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

Department of Computer Science

51 Citations (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 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.

Original language	English
Title of host publication	Computer Vision - ECCV 2010 : 11th European Conference on Computer Vision, Heraklion, Crete, Greece, September 5-11, 2010, Proceedings, Part VI
Editors	Kostas Daniilidis, Petros Maragos, Nikos Paragios
Number of pages	14
Volume	Part VI
Publisher	Springer
Publication date	2010
Pages	43-56
ISBN (Print)	978-3-642-15566-6
ISBN (Electronic)	978-3-642-15567-3
DOIs	https://doi.org/10.1007/978-3-642-15567-3_4
Publication status	Published - 2010
Event	11th European Conference on Computer Vision - Heraklion, Greece Duration: 5 Sept 2010 → 11 Sept 2010 Conference number: 11

Conference

Conference	11th European Conference on Computer Vision
Number	11
Country/Territory	Greece
City	Heraklion
Period	05/09/2010 → 11/09/2010

Series	Lecture notes in computer science
Number	6316
ISSN	0302-9743

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

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Sommer, S. H., Lauze, F. B., Hauberg, S., & Nielsen, M. (2010). Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations. In K. Daniilidis, P. Maragos, & N. Paragios (Eds.), Computer Vision - ECCV 2010: 11th European Conference on Computer Vision, Heraklion, Crete, Greece, September 5-11, 2010, Proceedings, Part VI (Vol. Part VI, pp. 43-56). Springer. Lecture notes in computer science No. 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. ed. / Kostas Daniilidis; Petros Maragos; Nikos Paragios. Vol. Part VI Springer, 2010. p. 43-56 (Lecture notes in computer science; No. 6316).

Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review

Sommer, SH , Lauze, FB, Hauberg, S & Nielsen, M 2010, Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations. in K Daniilidis, P Maragos & N Paragios (eds), Computer Vision - ECCV 2010: 11th European Conference on Computer Vision, Heraklion, Crete, Greece, September 5-11, 2010, Proceedings, Part VI. vol. Part VI, Springer, Lecture notes in computer science, no. 6316, pp. 43-56, 11th European Conference on Computer Vision, Heraklion, Greece, 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. In Daniilidis K, Maragos P, Paragios N, editors, Computer Vision - ECCV 2010: 11th European Conference on Computer Vision, Heraklion, Crete, Greece, September 5-11, 2010, Proceedings, Part VI. Vol. Part VI. Springer. 2010. p. 43-56. (Lecture notes in computer science; No. 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. editor / Kostas Daniilidis ; Petros Maragos ; Nikos Paragios. Vol. Part VI Springer, 2010. pp. 43-56 (Lecture notes in computer science; No. 6316).

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

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