Abstract We address the computation of paths globally minimizing an energy involving their curvature, with given endpoints and tangents at these endpoints, according to models known as the Reeds‐Shepp car (reversible and forward variants), the Euler‐Mumford elasticae, and the Dubins car. For that purpose, we numerically solve degenerate variants of the eikonal equation, on a three‐dimensional domain, in a massively parallel manner on a graphical processing unit. Due to the high anisotropy and nonlinearity of the addressed Partial Differential Equation, the discretization stencil is rather wide, has numerous elements, and is costly to generate, which leads to subtle compromises between computational cost, memory usage, and cache coherency. Accelerations by a factor 30 to 120 are obtained w.r.t a sequential implementation. The efficiency and the robustness of the method is illustrated in various contexts, ranging from motion planning to vessel segmentation and radar configuration.

The minimal geodesic models established upon the eikonal equation framework 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.

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 the image features in conjunction with geometric regularization terms (such as curve length or elastica length) for computing geodesic paths. In this paper, we consider a more complicated problem: finding simple and closed geodesic curves which are imposed a convexity shape prior. The proposed approach relies on an orientation-lifting strategy, by which a planar curve can be mapped to an high-dimensional orientation space. The convexity shape prior serves as a constraint for the construction of local metrics. The geodesic curves in the lifted space then can be efficiently computed through the fast marching method. In addition, we introduce a way to incorporate region-based homogeneity features into the proposed geodesic model so as to solve the region-based segmentation issues with shape prior constraints.

The Voronoi diagram-based dual-front scheme is known as a powerful and efficient technique for addressing the image segmentation and domain partitioning problems. In the basic formulation of existing dual-front approaches, the evolving contour can be considered as the interfaces of adjacent Voronoi regions. Among these dual-front models, a crucial ingredient is regarded as the geodesic metrics by which the geodesic distances and the corresponding Voronoi diagram can be estimated. In this paper, we introduce a new dual-front model based on asymmetric quadratic metrics. These metrics considered are built by the integration of the image features and a vector field derived from the evolving contour. The use of the asymmetry enhancement can reduce the risk for the segmentation contours being stuck at false positions, especially when the initial curves are far away from the target boundaries or the images have complicated intensity distributions. Moreover, the proposed dual-front model can be applied for image segmentation in conjunction with various region-based homogeneity terms. The numerical experiments on both synthetic and real images show that the proposed dual-front model indeed achieves encouraging results.

In this paper, we propose a new curvature penalized minimal path model for image segmentation via closed contour detection based on the weighted Euler elastica curves, firstly introduced to the field of computer vision in [22]. Our image segmentation method extracts a collection of curvature penalized minimal geodesics, concatenated to form a closed contour, by connecting a set of user-specified points. Globally optimal minimal paths can be computed by solving an Eikonal equation. This first order PDE is traditionally regarded as unable to penalize curvature, which is related to the path acceleration in active contour models. We introduce here a new approach that enables finding a global minimum of the geodesic energy including a curvature term. We achieve this through the use of a novel Finsler metric adding to the image domain the orientation as an extra space dimension. This metric is non-Riemannian and asymmetric, defined on an orientation lifted space, incorporating the curvature penalty in the geodesic energy. Experiments show that the proposed Finsler minimal path model indeed outperforms state-of-the-art minimal path models in both synthetic and real images.

International audience

Geodesic methods have been widely applied to image analysis [1]. They are particularly efficient to extract a tubular structure, such as a blood vessel, given its two endpoints in a 2D or 3D medical image [2]. We address here a more difficult problem: the extraction of a full vessel tree structure given a single initial root, by growing a collection of keypoints, connected by geodesic minimal paths as in [3]. Keypoints are iteratively added, using selection criteria which compare geodesic distances with the standard euclidean curve length and a path score. A weakness of existing approaches is that the geodesic length and the euclidean path length are locally proportional, due to the use of an isotropic geodesic potential P(x). In contrast, we use an anisotropic geodesic potential P(x, v), and develop new criteria for selecting keypoints and stopping the tree growth. Experimental results demonstrate that our method can extract vessel structures at a finer scale, with increased accuracy.