Structure tensor: Difference between revisions

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{{FeatureDetectionCompVisNavbox}}

{{Use American English|date = March 2019}}

In mathematics, the '''structure [[tensor]]''', also referred to as the '''second-moment matrix''', is a [[matrix (mathematics)|matrix]] derived from the [[gradient]] of a [[function (mathematics)|function]]. It describes the distribution of the gradient in a specified neighborhood around a point and makes the information invariant respectto the observing coordinates<!-- Example: if you have a 2D image with two components storing the gradient direction and a Gaussian blur is performed separately on each component, the result will be ill-formed (specially for the directions were vector orientations flip). On the other hand if the blur is performed component-wise on a 2x2 structure tensor the main eigenvector (scaled by its eigenvalue) will properly represent the gradient. -->. The structure tensor is often used in [[image processing]] and [[computer vision]].<ref name=bigun86>

J. Bigun and G. Granlund (1986), ''Optimal Orientation Detection of Linear Symmetry''. Tech. Report LiTH-ISY-I-0828, Computer Vision Laboratory, Linkoping University, Sweden 1986; Thesis Report, Linkoping studies in science and technology No. 85, 1986.

</ref><ref name=bigun87>

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Note that the average of the gradient <math>\nabla I</math> inside the window is '''not''' a good indicator of anisotropy. Aligned but oppositely oriented gradient vectors would cancel out in this average, whereas in the structure tensor they are properly added together.<ref>

{{cite techreporttech report|author1=T. Brox |author2= J. Weickert |author3=B. Burgeth |author4=P. Mrazek |name-list-style=amp |title=Nonlinear Structure Tensors|institution=Universität des Saarlandes|number=113|year=2004}}

</ref> This is a reason for why <math>(\nabla I)(\nabla I)^\text{T}</math> is used in the averaging of the structure tensor to optimize the direction instead of <math>\nabla I</math>.

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===Complex version===

The interpretation and implementation of the 2D structure tensor becomes particularly accessible using [[complex numbersnumber]]s.<ref name="bigun87" /> The structure tensor consists in 3 real numbers

<math display="block">

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===Interpretation===

As in the two-dimensional case, the eigenvalues <math>\lambda_1,\lambda_2,\lambda_3</math> of <math>S_w[p]</math>, and the corresponding eigenvectors <math>\hat{e}_1,\hat{e}_2,\hat{e}_3</math>, summarize the distribution of gradient directions within the neighborhood of ''p'' defined by the window <math>w</math>. This information can be visualized as an [[ellipsoid]] whose semi-axes are equal to the eigenvalues and directed along their corresponding eigenvectors.<ref name="Medioni"/><ref>{{Cite journal | last1=Westin|first1=C.-F. | last2=Maier|first2=S.E. | last3=Mamata|first3=H. | last4=Nabavi|first4=A. | last5=Jolesz|first5=F.A. | last6=Kikinis|first6=R. | date=June 2002 | title=Processing and visualization for diffusion tensor MRI | url=https://linkinghub.elsevier.com/retrieve/pii/S1361841502000531 | journal = Medical Image Analysis | language=en | volume=6 | issue=2 | pages=93–108 | doi=10.1016/S1361-8415(02)00053-1 | pmid=12044998| url-access=subscription }}</ref>

[[File:STgeneric.png|thumb|center|240px|Ellipsoidal representation of the 3D structure tensor.]]

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A. Almansa and T. Lindeberg (2000), ''[http://kth.diva-portal.org/smash/record.jsf?pid=diva2%3A338874&dswid=-9161 Enhancement of fingerprint images using shape-adaptated scale-space operators]''. IEEE Transactions on Image Processing, volume 9, number 12, pages 2027–2042.

</ref> [[diffusion-based image processing]],<ref>[http://www.mia.uni-saarland.de/weickert/book.html J. Weickert (1998), Anisotropic diffusion in image processing, Teuber Verlag, Stuttgart.]</ref><ref>

{{cite journal|author1=D. Tschumperle | author2= R. Deriche| name-list-style=amp | title=Diffusion PDEs on Vector-Valued Images|journal=IEEE Signal Processing Magazine|pages=16–25|volume=19|issue=5|doi=10.1109/MSP.2002.1028349 | date=September 2002| bibcode= 2002ISPM...19...16T}}

</ref><ref>

{{cite conference|author1=S. Arseneau |author2=J. Cooperstock |name-list-style=amp |chapter=An Asymmetrical Diffusion Framework for Junction Analysis|title=British Machine Vision Conference|volume=2|pages=689–698|date=September 2006}}

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</ref> corresponding to the transfer of [[affine shape adaptation]]<ref name=lingar97/> from spatial to spatio-temporal image data.

In combination with local spatio-temporal histogram descriptors,<ref>

{{cite conference|author1=I. Laptev |author2=T. Lindeberg |name-list-style=amp |title=Local descriptors for spatio-temporal recognition|conference=ECCV'04 Workshop on Spatial Coherence for Visual Motion Analysis (Prague, Czech Republic) Springer Lecture Notes in Computer Science|url=http://kth.diva-portal.org/smash/record.jsf?pid=diva2%3A445261&dswid=-1233| doi=10.1007/11676959|date=May 2004|volume=3667| pages=91–103|url-access=subscription}}

</ref>

these concepts together allow for Galilean invariant recognition of spatio-temporal events.<ref>