Scale Spaces on Lie Groups

Remco Duits, Bernhard Burgeth

Abstract: In the standard scale space approach one obtains a scale space representation u:ℝ of an image \(f \in \mathbb{L}_{2}(\mathbb{R}^d)\) by means of an evolution equation on the additive group (ℝd, + ). However, it is common to apply a wavelet transform (constructed via a representation \(\mathcal{U}\) 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 \(g \mapsto (\mathcal{U}_{g}\psi,f)_{\mathbb{L}_{2}(\mathbb{R}^2)}\) 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 \(\mathcal{U}\)-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 ℝd by a Lie-group with more structure.