Structure tensor: Difference between revisions

Revision as of 20:59, 19 May 2020

Revision as of 07:18, 20 October 2020

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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>

{{cite conference|author1=J. Bigun |author2=G. Granlund |lastauthorampname-list-style=yesamp |title=Optimal Orientation Detection of Linear Symmetry|___location=Piscataway|booktitle=First int. Conf. on Computer Vision, ICCV, (London) |publisher=IEEE Computer Society Press, Piscataway|pages=433–438|year=1987 }}

</ref><ref name=knutsson89>

{{cite conference|author=H. Knutsson|title=Representing local structure using tensors|___location=Oulu|booktitle=Proceedings 6th Scandinavian Conf. on Image Analysis|publisher=Oulu University|pages=244–251|year=1989}}

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{{cite book|author=B. Jahne|title=Spatio-Temporal Image Processing: Theory and Scientific Applications|___location=Berlin|publisher=Springer-Verlag|volume=751|year=1993}}

</ref><ref name=MedioniEA>

{{cite book|author1=G. Medioni, M. Lee |author2=C. Tang |lastauthorampname-list-style=yesamp |title=A Computational Framework for Feature Extraction and Segmentation|publisher=Elsevier Science|date=March 2000}}

</ref> This attribute can be quantified by the '''coherence''', defined as

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The elegance of the complex representation stems from that the two components of the structure tensor can be obtained as averages and independently. In turn, this means that <math>\kappa_{20}</math> and <math>\kappa_{11}</math> can be used in a scale space representation to describe the evidence for presence of unique orientation and the evidence for the alternative hypothesis, the presence of multiple balanced orientations, without computing the eigenvectors and eigenvalues. A functional, such as squaring the complex numbers have to this date not been shown to exist for structure tensors with dimensions higher than two. In Bigun 91, it has been put forward with due argument that this is because complex numbers are commutative algebras whereas quaternions, the possible candidate to construct such a functional by, constitute a non-commutative algebra.<ref name=bigun91>

{{cite journal|author1=J. Bigun |author2=G. Granlund |author3=J. Wiklund |lastauthorampname-list-style=yesamp |title=Multidimensional Orientation Estimation with Applications to Texture Analysis and Optical Flow| journal= IEEE Transactions on Pattern Analysis and Machine Intelligence|volume=13|number=8|pages=775–790|year=1991 |doi=10.1109/34.85668}}</ref>

The complex representation of the structure tensor is frequently used in fingerprint analysis to obtain direction maps containing certainties which in turn are used to enhance them, to find the locations of the global (cores and deltas) and local (minutia) singularities, as well as automatically evaluate the quality of the fingerprints.

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<ref name=lin94book/><ref name=lingar97>{{cite journal

|author1=T. Lindeberg |author2=J. Garding

|lastauthorampname-list-style=yesamp | title=Shape-adapted smoothing in estimation of 3-D depth cues from affine distortions of local 2-D structure

| journal=Image and Vision Computing

| year=1997

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==Applications==

The eigenvalues of the structure tensor play a significant role in many image processing algorithms, for problems like [[corner detection]], [[interest point detection]], and [[feature tracking]].<ref name="Medioni">

{{cite conference|author1=M. Nicolescu |author2=G. Medioni |lastauthorampname-list-style=yesamp |title=Motion Segmentation with Accurate Boundaries – A Tensor Voting Approach|booktitle=Proc. IEEE Computer Vision and Pattern Recognition|volume=1|pages=382–389|year=2003}}

</ref><ref>

{{cite journal

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|booktitle=International Archives of Photogrammetry and Remote Sensing|volume=26|pages=150–166|year=1986}}

</ref><ref>

{{cite conference|author1=C. Harris |author2=M. Stephens |lastauthorampname-list-style=yesamp |title=A Combined Corner and Edge Detector

|booktitle=Proc. of the 4th ALVEY Vision Conference|pages=147–151|year=1988}}

</ref><ref>

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|booktitle=Image and Vision Computing|volume=15|issue=3|pages=219–233|year=1997}}

</ref><ref>

{{cite conference|author1=I. Laptev |author2=T. Lindeberg |lastauthorampname-list-style=yesamp |title=Space–time interest points

|booktitle=International Conference on Computer Vision ICCV'03|url=ftp://ftp.nada.kth.se/CVAP/reports/LapLin03-ICCV.pdf|doi=10.1109/ICCV.2003.1238378|pages=432–439|volume=I|year=2003}}

</ref><ref>

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|booktitle=Proc. European Conference on Computer Vision|volume=4|pages=100–113|year=2004}}

</ref><ref>

{{cite conference|author1=C. Kenney, M. Zuliani |author2=B. Manjunath |lastauthorampname-list-style=yesamp |title=An Axiomatic Approach to Corner Detection|booktitle=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://www.nada.kth.se/cvap/abstracts/cvap226.html Enhancement of fingerprint images using shape-adaptated scale-space operators]''. IEEE Transactions on Image Processing, volume 9, number 12, pages 2027–2042.

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{{cite journal|author=D. Tschumperle and Deriche|title=Diffusion PDE's on Vector-Valued Images|booktitle=IEEE Signal Processing Magazine|pages=16–25|date=September 2002}}

</ref><ref>

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

</ref><ref>

{{cite conference|author1=S. Arseneau |author2=J. Cooperstock |lastname-authorlist-ampstyle=yesamp |title=An Improved Representation of Junctions through Asymmetric Tensor Diffusion|booktitle=International Symposium on Visual Computing|date=November 2006}}

</ref> and several other image processing problems. The structure tensor can be also applied in [[geology]] to filter [[Seismology|seismic]] data.<ref>{{Cite journal|last=Yang|first=Shuai|last2=Chen|first2=Anqing|last3=Chen|first3=Hongde|date=2017-05-25|title=Seismic data filtering using non-local means algorithm based on structure tensor|journal=Open Geosciences|volume=9|issue=1|pages=151–160|doi=10.1515/geo-2017-0013|issn=2391-5447|bibcode=2017OGeo....9...13Y}}</ref>

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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 |lastname-authorlist-ampstyle=yesamp |title=Galilean-corrected spatio-temporal interest operators|booktitle=International Conference on Pattern Recognition ICPR'04|url=ftp://ftp.nada.kth.se/CVAP/reports/LinAkhLap04-ICPR.pdf|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 |lastname-authorlist-ampstyle=yesamp |title=Velocity adaptation of space–time interest points|booktitle=International Conference on Pattern Recognition ICPR'04|url=http://www.csc.kth.se/cvap/abstracts/LapLin04-ICPR.html|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 |lastname-authorlist-ampstyle=yesamp |title=Local descriptors for spatio-temporal recognition|booktitle=ECCV'04 Workshop on Spatial Coherence for Visual Motion Analysis (Prague, Czech Republic) Springer Lecture Notes in Computer Science|url=http://www.csc.kth.se/cvap/abstracts/LapLin04-ECCVWS.html|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 |lastname-authorlist-ampstyle=yesamp |title=Local velocity-adapted motion events for spatio-temporal recognition|booktitle=Computer Vision and Image Understanding|url=http://www.csc.kth.se/cvap/abstracts/LapCapSchLin07-CVIU.html|doi=10.1016/j.cviu.2006.11.023|year=2007|volume=108| pages= 207–229}}</ref>

==See also==