Piecewise Polynomials Approximation Method Based on Voronoi Tessellation and Triangulation

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ABSTRACT

Abstract

The research of piecewise approximation is one of the most important part of the theory of function approximation, it is not only of importance in applied mathematics, but also at the core of many applications in graphics. Herein, we focus in handling the problem of piecewise approximation by using polynomials to construct the approximating function, while the flexible Voronoi tessellation and triangulation are adopted to partition the 2D domain. We use theL2 _metric that measures the distance between target function and approximating function to construct the objective function which tells how concisely the approximation is, while a minimizer of a least-squares problem is conducted to obtain the op-timal polynomials over each sub-domain. This allows us convert the aim of the approximation problem to minimizing the objective function. Based on differ-ent partition structures, Voronoi tessellation and triangulation, we propose two distinct objective functions which are similar in form and geometric meaning, while the optimization proceeds of minimizing are completely different. For the situation when Voronoi diagram is adopted, the objective function is only associ-ated with points’ position, we provide the explicit formula of the gradient of the objective function, which makes an efficient gradient-based algorithm workable for the function minimization responding to the optimal distribution of Voronoi sites. When a triangulation divides the domain, the positions and connectivity of the triangulation both influence significantly the objective function, and we use another effective optimization algorithm, Newton’s method, which needs the information of gradient and Hessian of the objective function those have been deducted herein to attain the optimal solution.

It is worth to point out that our piecewise polynomials approximation method based on Voronoi tessellation and triangulation possess the strong ability of ap-proximating piecewise functions, which helps us to apply our method to image approximation. To demonstrate the efficacy of our new approach, we conduct several experiments for generating piecewise polynomials approximation of

ana-III

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ABSTRACT

lytic functions and color images, and the results show that the resultant partitions are as favorable as that most features of the target function are maintained.

Keywords: Voronoi tessellation¶triangulation; piecewise polynomials

approx-imation; image approxapprox-imation; optimization algorithm

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CONTENTS

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CONTENTS

4 Approximation by Piecewise Polynomials on Triangulation 29

4.1 Mathematical Model . . . 29

4.2 Solution Mechanism . . . 38

4.3 Experimental Results . . . 44

4.4 Comparison . . . 48

4.5 Time Statistics . . . 52

5 Conclusion and Future Work 55 5.1 Conclusion . . . 55

5.2 Future Work . . . 56

References 59 Publications and Work 63 Acknowledgement 65 Appendix A More Experimental Results 67

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