Inspired by our own visual system we consider the construction of-and reconstruction from- an orientation bundle function (OBF) Uf : R2 ? S1 ?C as a local orientation score of an image, f : R2 ? R, via a wavelet transform W? corresponding to a representation of the Euclidean motion group onto L2(R2) and oriented wavelet ? ?L2(R2). This wavelet transform is a unitary mapping with stable inverse, which allows us to directly relate each operation ? on OBF’s 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 OBF’s are non-linear and shift-twist invariant. As an example we consider all linear 2nd order shift-twist invariant evolution equations on OBF’s corresponding to stochastic processes on the Euclidean motion group in order to construct nonlinear shift-twist invariant operations on OBF’s. 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.
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