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 8}| (0mC
function z = zernfun(n,m,r,theta,nflag) k|d+#u[Mj@
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. =odFmF
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N :*\P n!r
% and angular frequency M, evaluated at positions (R,THETA) on the x-3\Ls[I
% unit circle. N is a vector of positive integers (including 0), and ,zY$8y]
% M is a vector with the same number of elements as N. Each element i~J'% a<Qp
% k of M must be a positive integer, with possible values M(k) = -N(k) HyWCMK6b
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, PwLZkr@4^
% and THETA is a vector of angles. R and THETA must have the same M =r)I~
% length. The output Z is a matrix with one column for every (N,M) MFk5K
% pair, and one row for every (R,THETA) pair. V~5jfcd
% G'A R`"F
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ?5
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% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), [q #\D
% with delta(m,0) the Kronecker delta, is chosen so that the integral @sC`!Rmy'-
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, <e</m)j
% and theta=0 to theta=2*pi) is unity. For the non-normalized @I!0-OjL
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. k\GcHI-
% !Q0w\j h
% The Zernike functions are an orthogonal basis on the unit circle. ZzT9j~
% They are used in disciplines such as astronomy, optics, and j8lb~0JD
% optometry to describe functions on a circular domain. y_lU=(%Jd
% ;;N9>M?b
% The following table lists the first 15 Zernike functions. s,&Z=zt0R
% v^ VitLC
% n m Zernike function Normalization z ~/` 1
% -------------------------------------------------- v
z '&%(
% 0 0 1 1 W|63Ir67
% 1 1 r * cos(theta) 2 |_@>*Vmg
% 1 -1 r * sin(theta) 2 j+
0I-p
% 2 -2 r^2 * cos(2*theta) sqrt(6) b}TS0+TF
% 2 0 (2*r^2 - 1) sqrt(3) j HJ`,#
% 2 2 r^2 * sin(2*theta) sqrt(6) P\rg"
3
% 3 -3 r^3 * cos(3*theta) sqrt(8) Zba2d,8/
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) U|Ta4W`k\
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) `&c kZiq
% 3 3 r^3 * sin(3*theta) sqrt(8) {[?(9u7R
% 4 -4 r^4 * cos(4*theta) sqrt(10) q9r[$%G
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) i6Emhji
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) lp%pbx43s
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~%kkeh\j
% 4 4 r^4 * sin(4*theta) sqrt(10) H*'IK'O
% -------------------------------------------------- JO6)-U$7UG
% N~zdWnSZ@G
% Example 1: Od,qbU4O
% PP33i@G
% % Display the Zernike function Z(n=5,m=1) [~c|mOk
% x = -1:0.01:1; SbrecZ
% [X,Y] = meshgrid(x,x); o9yJf#-En
% [theta,r] = cart2pol(X,Y); z/2//mM
% idx = r<=1; '$]97b7G
% z = nan(size(X)); O) n~](sC\
% z(idx) = zernfun(5,1,r(idx),theta(idx)); y(yHt=r
% figure !9VY|&fHe
% pcolor(x,x,z), shading interp !Pfr,a
% axis square, colorbar 2B&