We solve and analyze PDEs on the homogeneous space of positions and orientations. This homogeneous space is given by $\mathbb M=SE(d)/H$ where $SE(d)$ is the roto-translation Lie group and $H\equiv SO(d−1)$ the subgroup of rotations around a reference axis. We consider $d\in{2,3}$ with emphasis on $d=3$.

We solve the following PDEs on $\mathbb M$ analytically:

– Degenerate and non-degenerate (convection-)diffusion systems on $\mathbb M$, cf. [1] – Forward Kolmogorov PDEs of $\alpha$-stable Lévy processes on $\mathbb M$, cf. [2].

this is done by a Fourier transform on $\mathbb M$, cf. [2].

We solve the following PDEs on $\mathbb M$ numerically:

– Nonlinear Diffusions on $\mathbb M$, cf. [3], – Mean Curvature Flows and Total Variation Flows on $\mathbb M$, cf. [4] ($d= 2,3$), [5, 6] ($d= 2$), – Eikonal PDEs for sub-Riemannian and Finslerian geodesic front propagation on $\mathbb M$, cf. [7, 8],

via anisotropic fast-marching [10], left-invariant finite difference techniques [11] or Monte-Carlo simula-tions [2] of the underlying SDEs. The numerics is tested to our new exact solutions of the PDEs [2, 12] and of the sub-Riemannian geodesics in $\mathbb M$ [13].

We show their applications in medical image analysis in enhancement of fibers/blood vessels in 2D and 3D medical images [3, 4], and in fiber-enhancement [14], denoising [4], fiber-tracking [15], and structuralconnectivity quantification [16] in DW-MRI.