Measurement of the 2D affine Lie group parameters for visual motion analysis

A class of linear operators is presented for estimating the local components of 2D translation, dilatation, rotation, and the shear/deformations which span the six degrees of freedom of motion of arbitrarily textured surfaces. This results in a model of visual motion analysis which proposes that the local transformations in the image are analysed by decomposing them into the six one-parameter subgroups of the 2D affine group. Each of the required invariant integral operators are easily specified by the characters of these six subgroups. The 2D affine group, however, does not have a simple structure. It is a Lie group which possesses a semi-direct product manifold, and classical harmonic analysis cannot proceed unless some mechanism is prescribed to isolate the 2D 'general linear' transformations from the 2D translations. It must also do so using measures which receive only local support from the image, since the global affine group model is only valid tangentially. A form of 'active perception' is thereby implicated; it is proposed that spatial indexing and 2D tracking is needed in order to form reliable estimates of 3D motion parameters using local operators in a data-driven fashion.