Scale Spaces on Lie Groups

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 (ℝ^d, + ). 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 (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 $\mathcal{U}$-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 ℝ^d by a Lie-group with more structure.

The Dutch Organization for Scientific Research is gratefully acknowledged for financial support

This article provides the theory and general framework we applied in [9],[5],[8].