Generalized structure tensor: Difference between revisions

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In image analysis, the '''generalized structure tensor (GST)''' is an extension of the Cartesian [[structure tensor]] to [[curvilinear coordinates]].<ref name="bigun04pami3">{{cite journal |last1=Bigun |first1=J. |last2=Bigun |first2=T. |last3=Nilsson |first3=K. |title=Recognition by symmetry derivatives and the generalized structure tensor |journal=IEEE Transactions on Pattern Analysis and Machine Intelligence |date=December 2004 |volume=26 |issue=12 |pages=1590–1605 |doi=10.1109/TPAMI.2004.126|pmid=15573820 |url=http://urn.kb.se/resolve?urn=urn:nbn:se:hh:diva-237 }}</ref>. 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,<ref name=fronthaler08tip>{{cite journal

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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=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.