Image processing via shift-twist invariant operations on orientation bundle functions

Inspired by our own visual system, we consider the construction of and reconstruction from an ori- entation bundle function (OBF) Uf : 2 S1 as a local orientation score of an image, f : 2 , via a wavelet transform corresponding to a representation of the Euclidean motion group onto 2(2) and ori- ented wavelet 2(2). This wavelet transform is a unitary mapping with stable inverse, which allows us to directly relate each operation on OBFs to an operation on images in a robust manner. We examine the geometry of the domain of an OBF and show that the only sensible operations on OBFs are nonlinear and shift- twist invariant. As an example, we consider all linear second-order shift-twist invariant evolution equations on OBFs corresponding to stochastic processes on the Euclidean motion group in order to construct nonlinear shift-twist invariant operations on OBFs. Given two such stochastic processes, we derive the probability density that particles of the different processes collide. As an application, we detect elongated structures in images and automatically close the gaps between them