Generalized structure tensor: Difference between revisions

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In mathematics, the '''Generalizedgeneralized Structurestructure Tensortensor (GST)''' is an extension of the Cartesian [[structure tensor]] to [[curvilinear coordinates]].,<ref name=bigun04pami3>{{cite news|

author = J. Bigun and T. Bigun and K. Nilsson|

title = Recognition by symmetry derivatives and the generalized structure tensor|

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pages = 1590–1605|

volume = 26|

year = 2004}}</ref> GST, is an extension of the Cartesian [[Structure Tensor]] to [[Curvilinear coordinates]]. It is mainly used to detect and to represent the "direction" parameters of curves, just likeas 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 applicatons of image and video processing including computer vision, such as biometric identification by fingerprints,<ref name=fronthaler08tip>{{cite news|

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volume= 74|

year=2010|

pages= 225–-243225–243

}}</ref>

<ref name=Schmitt2>{{cite news|

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1. The "lines" are ordinary lines in the curvilinear coordinate basis <math>\xi,\eta</math>

: <math> \begin{align}\cos(\theta) \xi(x,y)+\sin(\theta) \eta(x,y)= Constant \endtext{alignconstant} </math>

which are curves in Cartesian coordinates as depicted by the equation above. The error is measured in the <math>L^2</math> sense and the minimality of the error refers thereby to [[L2 norm]].

2. The functions <math>\xi(x,y), \eta(x,y)</math> constitute a harmonic pair, i.e. they fulfill [[Cauchy-RiemannCauchy–Riemann equations]],

: <math>

<math>\frac{\partial \xi}{\partial x}=-\frac{\partial \eta}{\partial y}</math> and <math>\frac{\partial \xi}{\partial y}=\frac{\partial \eta}{\partial x}</math>.

\begin{align}

<math>& \frac{\partial \xi}{\partial x}=-\frac{\partial \eta}{\partial y}</math>, and <math>\frac{\partial \xi}{\partial y}=\frac{\partial \eta}{\partial x}</math>.[4pt]

& \frac{\partial \xi}{\partial y}=\frac{\partial \eta}{\partial x}.

\end{align}

</math>

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

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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.

Generalized structure tensor can be used as an alternative to [[Hough Transformtransform]] in [[image processing]] and [[computer vision]] to detect patterns whose local orientations can be modelled, for example junction points. The main differences comprise:

*Negative, as well as complex voting are allowed;

*With one template multiple patterns belonging to the same family can be detected;

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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> \begin{align}\cos(\theta) x+\sin(\theta) y= Constant \endtext{alignconstant} </math>

where the amount is precisedspecified by the parameter <math>\theta</math>. Evidently <math>\theta</math> here represents the direction of the line.

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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== See also ==

*[[Hough Transformtransform]]

*[[Tensor]]

*[[Directional derivative]]