下面这个函数大家都不会陌生,计算zernike函数值的,并根据此可以还原出图像来, AJ)N?s-=  
我输入10阶的n、m,r,theta为38025*1向量,最后得到的z是29525*10阶的矩阵, aIklAj)=  
这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? drh,=M\F  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? s|-g)  
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function z = zernfun(n,m,r,theta,nflag) )C%S`d<%,  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ANXN.V  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 0{sYD*gK]  
%   and angular frequency M, evaluated at positions (R,THETA) on the uAv'%/  
%   unit circle.  N is a vector of positive integers (including 0), and !sav~dB)  
%   M is a vector with the same number of elements as N.  Each element >on'	y+  
%   k of M must be a positive integer, with possible values M(k) = -N(k) V; 1i/{  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, MFsW  
%   and THETA is a vector of angles.  R and THETA must have the same a\Dw*h?b~  
%   length.  The output Z is a matrix with one column for every (N,M) {#H'K*j{  
%   pair, and one row for every (R,THETA) pair. tnFhL&  
% !E9A=u{  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike c$~J7e6$  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), f}{Oj-:"CC  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral -ZBSkyMGy  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ?CZ*MMV  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized Pc=:j(  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. "Sd2VSLg  
% BnIZ+fg=  
%   The Zernike functions are an orthogonal basis on the unit circle. `&>CK`%Xu  
%   They are used in disciplines such as astronomy, optics, and m'5rzZP  
%   optometry to describe functions on a circular domain. J3AS"+]  
% 2jH&@g$cl;  
%   The following table lists the first 15 Zernike functions. $jL+15^N0+  
% 0A.9<&Lod  
%       n    m    Zernike function           Normalization VMV~K7%0  
%       -------------------------------------------------- bb"x^DtT  
%       0    0    1                                 1 a~O](/+p;  
%       1    1    r * cos(theta)                    2 y jY}o  
%       1   -1    r * sin(theta)                    2 JU RJN+)z  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) N="H
06t  
%       2    0    (2*r^2 - 1)                    sqrt(3) Rb_+C  
%       2    2    r^2 * sin(2*theta)             sqrt(6) I>45xVA  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) mY/x|)MmM  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) h/\/dp/tt  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) <!I^ xo[  
%       3    3    r^3 * sin(3*theta)             sqrt(8) vAo|o*  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) c9axzg
UA  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >}>cJh6  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Xsv^GmP+  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (vr
v-4  
%       4    4    r^4 * sin(4*theta)             sqrt(10) 	hPgDK.R'  
%       -------------------------------------------------- $_b^p=  
% ~Is-^k)y  
%   Example 1: * 2s(TW  
% ^%2S,3*0  
%       % Display the Zernike function Z(n=5,m=1) 6yPh0n  
%       x = -1:0.01:1; i`HXBq!|w  
%       [X,Y] = meshgrid(x,x); xgv&M:%D-  
%       [theta,r] = cart2pol(X,Y); oM)4""|  
%       idx = r<=1; yB,{:kq7D  
%       z = nan(size(X)); IL N0/eH  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); rdQ'#}Ix   
%       figure Vh;P,no#  
%       pcolor(x,x,z), shading interp O7GJg;>?  
%       axis square, colorbar y|[YEY	U)  
%       title('Zernike function Z_5^1(r,\theta)') O5 ?3nYHa  
% %!QY:[  
%   Example 2: </7_T<He.  
% :h60  
%       % Display the first 10 Zernike functions `]\:%+-  
%       x = -1:0.01:1; #\r5Q>  
%       [X,Y] = meshgrid(x,x); 0@*EwI  
%       [theta,r] = cart2pol(X,Y); _^cFdP)8|  
%       idx = r<=1; pG9qD2Cf  
%       z = nan(size(X));  R7-+@  
%       n = [0  1  1  2  2  2  3  3  3  3]; #ysSfM6  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; g7nqe~`{  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; Zi~-m]9U  
%       y = zernfun(n,m,r(idx),theta(idx)); u[2B0a  
%       figure('Units','normalized') k&8&