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

TY - GEN

T1 - Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations

AU - Sommer, Stefan Horst

AU - Lauze, Francois Bernard

AU - Hauberg, Søren

AU - Nielsen, Mads

N1 - Conference code: 11

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

U2 - 10.1007/978-3-642-15567-3_4

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

M3 - Article in proceedings

SN - 978-3-642-15566-6

VL - Part VI

T3 - Lecture notes in computer science

SP - 43

EP - 56

BT - Computer Vision - ECCV 2010

A2 - Daniilidis, Kostas

A2 - Maragos, Petros

A2 - Paragios, Nikos

PB - Springer

T2 - 11th European Conference on Computer Vision

Y2 - 5 September 2010 through 11 September 2010

ER -