Left-invariant Stochastic Evolution Equations on SE(2) and its Applications to Contour Enhancement and Contour Completion via Invertible Orientation Scores

Details Left-invariant Stochastic Evolution Equations on SE(2) and its Applications to Contour Enhancement and Contour Completion via Invertible Orientation Scores Left-invariant Stochastic Evolution Equations on SE(2) and its Applications to Contour Enhancement and Contour Completion via Invertible Orientation Scores

2007-11-06

arxiv.org/abs/0711.0951v4

Mathematics

Analysis of PDEs

Scientific paper

We provide the explicit solutions of linear, left-invariant, (convection)-diffusion equations and the corresponding resolvent equations on the 2D-Euclidean motion group SE(2). These diffusion equations are forward Kolmogorov equations for stochastic processes for contour enhancement and completion. The solutions are group-convolutions with the corresponding Green's function, which we derive in explicit form. We mainly focus on the Kolmogorov equations for contour enhancement processes which, in contrast to the Kolmogorov equations for contour completion, do not include convection. The Green's functions of these left-invariant partial differential equations coincide with the heat-kernels on SE(2), which we explicitly derive. Then we compute completion distributions on SE(2) which are the product of a forward and a backward resolvent evolved from resp. source and sink distribution on SE(2). On the one hand, the modes of Mumford's direction process for contour completion coincide with elastica curves minimizing $\int \kappa^{2} + \epsilon ds$, related to zero-crossings of 2 left-invariant derivatives of the completion distribution. On the other hand, the completion measure for the contour enhancement concentrates on geodesics minimizing $\int \sqrt{\kappa^{2} + \epsilon} ds$. This motivates a comparison between geodesics and elastica, which are quite similar. However, we derive more practical analytic solutions for the geodesics. The theory is motivated by medical image analysis applications where enhancement of elongated structures in noisy images is required. We use left-invariant (non)-linear evolution processes for automated contour enhancement on invertible orientation scores, obtained from an image by means of a special type of unitary wavelet transform.

Affiliated with

Also associated with

No associations

Rating

Left-invariant Stochastic Evolution Equations on SE(2) and its Applications to Contour Enhancement and Contour Completion via Invertible Orientation Scores does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Left-invariant Stochastic Evolution Equations on SE(2) and its Applications to Contour Enhancement and Contour Completion via Invertible Orientation Scores, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Left-invariant Stochastic Evolution Equations on SE(2) and its Applications to Contour Enhancement and Contour Completion via Invertible Orientation Scores will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-134994