In this paper, we propose a novel curvature-penalized minimal path model viaan orientation-lifted Finsler metric and the Euler elastica curve. The originalminimal path model computes the globally minimal geodesic by solving an Eikonalpartial differential equation (PDE). Essentially, this first-order model isunable to penalize curvature which is related to the path rigidity property inthe classical active contour models. To solve this problem, we present anEikonal PDE-based Finsler elastica minimal path approach to address thecurvature-penalized geodesic energy minimization problem. We were successful atadding the curvature penalization to the classical geodesic energy. The basicidea of this work is to interpret the Euler elastica bending energy via a novelFinsler elastica metric that embeds a curvature penalty. This metric isnon-Riemannian, anisotropic and asymmetric, and is defined over anorientation-lifted space by adding to the image domain the orientation as anextra space dimension. Based on this orientation lifting, the proposed minimalpath model can benefit from both the curvature and orientation of the paths.Thanks to the fast marching method, the global minimum of thecurvature-penalized geodesic energy can be computed efficiently. We introducetwo anisotropic image data-driven speed functions that are computed bysteerable filters. Based on these orientation-dependent speed functions, we canapply the proposed Finsler elastica minimal path model to the applications ofclosed contour detection, perceptual grouping and tubular structure extraction.Numerical experiments on both synthetic and real images show that theseapplications of the proposed model indeed obtain promising results.
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