Recent Geometric Flows in Multi-orientation Image Processing via a Cartan Connection

Applications of geometric flows to multi-orientation image processing require the choice of an (affine) connection on the Lie group G of roto-translations. Typical choices of such connections are called the (−), (0) and (+) connection. As the construction of these connections in standard references is quite involved, we provide an overview. We show that these connections are members of a larger, one-parameter class of connections, and we motivate that the (+) connection is most suited for our image analysis applications. The class ∇^[ν], with ν∈R, is given by ∇[ν]XY=ν[X,Y] for all left-invariant vector fields X, Y  on G. Their auto-parallel curves are the exponential curves. Their torsion is T[X, Y ] = (2ν − 1)[X, Y ], and the (−), (0) and (+) connections arise for ν=0,12,1.

We propose the case ν = 1, as then the Hamiltonian flows on T^∗(G) for Riemannian distance minimizers on G (induced by left-invariant metric tensor field G) reduce to ∇[1]γ˙λ=0 and γ˙=G|−1γλ, where γ˙is velocity and λ is momentum. So now ‘shortest curves’ have parallel momentum, whereas ‘straight curves’ have auto-parallel velocity. We also extend this idea to sub-Riemannian geometry via a partial connection.

The connection underlies PDE flows for crossing-preserving geodesic wavefront propagation and denoising in multi-orientation image processing, where we use:

1. The ‘shortest curves’ for tracking in multi-orientation image representations, 2. The ‘straight curve fits’ for locally adaptive frames in PDEs for crossing-preserving image denoising and enhancement.