A Variational Framework for Curve Shortening in Various Geometric Domains

Geodesics measure the shortest distance (either locally or globally) between two points on a curved surface and serve as a fundamental tool in digital geometry processing. Suppose that we have a parameterized path $\gamma (t)=\mathbf {x}(u(t),v(t))$ on a surface $\mathbf {x}=\mathbf {x}(u,v)$ with $\gamma (0)=p$ and $\gamma (1)=q$. We formulate the two-point geodesic problem into a minimization problem $\int _0^1 H(\Vert \mathbf {x}_uu^{\prime }(t)+\mathbf {x}_vv^{\prime }(t)\Vert)\text{d}t$, where $H(s)$ satisfies $H(0)=0,H^{\prime }(s)>0$ and $H^{\prime \prime }(s)\geq 0$ for $s>0$. In our implementation, we choose $H(s)=e^{s^2}-1$ and show that it has several unique advantages over other choices such as $H(s)=s^2$ and $H(s)=s$. It is also a minimizer of the traditional geodesic length variational and able to guarantee the uniqueness and regularity in terms of curve parameterization. In the discrete setting, we construct the initial path by a sequence of moveable points $\lbrace x_i\rbrace _{i=1}^n$ and minimize $\sum _{i=1}^{n} H(\Vert x_i - x_{i+1}\Vert)$. The resulting points are evenly spaced along the path. It’s obvious that our algorithm can deal with parametric surfaces. Considering that meshes, point clouds and implicit surfaces can be transformed into a signed distance function (SDF), we also discuss its implementation on a general SDF. Finally, we show that our method can be extended to solve a general least-cost path problem. We validate the proposed algorithm in terms of accuracy, performance and scalability, and demonstrate the advantages by extensive comparisons.