在简单形汤和点云之间求最优运输的算法
An algorithm for optimal transport between a simplex soup and a point cloud
AI总结：
本文提出一种数值方法，用于在低维子集上的测度与有限支持测度之间求最优运输映射，通过非线性系统求解来指定Laguerre图中各单元的质量，证明了阻尼牛顿算法的线性收敛性，并应用于三维空间中的运输计划计算及表面概率密度最优量化、重网格化和刚性点集注册。
摘要：
我们提出了一种数值方法，用于在低维子集上的测度与有限支持测度之间求最优运输映射。更具体地说，源测度被假设支持在一个简单形汤上，即在任意维度2到d之间的简单形的并集上。正如[Aurenhammer, Hoffman, Aronov, Algorithmica 20 (1), 1998, 61--76]所做的一样，我们将这个最优运输问题转化为求解一个非线性系统，其中希望指定Laguerre图中每个单元的质量。我们证明了阻尼牛顿算法以线性速度收敛于求解这个非线性系统。收敛性依赖于两个条件：(i)点云相对于简单形汤的通用性条件；(ii)源测度支持上的(强)连通性条件。最后，我们将在R^3中应用我们的算法来计算支撑在三角剖分上的测度与离散测度之间的最优运输计划。我们还详细说明了一些应用，如概率密度在表面上的最优量化、重网格化或在网格上的刚性点集注册。
英文摘要：
We propose a numerical method to find the optimal transport map between a measure supported on a lower-dimensional subset of R^d and a finitely supported measure. More precisely, the source measure is assumed to be supported on a simplex soup, i.e. on a union of simplices of arbitrary dimension between 2 and d. As in [Aurenhammer, Hoffman, Aronov, Algorithmica 20 (1), 1998, 61--76] we recast this optimal transport problem as the resolution of a non-linear system where one wants to prescribe the quantity of mass in each cell of the so-called Laguerre diagram. We prove the convergence with linear speed of a damped Newton's algorithm to solve this non-linear system. The convergence relies on two conditions: (i) a genericity condition on the point cloud with respect to the simplex soup and (ii) a (strong) connectedness condition on the support of the source measure defined on the simplex soup. Finally, we apply our algorithm in R^3 to compute optimal transport plans between a measure supported on a triangulation and a discrete measure. We also detail some applications such as optimal quantization of a probability density over a surface, remeshing or rigid point set registration on a mesh.
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