Structure tensor: Difference between revisions

Revision as of 19:14, 14 February 2021

Revision as of 19:18, 14 February 2021

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{{cite conference|author1=C. Kenney, M. Zuliani |author2=B. Manjunath |name-list-style=amp |title=An Axiomatic Approach to Corner Detection|book-title=Proc. IEEE Computer Vision and Pattern Recognition|pages=191–197|year=2005}}

</ref> The structure tensor also plays a central role in the [[Lucas–Kanade Optical Flow Method|Lucas-Kanade optical flow algorithm]], and in its extensions to estimate [[affine shape adaptation]];<ref name=lingar97/> where the magnitude of <math>\lambda_2</math> is an indicator of the reliability of the computed result. The tensor has been used for [[scale space]] analysis,<ref name=lin94book/> estimation of local surface orientation from monocular or binocular cues,<ref name=garlin96/> non-linear [[fingerprint enhancement]],<ref>

A. Almansa and T. Lindeberg (2000), ''[http://wwwkth.nadadiva-portal.kth.se/cvaporg/abstractssmash/cvap226record.htmljsf?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|author=D. Tschumperle and Deriche|title=Diffusion PDE's on Vector-Valued Images|book-title=IEEE Signal Processing Magazine|pages=16–25|date=September 2002}}

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:<math> \begin{bmatrix} x' \\ y' \\ t' \end{bmatrix} = G \begin{bmatrix} x \\ y \\ t \end{bmatrix} = \begin{bmatrix} x - v_x \, t \\ y - v_y \, t \\ t \end{bmatrix} </math>,

it is, however, from a computational viewpoint preferable to parameterize the components in the structure tensor/second-moment matrix <math>S</math> using the notion of ''Galilean diagonalization''<ref name=lin04icpr>

{{cite conference|author1=T. Lindeberg |author2=A. Akbarzadeh |author3=I. Laptev |name-list-style=amp |title=Galilean-corrected spatio-temporal interest operators|book-title=International Conference on Pattern Recognition ICPR'04|url=ftphttp://ftpkth.nadadiva-portal.kth.seorg/CVAPsmash/reports/LinAkhLap04record.jsf?pid=diva2%3A441205&dswid=-ICPR.pdf545 |doi=10.1109/ICPR.2004.1334004|date=August 2004|volume=I| pages=57–62}}

</ref>

:<math> S' = R_\text{space}^{-\text{T}} \, G^{-\text{T}} \, S \, G^{-1} \, R_\text{space}^{-1} = \begin{bmatrix} \nu_1 & \, & \, \\ \, & \nu_2 & \, \\ \, & \, & \nu_3 \end{bmatrix} </math>

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:<math> S'' = R_\text{spacetime}^{-\text{T}} \, S \, R_\text{spacetime}^{-1} = \begin{bmatrix} \lambda_1 & & \\ & \lambda_2 & \\ & & \lambda_3 \end{bmatrix} </math>.

To obtain true Galilean invariance, however, also the shape of the spatio-temporal window function needs to be adapted,<ref name=lin04icpr/><ref>

{{cite conference|author1=I. Laptev |author2=T. Lindeberg |name-list-style=amp |title=Velocity adaptation of space–time interest points|book-title=International Conference on Pattern Recognition ICPR'04|url=http://wwwkth.cscdiva-portal.kth.seorg/cvapsmash/abstracts/LapLin04record.jsf?pid=diva2%3A451230&dswid=-ICPR.html7763|doi=10.1109/ICPR.2004.971|date=August 2004|volume=I| pages=52–56}}

</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|book-title=ECCV'04 Workshop on Spatial Coherence for Visual Motion Analysis (Prague, Czech Republic) Springer Lecture Notes in Computer Science|url=http://wwwkth.cscdiva-portal.kth.seorg/cvapsmash/abstracts/LapLin04record.jsf?pid=diva2%3A445261&dswid=-ECCVWS.html1233|doi=10.1007/11676959|date=May 2004|volume=3667| pages=91–103}}

</ref>

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

{{cite conference|author1=I. Laptev |author2=B. Caputo |author3=C. Schuldt |author4=T. Lindeberg |name-list-style=amp |title=Local velocity-adapted motion events for spatio-temporal recognition|book-title=Computer Vision and Image Understanding|url=http://wwwkth.cscdiva-portal.kth.seorg/cvapsmash/abstracts/LapCapSchLin07-CVIUrecord.htmljsf?pid=diva2%3A335153&dswid=7950 |doi=10.1016/j.cviu.2006.11.023|year=2007|volume=108| pages= 207–229}}</ref>

==See also==