Fitting Curves with Fractional Implicit Polynomials: A PSO-Assisted Monomial Combination Optimization Framework

Implicit polynomial can efficiently represent the object contour for the curve fitting, and fractional implicit polynomial (FIP) is capable of describing complex objects at lower degree. However, both of IP and FIP based curve fitting methods suffer from the problem of insufficient fitting accuracy because of spurious zero sets. In this paper, we propose a particle swarm optimization (PSO)-assisted monomial combination optimization framework to fit curves based on FIP. In the proposed PSO-FIP framework, we adopt a monomial combination optimization strategy to remove redundant monomials and reduce spurious zero sets accordingly. And then, we solve the combinatorial explosion problem by using PSO in the monomial combination optimization. We further propose a geometric distance metric to serve as the fitness function in PSO which overcomes the trivial solution problem of algebraic distance. We conduct extensive experiments to validate the PSO-FIP framework and the results demonstrate that our approach can achieve higher accuracy with fewer coefficients in curve fitting.