Image Analysis and Reconstruction using a Wavelet Transform Constructed from a Reducible Representation of the Euclidean Motion Group

摘要

Inspired by the early visual system of many mammalians we consider the construction of-and reconstruction from- an orientation score \({\it U_f}:\mathbb{R}^2 \times S^{1} \to \mathbb{C}\) as a local orientation representation of an image, \(f:\mathbb{R}^2 \to \mathbb{R}\). The mapping \(f\mapsto {\it U_f}\) is a wavelet transform \(\mathcal{W}_{\psi}\) corresponding to a reducible representation of the Euclidean motion group onto \(\mathbb{L}_{2}(\mathbb{R}^2)\) and oriented wavelet \(\psi \in \mathbb{L}_{2}(\mathbb{R}^2)\). This wavelet transform is a special case of a recently developed generalization of the standard wavelet theory and has the practical advantage over the usual wavelet approaches in image analysis (constructed by irreducible representations of the similitude group) that it allows a stable reconstruction from one (single scale) orientation score. Since our wavelet transform is a unitary mapping with stable inverse, we directly relate operations on orientation scores to operations on images in a robust manner.