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 :x5O1Zn/t
function z = zernfun(n,m,r,theta,nflag) IC8%E3
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ypGt6t(;
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N =-r); d
% and angular frequency M, evaluated at positions (R,THETA) on the /d!
% unit circle. N is a vector of positive integers (including 0), and fE)o-q6Z
% M is a vector with the same number of elements as N. Each element XpkOC o 02
% k of M must be a positive integer, with possible values M(k) = -N(k) ~b
X~_\
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, \o72VHG66
% and THETA is a vector of angles. R and THETA must have the same mvTp,^1
% length. The output Z is a matrix with one column for every (N,M) 5a@9PX^.J
% pair, and one row for every (R,THETA) pair. E^c*x^
% 9;\mq'v%
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike r_,;[+!
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), X6(s][Wn
% with delta(m,0) the Kronecker delta, is chosen so that the integral )[M:#;,L
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3iX\):4
% and theta=0 to theta=2*pi) is unity. For the non-normalized |6^%_kO!|
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. cPAR.h,b?
% }a9G,@:k
% The Zernike functions are an orthogonal basis on the unit circle. P,3w
b
% They are used in disciplines such as astronomy, optics, and |#SZdXg
% optometry to describe functions on a circular domain. wYV>Qd
Z
% aHYISjZ]>
% The following table lists the first 15 Zernike functions. [.Kp/,JY
% IFS_DW
% n m Zernike function Normalization y5O &9Ckw
% -------------------------------------------------- Ar,n=obG
% 0 0 1 1 f.66N9BHL,
% 1 1 r * cos(theta) 2 7OG:G z+)x
% 1 -1 r * sin(theta) 2
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% 2 -2 r^2 * cos(2*theta) sqrt(6) rMZuiRz*
% 2 0 (2*r^2 - 1) sqrt(3) XQfmD;U
% 2 2 r^2 * sin(2*theta) sqrt(6) <;~u@^>
% 3 -3 r^3 * cos(3*theta) sqrt(8) b8YdONdy
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ~7*2Jp'
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Q@NFfJJ
% 3 3 r^3 * sin(3*theta) sqrt(8) o59$vX,
% 4 -4 r^4 * cos(4*theta) sqrt(10) `JPkho
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) V?wV*]c
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 1 ^= QIX
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f38e(Q];m
% 4 4 r^4 * sin(4*theta) sqrt(10) d(ypFd9z
% -------------------------------------------------- 3/Z>W|w#w
% +`{OOp=
% Example 1: a@qc?
% 2u!&Te(!9
% % Display the Zernike function Z(n=5,m=1) v0E6i!D/
% x = -1:0.01:1; DC-d@N+
% [X,Y] = meshgrid(x,x);
#C?M-
% [theta,r] = cart2pol(X,Y); 66" 6>
% idx = r<=1; $8HiX6r
% z = nan(size(X)); %Pt){9b
% z(idx) = zernfun(5,1,r(idx),theta(idx)); SUUN_w~
% figure 9:VUtx#}2
% pcolor(x,x,z), shading interp xb9+- {<J
% axis square, colorbar : N$-SV
% title('Zernike function Z_5^1(r,\theta)') >-<iY4|[d
% 1TGRIe)
% Example 2:
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% 1,bE[_
% % Display the first 10 Zernike functions [?KGLUmTAI
% x = -1:0.01:1; "UNFB3
% [X,Y] = meshgrid(x,x); pb)8?1O|s
% [theta,r] = cart2pol(X,Y); SZHgXl3:
% idx = r<=1; b"N!#&O