Abstract: 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.

Key words: generalized barycentric coordinates, function approximation, Voronoi tessellation, image
approximation