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