Generalized structure tensor: Difference between revisions

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

<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>.

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.

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

The Generalized structure tensor can be used as an alternative to [[Hough Transform]] in [[image processing]] and [[computer vision]] to detect for example, circles, or junction points. The main differences comprise:

*Negative, as well as complex voting are allowed,

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) images or in a binary image, i.e. thelocations imageof must notedge beelements aof binarythe imageconcerned patterns, such noras mustcontours itsof edgescircles or spirals, must not be known or marked otherwise.

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

The Generalized structure tensor can be used as an alternative to [[Hough Transform]] in [[image processing]] and [[computer vision]] to detect forpatterns example,whose circleslocal orientations can be modelled, for orexample 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.;

*Image binarization is not required.

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).