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,
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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, ?R282l
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? hg4J2m
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? u]0!|Jd0
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function z = zernfun(n,m,r,theta,nflag) %]!?{U\*k
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. H(?e&Qkg
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N %;
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% and angular frequency M, evaluated at positions (R,THETA) on the soDfi-2o3
% unit circle. N is a vector of positive integers (including 0), and kR_E6Fl
% M is a vector with the same number of elements as N. Each element &*jixqzvn
% k of M must be a positive integer, with possible values M(k) = -N(k) 6# R;HbkO
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, v|>BDN@,6
% and THETA is a vector of angles. R and THETA must have the same f@L\E>t
% length. The output Z is a matrix with one column for every (N,M) LPMb0F}"5
% pair, and one row for every (R,THETA) pair. `!_? uT
% eiOi3q
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike \wTW?>oZ
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), yG4 MUf6
% with delta(m,0) the Kronecker delta, is chosen so that the integral WFXx70n
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, q4 $sc_0i
% and theta=0 to theta=2*pi) is unity. For the non-normalized I'P!,Y/>
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. |NXFla
% m8p4U-*j
% The Zernike functions are an orthogonal basis on the unit circle. 4@gl4&<h
% They are used in disciplines such as astronomy, optics, and CO7CNN
% optometry to describe functions on a circular domain. uQ-WTz|*
% X=\x&Wt
% The following table lists the first 15 Zernike functions. oUCVd}wH
% } cRi
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% n m Zernike function Normalization ga,A'Z
% -------------------------------------------------- L-SdQTx_
% 0 0 1 1 E|\3f(aF
% 1 1 r * cos(theta) 2 WGluZhRuT3
% 1 -1 r * sin(theta) 2 Xp.|.)Od
% 2 -2 r^2 * cos(2*theta) sqrt(6) []!tT-Gzy
% 2 0 (2*r^2 - 1) sqrt(3) gZ=)qT]Pj
% 2 2 r^2 * sin(2*theta) sqrt(6) 2zwuvgiZ
% 3 -3 r^3 * cos(3*theta) sqrt(8) v#w4{.8)
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ?! !;XW
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) MV7}
% 3 3 r^3 * sin(3*theta) sqrt(8) 0GF%~6
% 4 -4 r^4 * cos(4*theta) sqrt(10) 3KbUHSx
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) N IO;
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) UqY J#&MqY
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) P"U>tsHK:
% 4 4 r^4 * sin(4*theta) sqrt(10) A5`#Ot*3
% -------------------------------------------------- Gt?!E6^!
% _$~ex ~v
% Example 1: 1V#B]x:
% X*(gT1"t
% % Display the Zernike function Z(n=5,m=1) 'Wd3`4V$
% x = -1:0.01:1; 9(V=Ubj
% [X,Y] = meshgrid(x,x); }Z<D^Z~w
% [theta,r] = cart2pol(X,Y); (8+.#1!*
% idx = r<=1; 'cWlY3%t
% z = nan(size(X)); 8s\8`2=
% z(idx) = zernfun(5,1,r(idx),theta(idx)); PL9zNCr-[
% figure 9N`+ O
% pcolor(x,x,z), shading interp Fa Qu$q
% axis square, colorbar _gis+f/8h
% title('Zernike function Z_5^1(r,\theta)') Z:W')Nd(
% g9RzzE!
% Example 2: sqgD?:@J
% 9CgXc5
% % Display the first 10 Zernike functions =P@M&Yy'
% x = -1:0.01:1; ayB=|*Q"
% [X,Y] = meshgrid(x,x);
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% [theta,r] = cart2pol(X,Y); WMt&8W5
% idx = r<=1; ]0at2
% z = nan(size(X)); &6=TtTp"9
% n = [0 1 1 2 2 2 3 3 3 3]; XY&]T'A
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; (Q*2dd>
% Nplot = [4 10 12 16 18 20 22 24 26 28]; yHV^a0e7EH
% y = zernfun(n,m,r(idx),theta(idx)); /1s 9;'I
% figure('Units','normalized') $_%2D3-;D
% for k = 1:10 eP-R""uPw
% z(idx) = y(:,k); |:J*>"sq
% subplot(4,7,Nplot(k)) ~)oWSo5ll
% pcolor(x,x,z), shading interp b7F3]W<`&