Generalized structure tensor: Difference between revisions

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Revision as of 12:22, 20 July 2018

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Then GST consists in

: <math> GST=(\lambda_{max}-\lambda_{min})

\int \omega(\xi,\eta)\left[

\begin{array}{c}

\frac{\partial f}{\partial \xi} \\

\cos(\theta) \\

\frac{\partial f}{\partial \eta} \\

\sin(\theta) \\

\end{array}

\right]

[\cos(frac{\theta)partial f}{\partial \xi}, \sin(frac{\theta)partial f}{\partial \eta}] d\xi d\eta +\lambda_{min} I </math>

where <math> 0\le \lambda_{min}\le \lambda_{max}</math> are the errors of (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 function <math> \omega </math> is the window function defining the "outer scale" wherein the detection of <math>\theta</math> will be carried out, which can be omitted if it is already included in <math>f</math> or if <math>f</math> is the full image (rather than local). The matrix <math> I </math> is the identity matrix. Using the chain rule, it can be shown that the integration above can be implemented as convolutions in Cartesian coordinates applied to the ordinary structure tensor when <math>\xi,\eta</math> pair the real and imaginary parts of an analytic function <math>g(z)</math>,

:<math>

\begin{array}{c}

\xi(x,y)=\Re g(z)\\

\eta(x,y)=\Im g(z)\\

\end{array}

</math>

where <math>z=x+iy</math> <ref>{{cite journal |last1=Bigun |first1=Josef |title=Pattern Recognition in Images by Symmetries and Coordinate Transformations |journal=Computer Vision and Image Understanding |date=December 1997 |volume=68 |issue=3 |pages=290–307 |doi=https://doi.org/10.1006/cviu.1997.0556}}</ref>.

Examples of analytic functions include <math> g(z)=\log z=\log(x+iy)</math>, as well as monomials <math> g(z)=z^n=(x+iy)^n</math>, <math> g(z)=z^(n/2)=(x+iy)^(n/2)</math>, where <math> n</math> is an arbitrary positive or negative integer. The monomials <math> g(z)=z^n</math> are also referred to as Harmonic functions in Computer Vision, and Image Processing.

Thereby, Cartesian [[Structure tensor]] is a special case of GST where <math> \xi=x</math>, and <math> \eta=y</math>, i.e. the harmonic function is simply <math> g(z)= z=(x+iy)</math>. Thus by choosing a harmonic function <math>g</math>, one can detect all curves that are linear combinations of its real and imaginary parts by convolutions on (rectangular) image grids only, even if <math>\xi,\eta</math> are non-Cartesian. Furthermore, the convolution computations can be done by using complex filters applied to the complex version of the structure tensor. Thus, GST implementations have frequently been done using complex version of the structure tensor, rather than using the (1,1) tensor.

Thereby, Cartesian [[Structure tensor]] is a special case of GST where <math> \xi=x</math>, and <math> \eta=y</math>.

==Basic concept for its use in image processing and computer vision ==