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, "E6*.EtTN#
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, Zdm7As]
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? "P@jr{zvMd
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? E76#xsyhF
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function z = zernfun(n,m,r,theta,nflag) -%g$~MZ?'
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. DUAI
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N OX
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% and angular frequency M, evaluated at positions (R,THETA) on the d
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% unit circle. N is a vector of positive integers (including 0), and p1blPBlp
% M is a vector with the same number of elements as N. Each element /3!c
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% k of M must be a positive integer, with possible values M(k) = -N(k) V*C%r:5 ,v
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, lDV}vuM<4
% and THETA is a vector of angles. R and THETA must have the same >,&@j,?']
% length. The output Z is a matrix with one column for every (N,M) SFiK_;
% pair, and one row for every (R,THETA) pair. v95O)cC:W
% bRhc8#kw)
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike k,kr7'Q
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 1c%ee$Q
% with delta(m,0) the Kronecker delta, is chosen so that the integral !L=RhMI
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, DMc H, _(
% and theta=0 to theta=2*pi) is unity. For the non-normalized ],3#[n[ m
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 3rUuRsXn
% .:nV^+)
% The Zernike functions are an orthogonal basis on the unit circle. \D<w:\P
% They are used in disciplines such as astronomy, optics, and /ta5d;@
% optometry to describe functions on a circular domain. 0<n*8t?A-
% PE\.J U
% The following table lists the first 15 Zernike functions. uDWxIP,m
% /M3UK
% n m Zernike function Normalization U=G}@Y
% -------------------------------------------------- E;vF
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% 0 0 1 1 ~:ldGfb|
% 1 1 r * cos(theta) 2 e0nr dM[i
% 1 -1 r * sin(theta) 2 ;
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% 2 -2 r^2 * cos(2*theta) sqrt(6) gl$ Ks+od
% 2 0 (2*r^2 - 1) sqrt(3) + bU*"5"
% 2 2 r^2 * sin(2*theta) sqrt(6) @}8~TbP
% 3 -3 r^3 * cos(3*theta) sqrt(8) G)S(a4
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) : =J^ "c
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ,+Bp>=pvs
% 3 3 r^3 * sin(3*theta) sqrt(8) Bw`7ND}&
% 4 -4 r^4 * cos(4*theta) sqrt(10) @|i
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% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .GM}3(1fX`
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) RY4b<i3
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /KCJ)0UU
% 4 4 r^4 * sin(4*theta) sqrt(10) bFv,.(h'
% -------------------------------------------------- ))<1"7D^^
% z/Kjz$l!
% Example 1: {=q$k=ib
% ui[E,W~
% % Display the Zernike function Z(n=5,m=1) @'ln)RT,
% x = -1:0.01:1; Tx|}ke~
% [X,Y] = meshgrid(x,x); - UMPt"o
% [theta,r] = cart2pol(X,Y); iYE7BUH=
% idx = r<=1; _Dv<
% z = nan(size(X)); |vI1C5e
% z(idx) = zernfun(5,1,r(idx),theta(idx)); s&