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 mGQgy[gX
function z = zernfun(n,m,r,theta,nflag) Tl#Jf3XY}
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. +s6wF{
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1MtvnPY
% and angular frequency M, evaluated at positions (R,THETA) on the -DO*,Eecv
% unit circle. N is a vector of positive integers (including 0), and 7k<4/|CQ{
% M is a vector with the same number of elements as N. Each element dVDQ^O&
% k of M must be a positive integer, with possible values M(k) = -N(k) kT(}>=]g
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, K>k MKd1
% and THETA is a vector of angles. R and THETA must have the same &`a$n2ycy
% length. The output Z is a matrix with one column for every (N,M) SL;\S74
% pair, and one row for every (R,THETA) pair. Z\=].[,w4
% jafq(t
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike wz*QB6QtU
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), H=vrF - #
% with delta(m,0) the Kronecker delta, is chosen so that the integral Lw=.LN
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, q Yg4H|6
% and theta=0 to theta=2*pi) is unity. For the non-normalized (89NK]2x
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. b$sw`Rsw
% k_9tz}Z
% The Zernike functions are an orthogonal basis on the unit circle. ZQvpkO7}M
% They are used in disciplines such as astronomy, optics, and YyX/:1 sg>
% optometry to describe functions on a circular domain. '676\2.
% l`2X'sw[/
% The following table lists the first 15 Zernike functions. eNlE]W,=
% 6 ^X$;
% n m Zernike function Normalization 5/Ng!bW
% -------------------------------------------------- oZ1#.o{
% 0 0 1 1 r}i<cyL
% 1 1 r * cos(theta) 2 %/dYSC
% 1 -1 r * sin(theta) 2 }>JFO:v&
% 2 -2 r^2 * cos(2*theta) sqrt(6) D4yJ:ATO&
% 2 0 (2*r^2 - 1) sqrt(3) [y
y D-
% 2 2 r^2 * sin(2*theta) sqrt(6) TB] %?L:
% 3 -3 r^3 * cos(3*theta) sqrt(8) JMu|$"o&{
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Q? a&