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

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.Furthermore, by geometrical examination of the Euclidean motion group $$G=\mathbb{R}^2 \mathbb{R}\times \mathbb{T}$$, which is the domain of our orientation scores, we deduce that an operator Φ on orientation scores must be left invariant to ensure that the corresponding operator $$\mathcal{W}_{\psi}^{-1}\Phi \mathcal{W}_{\psi}$$ on images is Euclidean invariant. As an example we consider all linear second order left invariant evolutions on orientation scores corresponding to stochastic processes on G. As an application we detect elongated structures in (medical) images and automatically close the gaps between them.Finally, we consider robust orientation estimates by means of channel representations, where we combine robust orientation estimation and learning of wavelets resulting in an auto-associative processing of orientation features. Here linear averaging of the channel representation is equivalent to robust orientation estimation and an adaptation of the wavelet to the statistics of the considered image class leads to an auto-associative behavior of the system.