结合广义重心坐标与Voronoi 剖分的函数分片逼近

结合广义重心坐标理论，提出了一个新方法，以解决在平面区域上的函数逼近问题。该方法通过构建基于广义重心坐标的最优分片函数来逼近目标函数。采用Voronoi 图来划分区域，并提出一个度量逼近误差的能量函数。推导出该函数的导数后，采用一种高效的Voronoi 节点更新方法来获得区域的最优剖分，并通过最优剖分构建最优分片函数。由于该方法对不连续函数具有良好地逼近能力，因此将其应用在图像逼近问题中。分别在解析函数和彩色图像上对该方法进行实验，均获得了很好的逼近效果。

Approximation by Piecewise Function Based on Generalized BarycentricCoordinates and Voronoi Tessellation

Under the generalized barycentric coordinates theory, we propose a new method to solve theproblem of approximating a given function on the planar domain. To accomplishing this, an optimalpiecewise function which based on the generalized barycentric coordinates is constructed. We use theVoronoi tessellation to create a partition of the domain, then an energy function that measures theapproximation error is built. After deriving the gradient of the energy function, an efficient optimizationmethod is adopted to update the tessellation. The optimal piecewise function will be constructed fromthe optimal tessellation. Due to its good ability of approximating discontinuous functions, our methodcan be applied to image approximation field. In order to demonstrate its efficacy, some experiments onanalytic functions and color images are designed, which have produced good results.