Geodesic Models with Convexity Shape Prior

The minimal geodesic models based on the Eikonal equations are capable of finding suitable solutions in various image segmentation scenarios. Existing geodesic-based segmentation approaches usually exploit image features in conjunction with geometric regularization terms, such as Euclidean curve length or curvature-penalized length, for computing geodesic curves. In this paper, we take into account a more complicated problem: finding curvature-penalized geodesic paths with a convexity shape prior. We establish new geodesic models relying on the strategy of orientation-lifting, by which a planar curve can be mapped to an high-dimensional orientation-dependent space. The convexity shape prior serves as a constraint for the construction of local geodesic metrics encoding a particular curvature constraint. Then the geodesic distances and the corresponding closed geodesic paths in the orientation-lifted space can be efficiently computed through state-of-the-art Hamiltonian fast marching method. In addition, we apply the proposed geodesic models to the active contours, leading to efficient interactive image segmentation algorithms that preserve the advantages of convexity shape prior and curvature penalization.Comment: This paper has been accepted by TPAM

Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D

We propose an efficient approach for the grouping of local orientations (points on vessels) via nilpotent approximations of sub-Riemannian distances in the 2D and 3D roto-translation groups $SE(2)$ and $SE(3)$. In our distance approximations we consider homogeneous norms on nilpotent groups that locally approximate $SE(n)$, and which are obtained via the exponential and logarithmic map on $SE(n)$. In a qualitative validation we show that the norms provide accurate approximations of the true sub-Riemannian distances, and we discuss their relations to the fundamental solution of the sub-Laplacian on $SE(n)$. The quantitative experiments further confirm the accuracy of the approximations. Quantitative results are obtained by evaluating perceptual grouping performance of retinal blood vessels in 2D images and curves in challenging 3D synthetic volumes. The results show that 1) sub-Riemannian geometry is essential in achieving top performance and 2) that grouping via the fast analytic approximations performs almost equally, or better, than data-adaptive fast marching approaches on $\mathbb{R}^n$ and $SE(n)$.Comment: 18 pages, 9 figures, 3 tables, in review at JMI

Asymmetric Geodesic Distance Propagation for Active Contours

This is the final version. Available from British Machine Vision Association (BMVA) via the link in this record. The dual-front scheme is a powerful curve evolution tool for active contours and image segmentation, which has proven its capability in dealing with various segmentation tasks. In its basic formulation, a contour is represented by the interface of two adjacent Voronoi regions derived from the geodesic distance map which is the solution to an Eikonal equation. The original dual-front model [17] is based on isotropic metrics, and thus cannot take into account the asymmetric enhancements during curve evolution. In this paper, we propose a new asymmetric dual-front curve evolution model through an asymmetric Finsler geodesic metric, which is constructed in terms of the extended normal vector field of the current contour and the image data. The experimental results demonstrate the advantages of the proposed method in computational efficiency, robustness and accuracy when compared to the original isotropic dual-front model.Roche pharmaAgence Nationale de la Recherch