  In this paper, we propose a novel curvature-penalized minimal path model via
an orientation-lifted Finsler metric and the Euler elastica curve. The original
minimal path model computes the globally minimal geodesic by solving an Eikonal
partial differential equation (PDE). Essentially, this first-order model is
unable to penalize curvature which is related to the path rigidity property in
the classical active contour models. To solve this problem, we present an
Eikonal PDE-based Finsler elastica minimal path approach to address the
curvature-penalized geodesic energy minimization problem. We were successful at
adding the curvature penalization to the classical geodesic energy. The basic
idea of this work is to interpret the Euler elastica bending energy via a novel
Finsler elastica metric that embeds a curvature penalty. This metric is
non-Riemannian, anisotropic and asymmetric, and is defined over an
orientation-lifted space by adding to the image domain the orientation as an
extra space dimension. Based on this orientation lifting, the proposed minimal
path model can benefit from both the curvature and orientation of the paths.
Thanks to the fast marching method, the global minimum of the
curvature-penalized geodesic energy can be computed efficiently. We introduce
two anisotropic image data-driven speed functions that are computed by
steerable filters. Based on these orientation-dependent speed functions, we can
apply the proposed Finsler elastica minimal path model to the applications of
closed contour detection, perceptual grouping and tubular structure extraction.
Numerical experiments on both synthetic and real images show that these
applications of the proposed model indeed obtain promising results.
