Two Kinds Of Numerical Differential Algorithms And Their Applications In Images Processing

Numerical differentiation,in essence,is to use the measured value of a function at some discrete points to obtain its approximate derivative through some calculation method,which is a typical ill-posed problem.In general,the measurement data will inevitably contain error,which makes the calculation error when calculating the approximate derivative can be arbitrarily large.In order to overcome the instability of numerical differential problems,especially the ill-posed problems,a stabilization algorithm must be introduced to solve the ill-posed problems.That is,how to construct a stable differential algorithm has become the key content of numerical differential problems.The common stabilization algorithms include integral operator method,regularization method,finite difference method,polishing method and so on.In this paper,the method of integral operator and regularization in frequency domain are introduced to solve numerical differential problems,and the construction,error analysis and numerical implementation of the two algorithms are given.The integral operator method is mainly a generalization based on Lanczos integral method,that is,using the method of undetermined coefficients to construct the integral operator format of the first or arbitrary derivative,and also gives the strategy of selecting the step size in the integral operator and the error analysis.In the regularization method in the frequency domain,several(class)methods for obtaining the modified "kernel" are given based on the Fourier transform,and an error estimate and numerical realization before and after the modified "kernel" are given.The numerical validity and feasibility of these two algorithms are illustrated by numerical experiments.In addition,this paper uses the first-order Lanczos integral operator to make a small improvement on the gradient amplitude of Canny edge detection algorithm,which is applied to the edge detection in image processing,and implemented numerically by MATLAB.The experimental results show that the improved method proposed in this paper is feasible.