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 $$\nu \in \mathbb {R}$$ , is given by ∇ X [ ν ] Y = ν [ X , Y ] $$\nabla _{X}^{[\nu ]}Y=\nu \, [X,Y]$$ 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 , 1 2 , 1 $$\nu =0, { \frac {1}{2}}, 1$$ . 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 $$\mathcal {G}$$ ) reduce to ∇ γ ̇ [ 1 ] λ = 0 $$\nabla ^{[1]}_{\dot {\gamma }} \lambda =0$$ and γ ̇ = G γ − 1 λ $$\dot {\gamma }=\left .\mathcal {G}\right |{ }_{\gamma }^{-1} \lambda $$ , where γ ̇ $$\dot {\gamma }$$ 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.