3D Rotation and Its Irreducible Representations

  • Kenichi Kanatani
Part of the Springer Series in Information Sciences book series (SSINF, volume 20)


In Chap. 2, we studied invariance properties of image characteristics under image coordinate rotation. In this chapter, we study invariance properties under the camera rotation we introduced in Chap. 1. Just as invariants for the image coordinate rotation are obtained by irreducibly reducing representations of SO(2), invariants for the camera rotation are obtained by irreducibly reducing representations of SO(3). However, direct analysis is very difficult due to the fact that SO(3) is not Abelian. Fortunately, a powerful tool is available: we only need to analyze “infinitesimal transformations”. This is because the structure of a compact Lie group is completely determined by its Lie algebra. To demonstrate our technique, we analyze the transformation of optical flow under camera rotation.


Irreducible Representation Commutation Relation Optical Flow Transformation Rule Casimir Operator 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Kenichi Kanatani
    • 1
  1. 1.Department of Computer ScienceGunma University KiryuJapan

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