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 _4jRUsvjY
function z = zernfun(n,m,r,theta,nflag) <kr%ylhIu
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. L0O},O
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 5>'1[e45
% and angular frequency M, evaluated at positions (R,THETA) on the h tn?iLq
% unit circle. N is a vector of positive integers (including 0), and ~&Gw[Nd1
% M is a vector with the same number of elements as N. Each element %}asw/WiUa
% k of M must be a positive integer, with possible values M(k) = -N(k) LE:nmo
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, gLef6q{}
% and THETA is a vector of angles. R and THETA must have the same XVKR}I
% length. The output Z is a matrix with one column for every (N,M) lIj2w;$v
% pair, and one row for every (R,THETA) pair. P}+-))J
% %2)'dtPD~
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "e\:Cq>\
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), v&GBu
% with delta(m,0) the Kronecker delta, is chosen so that the integral |tU4(hC
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, } 1> i
% and theta=0 to theta=2*pi) is unity. For the non-normalized ."m2/Ks7
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. T>ds<MaLP
% \Q+<G-Kb.
% The Zernike functions are an orthogonal basis on the unit circle. D20n'>ddg
% They are used in disciplines such as astronomy, optics, and j7|r^
% optometry to describe functions on a circular domain. C4 &1M
% ;-1yG@KG
% The following table lists the first 15 Zernike functions. /M;A)z
% SDTX3A1
% n m Zernike function Normalization W c"f
% -------------------------------------------------- p Rn vd|
% 0 0 1 1 g6kVHxh-
% 1 1 r * cos(theta) 2 QDg\GA8|
% 1 -1 r * sin(theta) 2 %usy`4
2
% 2 -2 r^2 * cos(2*theta) sqrt(6) ]_yk,}88d
% 2 0 (2*r^2 - 1) sqrt(3) eVZ/3o
% 2 2 r^2 * sin(2*theta) sqrt(6) TrHz(no
% 3 -3 r^3 * cos(3*theta) sqrt(8) n3t0Qc
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) b[3K:ot+
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Ih]'OaE
% 3 3 r^3 * sin(3*theta) sqrt(8) Jm|eZDp
% 4 -4 r^4 * cos(4*theta) sqrt(10) 4$oX,Q`#
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) a~_5N&~pi
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) -$#'
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u[_~ !y
% 4 4 r^4 * sin(4*theta) sqrt(10)
9I:H=5c
% -------------------------------------------------- :6 ?&L
% Pd@y+|
% Example 1: e{~s\G8g
% p
xrd D7
% % Display the Zernike function Z(n=5,m=1) > !thxG/_
% x = -1:0.01:1; zice0({iJ
% [X,Y] = meshgrid(x,x); ei>8{v&g
% [theta,r] = cart2pol(X,Y); xG05OqKpE
% idx = r<=1; gu[3L
% z = nan(size(X)); &