In image analysis, the generalized structure tensor (GST) is an extension of the Cartesian structure tensor to curvilinear coordinates.
[1] It is mainly used to detect and to represent the "direction" parameters of curves, just as 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 applications of image and video processing including computer vision, such as biometric identification by fingerprints,[2] and studies of human tissue sections.
can undergo an infinitesimal translation with minimal (total least squares) error, along the "lines" fulfilling the following conditions: 1.
sense and the minimality of the error refers thereby to L2 norm.
constitute a harmonic pair, i.e. they fulfill Cauchy–Riemann equations, Accordingly, such curvilinear coordinates
are errors of (infinitesimal) translation in the best direction (designated by the angle
is the window function defining the "outer scale" wherein the detection of
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
pair the real and imaginary parts of an analytic function
are also referred to as harmonic functions in computer vision, and image processing.
Thereby, Cartesian Structure tensor is a special case of GST where
, one can detect all curves that are linear combinations of its real and imaginary parts by convolutions on (rectangular) image grids only, even if
This simplicity is a reason for why GST implementations have predominantly used the complex version above.
, it can be shown that, [1] the neighborhood defining function is complex valued, a so called symmetry derivative of a Gaussian.
Thus, the orientation wise variation of the pattern to be looked for is directly incorporated into the neighborhood defining function, and the detection occurs in the space of the (ordinary) structure tensor.
Complex convolutions (or the corresponding matrix operations) and point-wise non-linear mappings are the basic computational elements of GST implementations.
can be used as a quality (confidence, certainty) measure for the angle estimation.
Logarithmic spirals, including circles, can for instance be detected by (complex) convolutions and non-linear mappings.
[1] 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 transform in image processing and computer vision to detect patterns whose local orientations can be modelled, for example junction points.
The main differences comprise: The curvilinear coordinates of GST can explain physical processes applied to images.
A well known pair of processes consist in rotation, and zooming.
is any real valued differentiable function defined on 1D, the image is invariant to rotations (around the origin).
Zooming (comprising unzooming) operation is modeled similarly.
If the image has iso-curves that look like a "star" or bicycle spokes, i.e.
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
coordinates) along which infinitesimal translations leave the image invariant, in practice least variant.