非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 - K"L6m|
function z = zernfun(n,m,r,theta,nflag) M?<iQxtyb}
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. mq(K_
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N hYA1N&yz@
% and angular frequency M, evaluated at positions (R,THETA) on the cg_tJ^vrY
% unit circle. N is a vector of positive integers (including 0), and !c0x^,iE
% M is a vector with the same number of elements as N. Each element \<y|[
% k of M must be a positive integer, with possible values M(k) = -N(k) >}C:EnECy
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, muBl~6_mb2
% and THETA is a vector of angles. R and THETA must have the same 1Mx2%
% length. The output Z is a matrix with one column for every (N,M) hv#LKyp%
% pair, and one row for every (R,THETA) pair. vS:=%@c>ta
% qC=ZH#
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike e(OKE7
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), uKJo5%>
% with delta(m,0) the Kronecker delta, is chosen so that the integral $bBUL C
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 2$2@?]|?
% and theta=0 to theta=2*pi) is unity. For the non-normalized zP@\rZ @4
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. P8w56
% jd"YaZOQ
% The Zernike functions are an orthogonal basis on the unit circle. V,\}|_GY
% They are used in disciplines such as astronomy, optics, and \[8uE,=|
% optometry to describe functions on a circular domain. An,TunX
% DGz}d,ie
% The following table lists the first 15 Zernike functions. Lm0q/d2|\X
% bIk4?S
% n m Zernike function Normalization 63t'|9^5
% -------------------------------------------------- V4W(>g
% 0 0 1 1 S3QX{5t\
% 1 1 r * cos(theta) 2 DIhV;[\
% 1 -1 r * sin(theta) 2 /R(
.7 N
% 2 -2 r^2 * cos(2*theta) sqrt(6) w2`JFxQ^x
% 2 0 (2*r^2 - 1) sqrt(3) _?bF;R
% 2 2 r^2 * sin(2*theta) sqrt(6) {t:*Xu
% 3 -3 r^3 * cos(3*theta) sqrt(8) O\@0o|NM
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Tv%
Z|%*
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) JiXN"s^mcb
% 3 3 r^3 * sin(3*theta) sqrt(8) Z%SDN"+'g
% 4 -4 r^4 * cos(4*theta) sqrt(10) 9/R=_y-
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 3#F"UG2,_
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) [W dxMU
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) wNh\pWA
% 4 4 r^4 * sin(4*theta) sqrt(10) sd*NY
% -------------------------------------------------- w'mn O'%
% [LbCG
% Example 1: wc}4:~
% Oe k$f,J-
% % Display the Zernike function Z(n=5,m=1) aLQ]2m
% x = -1:0.01:1; xP'"!d4^i
% [X,Y] = meshgrid(x,x);
g\a q#QV
% [theta,r] = cart2pol(X,Y); &