Lie Groups for 2D and 3D Transformations - Ethan Eade

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LieGroupsfor2Dand3DTransformationsEthanE adeUp datedMay20,2017*1Intro ductionThisdo ologicalgroupthatisalsoasmo othmanifold,withsomeotherniceprop ciatedwitheveryLiegroupisaLiealgebra,whi chisavectorspacediscussedb ortantly,aLiegroupanditsLiealgebraareint imatelyrelated,allowingcalculationsinone tob emapp cumentdo esnotgivearigorousintro ductiontoLiegroups,nordo esattempttoprovideenoughinformationthatt heLiegroupsrepresentingspatialtransforma tionscanb eemployedusefullyinrob (3)3DRotations33DrotationmatrixSE(3)3DRi gidtransformations6Lineartransformationo nhomogeneous4-vectorsSO(2)2DRotations12D rotationmatrixSE(2)2DRigidtransformation s3Lineartransformationonhomogeneous3-vec torsSim(3)3DSimilaritytransformations(ri gidmotion+scale)7Lineartransformationonh omogeneous4-vectorsForeachofthesegroups, therepresentationisdescrib ed,andtheexp oticsorcomputervision?

May 20, 2017 · Then di erentiation by the vector is straightforward, as fis linear in x: @y @x = R (27) Di erentiation by the rotation parameters is performed by implicitly left multiplying the rotation by the exponential of a tangent vector and di erentiating the resulting expression around the zero perturbation.

  Di erentiation, Erentiation