Curve Approximation Algorithm Based On Parameterization

Curve approximation is a common and important problem in digital image processing and digital signal processing.It is widely used in image denoising,image restoration,image compression,et al.Most curve approximation problems can be transformed into a series of nonlinear equation solving problems.This thesis studies the curve approximation algorithm based on parameterization and its application.The main contents include:(1)Arctangent curve approximation algorithm based on two parametersA curve approximation algorithm based on two parameters is proposed to solve the problem of low efficiency of arctan(x)2 curve approximation.Different from other approximation algorithms,this algorithm introduces an extra parameter α.By modifying the parameter value of α,this algorithm can flexibly generate approximation errors of different precision to meet the requirements of various application scenarios.By comparing with the results in the literature,the experiment shows that the proposed algorithm has higher approximation performance,and the good approximation effect of the algorithm is expected to be used in image denoising and other aspects.(2)Sinc curve approximation algorithm based on reparameterizationTo solve the poor approximation of Sinc curve,a new trigonometric boundary is found based on the reparameterization method.Numerical experiments show that the boundary obtained by this algorithm can approximate the Sinc curve more accurately than the conclusions proposed in other references,and has a good approximation effect,which can be applied in the field of digital signal processing.(3)Progressive root-finding based on reparameterizationTo solve the problems of low efficiency in the existing root-finding algorithms and the clipping method can not deal with the non-polynomial function,a progressive algorithm based on reparameterization was proposed.The real roots of the equations are iteratively approximated by searching for reparameterized functions by interpolation,the algorithm can reach the second order of convergence by adding a function calculation.In addition,based on the idea of the algorithm,this paper also extends it to the solution of binary equations.A large number of numerical experiments show that compared with the Newton-like method,polynomial clipping method,and previous asymptotic method,the algorithm has better performance in approximation error,computational efficiency and robustness,and can be applied to CAD modeling systems.