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, ]IoS-)$Z/
我输入10阶的n、m,r,theta为38025*1向量,最后得到的z是29525*10阶的矩阵, ciXAyT cG
这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? JY_' d,O
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? lc'Jn$O@
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function z = zernfun(n,m,r,theta,nflag) !X,=RR`zT
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ME7JU|@Z
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N =6%0pu]0
% and angular frequency M, evaluated at positions (R,THETA) on the v8WoV*
% unit circle. N is a vector of positive integers (including 0), and TQ>1u
% M is a vector with the same number of elements as N. Each element @ 8SYV}0H
% k of M must be a positive integer, with possible values M(k) = -N(k) ,R]7{7$
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Karyipn}
% and THETA is a vector of angles. R and THETA must have the same IYrO;GQ
% length. The output Z is a matrix with one column for every (N,M) i.'f<z$<
% pair, and one row for every (R,THETA) pair. {j(,Q qB;f
% "%sW/ph
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike $w65/
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), x JepDCUJ>
% with delta(m,0) the Kronecker delta, is chosen so that the integral 3L;)asF
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, rA_e3L@v#[
% and theta=0 to theta=2*pi) is unity. For the non-normalized ;,F}!R
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 7.]xcJmt>'
% !e%#Zb
MIo
% The Zernike functions are an orthogonal basis on the unit circle. \K>6-0r|
% They are used in disciplines such as astronomy, optics, and /njN*rhx&Z
% optometry to describe functions on a circular domain.
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% W;o\}irep
% The following table lists the first 15 Zernike functions. :,cSEST
% )!OEa]
% n m Zernike function Normalization ty"k
% -------------------------------------------------- J
\G8g,@
% 0 0 1 1 z43 H]
% 1 1 r * cos(theta) 2 x2tx{Z
% 1 -1 r * sin(theta) 2 WJhI6lu
% 2 -2 r^2 * cos(2*theta) sqrt(6) 4sG^bZ,
% 2 0 (2*r^2 - 1) sqrt(3) qf'uXH
% 2 2 r^2 * sin(2*theta) sqrt(6) O! ;!amvz
% 3 -3 r^3 * cos(3*theta) sqrt(8) +nZx{d,wt
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 2"2b\b}my
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 5Rc
5/ m
% 3 3 r^3 * sin(3*theta) sqrt(8) 9GCxF`OB
% 4 -4 r^4 * cos(4*theta) sqrt(10) UW40Y3W0
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /#.6IV(
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) j'v2m 6/
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) E5Z,4B
% 4 4 r^4 * sin(4*theta) sqrt(10) eH75:`
% -------------------------------------------------- Xd{"+'29
% r`mfLA]d
% Example 1: k(V#{
YP
% yht_*7.lM
% % Display the Zernike function Z(n=5,m=1) MQLa+I,S4
% x = -1:0.01:1; w+[r$+z!k
% [X,Y] = meshgrid(x,x); )x8Izn
% [theta,r] = cart2pol(X,Y); nIdvff
% idx = r<=1; o-49o5:1
% z = nan(size(X)); 5a_1x|Fhi
% z(idx) = zernfun(5,1,r(idx),theta(idx)); <r_ldkZ
% figure J6=*F;x6E
% pcolor(x,x,z), shading interp E{1O<qO<