Wasserstein Barycentric Coordinates: Histogram Regression Using Optimal Transport
Wasserstein Barycentric Coordinates: Histogram Regression Using Optimal Transport
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DOI：
10.1145/2897824.2925918
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发表时间：
2016-07-01
影响因子：
6.2
通讯作者：
Cuturi, Marco
中科院分区：
文献类型：
--
作者：
Bonneel, Nicolas;Peyre, Gabriel;Cuturi, Marco
本文定义了一种新的方法来对直方图值数据执行直观和几何上忠实的回归。它利用最优运输理论，特别是Wasserstein重心的定义，首次引入直方图重心坐标的概念。这些坐标考虑了定义直方图的地面空间的基本几何形状，因此对于图形到形状、颜色或材料修改的应用特别有意义。除了这个抽象的建设，我们提出了一个快速的数值优化方案来解决这个向后的问题（找到一个给定的直方图的重心坐标）与低的计算开销相对于向前的问题（计算重心）。该方案依赖于Sinkhorn算法的向后算法微分，该算法用于优化Wasserstein重心的熵正则化。我们展示了一组说明性的应用程序，这些Wasserstein坐标在计算机图形学中的各种问题：形状近似，BRDF采集和颜色编辑。
This article defines a new way to perform intuitive and geometrically faithful regressions on histogram-valued data. It leverages the theory of optimal transport, and in particular the definition of Wasserstein barycenters, to introduce for the first time the notion of barycentric coordinates for histograms. These coordinates take into account the underlying geometry of the ground space on which the histograms are defined, and are thus particularly meaningful for applications in graphics to shapes, color or material modification. Beside this abstract construction, we propose a fast numerical optimization scheme to solve this backward problem (finding the barycentric coordinates of a given histogram) with a low computational overhead with respect to the forward problem (computing the barycenter). This scheme relies on a backward algorithmic differentiation of the Sinkhorn algorithm which is used to optimize the entropic regularization of Wasserstein barycenters. We showcase an illustrative set of applications of these Wasserstein coordinates to various problems in computer graphics: shape approximation, BRDF acquisition and color editing.