Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations

Stefan Horst Sommer; Francois Bernard Lauze; Søren Hauberg; Mads Nielsen

doi:10.1007/978-3-642-15567-3_4

Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations

Stefan Horst Sommer, Francois Bernard Lauze, Søren Hauberg, Mads Nielsen

Datalogisk Institut

51 Citationer (Scopus)

Abstract

Manifolds are widely used to model non-linearity arising in
a range of computer vision applications. This paper treats statistics on
manifolds and the loss of accuracy occurring when linearizing the manifold
prior to performing statistical operations. Using recent advances in
manifold computations, we present a comparison between the non-linear
analog of Principal Component Analysis, Principal Geodesic Analysis,
in its linearized form and its exact counterpart that uses true intrinsic
distances. We give examples of datasets for which the linearized version
provides good approximations and for which it does not. Indicators for
the dierences between the two versions are then developed and applied
to two examples of manifold valued data: outlines of vertebrae from a
study of vertebral fractures and spacial coordinates of human skeleton
end-eectors acquired using a stereo camera and tracking software.

Originalsprog	Engelsk
Titel	Computer Vision - ECCV 2010 : 11th European Conference on Computer Vision, Heraklion, Crete, Greece, September 5-11, 2010, Proceedings, Part VI
Redaktører	Kostas Daniilidis, Petros Maragos, Nikos Paragios
Antal sider	14
Vol/bind	Part VI
Forlag	Springer
Publikationsdato	2010
Sider	43-56
ISBN (Trykt)	978-3-642-15566-6
ISBN (Elektronisk)	978-3-642-15567-3
DOI	https://doi.org/10.1007/978-3-642-15567-3_4
Status	Udgivet - 2010
Begivenhed	11th European Conference on Computer Vision - Heraklion, Grækenland Varighed: 5 sep. 2010 → 11 sep. 2010 Konferencens nummer: 11

Konference

Konference	11th European Conference on Computer Vision
Nummer	11
Land/Område	Grækenland
By	Heraklion
Periode	05/09/2010 → 11/09/2010

Navn	Lecture notes in computer science
Nummer	6316
ISSN	0302-9743

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10.1007/978-3-642-15567-3_4

https://link.springer.com/content/pdf/10.1007/978-3-642-15567-3_4.pdf

Citationsformater

Sommer, S. H., Lauze, F. B., Hauberg, S., & Nielsen, M. (2010). Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations. I K. Daniilidis, P. Maragos, & N. Paragios (red.), Computer Vision - ECCV 2010: 11th European Conference on Computer Vision, Heraklion, Crete, Greece, September 5-11, 2010, Proceedings, Part VI (Bind Part VI, s. 43-56). Springer. Lecture notes in computer science Nr. 6316 https://doi.org/10.1007/978-3-642-15567-3_4

Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations. / Sommer, Stefan Horst ; Lauze, Francois Bernard; Hauberg, Søren et al.

Computer Vision - ECCV 2010: 11th European Conference on Computer Vision, Heraklion, Crete, Greece, September 5-11, 2010, Proceedings, Part VI. red. / Kostas Daniilidis; Petros Maragos; Nikos Paragios. Bind Part VI Springer, 2010. s. 43-56 (Lecture notes in computer science; Nr. 6316).

Publikation: Bidrag til bog/antologi/rapport › Konferencebidrag i proceedings › Forskning › peer review

Sommer, SH , Lauze, FB, Hauberg, S & Nielsen, M 2010, Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations. i K Daniilidis, P Maragos & N Paragios (red), Computer Vision - ECCV 2010: 11th European Conference on Computer Vision, Heraklion, Crete, Greece, September 5-11, 2010, Proceedings, Part VI. bind Part VI, Springer, Lecture notes in computer science, nr. 6316, s. 43-56, 11th European Conference on Computer Vision, Heraklion, Grækenland, 05/09/2010. https://doi.org/10.1007/978-3-642-15567-3_4

Sommer SH , Lauze FB, Hauberg S, Nielsen M. Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations. I Daniilidis K, Maragos P, Paragios N, red., Computer Vision - ECCV 2010: 11th European Conference on Computer Vision, Heraklion, Crete, Greece, September 5-11, 2010, Proceedings, Part VI. Bind Part VI. Springer. 2010. s. 43-56. (Lecture notes in computer science; Nr. 6316). doi: 10.1007/978-3-642-15567-3_4

Sommer, Stefan Horst ; Lauze, Francois Bernard ; Hauberg, Søren et al. / Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations. Computer Vision - ECCV 2010: 11th European Conference on Computer Vision, Heraklion, Crete, Greece, September 5-11, 2010, Proceedings, Part VI. red. / Kostas Daniilidis ; Petros Maragos ; Nikos Paragios. Bind Part VI Springer, 2010. s. 43-56 (Lecture notes in computer science; Nr. 6316).

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abstract = "Manifolds are widely used to model non-linearity arising in a range of computer vision applications. This paper treats statistics on manifolds and the loss of accuracy occurring when linearizing the manifold prior to performing statistical operations. Using recent advances in manifold computations, we present a comparison between the non-linear analog of Principal Component Analysis, Principal Geodesic Analysis, in its linearized form and its exact counterpart that uses true intrinsic distances. We give examples of datasets for which the linearized version provides good approximations and for which it does not. Indicators for the differences between the two versions are then developed and applied to two examples of manifold valued data: outlines of vertebrae from a study of vertebral fractures and spacial coordinates of human skeleton end-effectors acquired using a stereo camera and tracking software.",

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Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations

Abstract

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