Affine group

Affine group

An affine group is a group of affine automorphisms, which are linear combinations from a set to itself, induced by a vector space. It is used to describe both linear shapes and reflection-affine shapes, with the general linear group GL(m,R) being considered for linear shapes and the affine group in m dimensions, Aff(m), being considered for reflection-affine shapes.From: Computational Topology for Biomedical Image and Data Analysis [2019], Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis [2019]

Functional Equations on Affine Groups

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Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019

László Székelyhidi

If we take a closed subgroup H of GL(V) then the semidirect product H ⋉ V is called the affine group of H on V, denoted by Aff H. For instance, in the trivial case H = {id} the affine group Aff H is isomorphic to V, this is the canonical representation of V on itself by translations. On the other hand, in the case H = SL(V) we obtain the special affine group: Aff SL(V)=SL(V)⋉V.

Elements of topology and homology

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Published in Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama, Computational Topology for Biomedical Image and Data Analysis, 2019

Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama

An affine combination induces a linear subspace called an affine space which is nothing more than a vector space under affine combinations. The vector space and the affine automorphisms (linear combinations from a set to itself) induce a group called the affine group.

Cooperative processing based on posture change detection and trajectory estimation for unknown multi-object tracking

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Published in International Journal of Systems Science, 2019

Houssem Eddine Rouabhia, Brahim Farou, Zine Eddine Kouahla, Hamid Seridi, Herman Akdag

Another approach, focusing on geometric information (Kwon, Lee, & Park, 2009), used an optimal importance function obtained with Taylor PCA expansion on the 2D affine group to track the object. Despite its effectiveness to deal with abrupt motion and appearance changes, this approach cannot detect the target in case of occlusions.

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Functional Equations on Affine Groups

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