We consider morphological and linear scale spaces on the space R 3 S 2 of 3D positions and orientations naturally embedded in the group SE(3) of 3D rigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossing-preserving fiber enhancement on probability densities defined on the space of positions and orientations. The strength of these enhancements is that they are expressed in a moving frame of reference attached to fiber fragments, allowing us to diffuse along the fibers and to erode orthogonal to them.The linear scale spaces are described by forward Kolmogorov equations of Brownian motions on R 3 S 2 and can be solved by convolution with the corresponding Green's functions. The morphological scale spaces are Bellman equations of cost processes on R 3 S 2 and we show that their viscosity solutions are given by a morphological convolution with the corresponding morphological Green's function. For theoretical underpinning of our scale spaces on R 3 S 2 we introduce Lagrangians and Hamiltonians on R 3 S 2 indexed by a parameter η ∈ [1, ∞). The Hamiltonian induces a Hamilton-Jacobi-Bellman system that coincides with our morphological scale spaces on R 3 S 2 .By means of the logarithm on SE(3) we provide tangible estimates for both the linear-and the morphological Green's functions. We also discuss numerical finite difference upwind schemes for morphological scale spaces (erosions) of Diffusion-Weighted Magnetic Resonance Imaging (DW-MRI), which allow extensions to data-adaptive erosions of DW-MRI.We apply our theory to the enhancement of (crossing) fibres in DW-MRI for imaging water diffusion processes in brain white matter.

Abstract. We consider the linear and non-linear enhancement of diffusion weighted magnetic resonance images (DW-MRI) to use contextual information in denoising and inferring fiber crossings. We describe the space of DW-MRI images in a moving frame of reference, attached to fiber fragments which allows for convection-diffusion along the fibers. Because of this approach, our method is naturally able to handle crossings in data. We will perform experiments showing the ability of the enhancement to infer information about crossing structures, even in diffusion tensor images (DTI) which are incapable of representing crossings themselves. We will present a novel non-linear enhancement technique which performs better than linear methods in areas around ventricles, thereby eliminating the need for additional preprocessing steps to segment out the ventricles. We pay special attention to the details of implementation of the various numeric schemes.

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In this article we study both left-invariant (convection-)diffusions and left-invariant Hamilton-Jacobi equations (erosions) on the space R 3 S 2 of 3D-positions and orientations naturally embedded in the group SE(3) of 3Drigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossingpreserving fiber enhancement on probability densities defined on the space of positions and orientations.The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on R 3 S 2 and can be solved by R 3 S 2 -convolution with the corresponding Green's functions or by a finite difference scheme. The left-invariant Hamilton-Jacobi equations are Bellman equations of cost processes on R 3 S 2 and they are solved by a morphological R 3 S 2 -convolution with the corresponding Green's functions. We will reveal the remarkable analogy between these erosions/dilations and diffusions. Furthermore, we consider pseudo-linear scale spaces on the space of positions and orientations that combines dilation and diffusion in a single evolution.In our design and analysis for appropriate linear, non-linear, morphological and pseudo-linear scale spaces on R 3 S 2 we employ the underlying differential geometry on SE(3), where the frame of left-invariant vector fields serves as a moving frame of reference. Furthermore, we will present new and simpler finite difference schemes for our diffusions, which are clear improvements of our previous finite difference schemes.We apply our theory to the enhancement of fibres in magnetic resonance imaging (MRI) techniques (HARDI and DTI) for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. We provide experiments of our crossing-preserving (non-linear) left-invariant evolutions on neural images of a human brain containing crossing fibers.

In the type-setting of this article, an unfortunate error has occurred. Within this article the captions of Fig. 22 (experiment on human dataset) and Fig. 23 (experiment on artificial dataset) were accidently interchanged.