Approximation by piecewise polynomials on Voronoi tessellation

Abstract

We propose a novel method to approximate a function on 2D domain by piecewise polynomials. The Voronoi tessellation is used as a partition of the domain, on which the best fitting polynomials in L 2 metric are constructed. Our method optimizes the domain partition and the fitting polynomials simultaneously by minimizing an objective function indicating the approximation quality. We also provide the explicit formula of the gradient of the objective function, which makes an efficient gradient-based algorithm workable for the function minimization. We conduct several experiments to demonstrate the efficacy of our new approach for generating piecewise polynomial approximations of analytic functions and color images.

Image Representation on Curved Optimal Triangulation

“…The method proposed in Nivoliers and Lévy[NL13] was further extended by Chen et al . [CXC14] to piecewise polynomial approximation with arbitrary degrees, while Cao et al . [CXC*18] adopted barycentric coordinates to construct the approximation.…”

“…Error‐based initialization is a common strategy, which starts from a coarse triangulation and fine‐tunes the results by iteratively inserting a new vertex into the triangle with maximum approximation error [CXC14, XCC*18]. In this case, regardless of its number of iterations, the optimization can easily get stuck in the local minima for most testing images, especially for those features that are located close to one another (Figures 5(b) and 5(c)).…”

“…Given its advantages, such as resolution independence, data compression and editing, geometric representation has attracted much attention in various fields, including image approximation (e.g. [LL06, LA13, CXC14, LG19]), vectorization (e.g. [XLY09, ZZW14, FLB17]) and the artist community [Bit14, Bry17].…”

Functional data approximation on bounded domains using polygonal finite elements

“…On the other hand, with the fixed functions {Qk∗}k=1N, the gradient of the objective function in Equation (3) can be derived as follows : ∂Efalse(Xfalse)∂xi=true∑j∈Jitrue∫Ωi,j(|ffalse(boldxfalse)−Qi∗false(boldxfalse)|2−|ffalse(boldxfalse)−Qj∗false(boldxfalse)|2)x−boldxi|xj−xi|ds, where J i is the set of indices of the sites whose Voronoi cells are adjacent to Ω i . Then we adopt a modified gradient descent method proposed in [8] for an efficient solution to the L 2 optimization. For the sake of completeness, we give a brief description of the L 2 optimization method here.…”

“…Instead of using uniform Voronoi diagrams, the focus of recent attempts has been on generating non-uniform Voronoi or optimized diagrams to achieve adaptive piecewise approximations. In particular, optimization methods were proposed in [33] and [8] to generate non-uniform Voronoi tessellations adapting to the features of the target functions or images, where the features or discontinuities were well preserved by aligning the Voronoi cells along them. However, these methods find it difficult to achieve geometric continuities across cell boundaries as they use full power order polynomials on each Voronoi cell.…”