Generalized structure tensor: Difference between revisions

Revision as of 14:17, 30 December 2015

Revision as of 14:35, 30 December 2015

Line 3:

{{FeatureDetectionCompVisNavbox}}

Let the term image represent a function <math>f(\xi(x,y),\eta(x,y))</math> where <math>\xi,\eta,x,y </math>, and <math>f</math>, are real valued.

Generalized Structure Tensor<ref name=bigun04pami3> {{cite article|

Generalized Structure Tensor, GST, is an extension of the Cartesian [[Structure Tensor]] to the [[Curvilinear coordinates]]. It is mainly used as a way to detect and to represent the "direction" parameters of curves, just like the Cartesian structure tensor detects and represents the direction in Cartesian coordinates. Detection of curve families generated by pairs of locally orthogonal functions are best studied.

To be precise, GST represents the direction along which the image <math>f</math> can undergo an infinitesimal translation with minimal error, along the "lines" fulfilling the following conditions<ref name=bigun04pami3> {{cite article|

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

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

Line 14 ⟶ 11:

volume = 26|

year = 2004|

Generalized}}</ref>, Structure Tensor, GST, is an extension of the Cartesian [[Structure Tensor]] to the [[Curvilinear coordinates]]. It is mainly used as a way to detect and to represent the "direction" parameters of curves, just like the Cartesian structure tensor detects and represents the direction in Cartesian coordinates. Detection of curve families generated by pairs of locally orthogonal functions are best studied.

}}</ref>

:

Let the term image represent a function

<math>f(\xi(x,y),\eta(x,y))</math>

where <math>x,y </math> are real variables and

<math>\xi,\eta </math>, and <math>f</math>, are real valued functions.

To be precise, GST represents the direction along which the image <math>f</math> can undergo an infinitesimal translation with minimal (total least squares) error, along the "lines" fulfilling the following conditions<ref name=bigun04pami3>: {{cite article|

1. The "lines" are ordinary lines in the curvilinear coordinate basis

Line 23 ⟶ 25:

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-Riemann conditions, i.e. Thus,<math>\frac{\partial the\xi}{\partial curvilinearx}=-\frac{\partial coordinates,\eta}{\partial ofy}</math> theand Generalized<math>\frac{\partial Structure\xi}{\partial Tensory}=\frac{\partial using\eta}{\partial harmonicx}</math>. pairsAccordingly, such curvilinear coordinates <math>\xi,\eta</math> are locally orthogonal coordinates.

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 iso-curves in a gray valued image i.e. the image must not be a binary image, nor must its edges be marked.