Scale Spaces on Lie Groups

In the standard scale space approach one obtains a scale space representation u : Rd R+ → R of an image f ∈ L2(Rd ) by means of an evolution equation on the additive group (Rd , +). However, it is common to apply a wavelet transform (constructed via a representation U of a Lie-group G and admissible wavelet ψ) to an image which provides a detailed overview of the group structure in an image. The result of such a wavelet transform provides a function g → (Ugψ, f)L2(R2) on a group G (rather than (Rd , +)), which we call a score. Since the wavelet transform is unitary we have stable reconstruction by its adjoint. This allows us to link operators on images to operators on scores in a robust way. To ensure U-invariance of the corresponding operator on the image the operator on the wavelet transform must be left-invariant. Therefore we focus on leftinvariant evolution equations (and their resolvents) on the Lie-group G generated by a quadratic form Q on left invariant vector fields....

• Type: Conference
• Year: 2007
• Where: SCALESPACE
• Authors: Remco Duits, Bernhard Burgeth
• Keywords: Evolution Equations, Scale Space, SCALESPACE 2007, Wavelet Transform