xi(x,y)+\sin(\theta) \eta(x,y))$ is invariant to a certain amount of rotation combined with scaling, where the amount is precised by the parameter $\theta$.
The ordinary structure tensor is thus a representation of a translation too. Here the physical process is the ordinary translation, i.e. $\xi,\eta$ are the trival identity transformations, $\xi=x$, $\eta=y$.
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.
<ref name=bigun86>
{{cite conference|
author=J. Bigun|
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year=1988
}}</ref>
<ref name=bigun04pami3>
{{cite article|
author = J. Bigun and T. Bigun and K. Nilsson|
title = Recognition by symmetry derivatives and
the generalized structure tensor|
journal = IEEE trans. Pattern Analysis and Machine Intelligence|
pages = 1590--1605|
volume = 26|
year = 2004|
}}</ref>
The Generalized structure tensor can be used as an alternative to [[Hough Transform]] in [[image processing]] and [[computer vision]]. The main differences comprise:
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