Scale Spaces on Lie Groups

摘要

In the standard scale space approach one obtains a scale space representation u : R~d × R~+ → R of an image f ∈L_2(R~d) by means of an evolution equation on the additive group (R~d, +). 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 → (U_gψ, f)_(L_2(R~2)) on a group G (rather than (R~d, +)), 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 left-invariant evolution equations (and their resolvents) on the Lie-group G generated by a quadratic form Q on left invariant vector fields. These evolution equations correspond to stochastic processes on G and their solution is given by a group convolution with the corresponding Green's function, for which we present an explicit derivation in two particular image analysis applications. In this article we describe a general approach how the concept of scale space can be extended by replacing the additive group R~d by a Lie-group with more structure.