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, OTFu4"]M
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, \%4+mgiD
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? zC>(!fJqq
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? [2j(\vC!
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function z = zernfun(n,m,r,theta,nflag) HI?~t|[y
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. %Pvb>U(Xs
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N PS<tS_.
% and angular frequency M, evaluated at positions (R,THETA) on the C2,cyhr
% unit circle. N is a vector of positive integers (including 0), and Mp@(/
% M is a vector with the same number of elements as N. Each element vM3|Ti>a'
% k of M must be a positive integer, with possible values M(k) = -N(k) Ynh4oWUp
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, (XIq?c1T
% and THETA is a vector of angles. R and THETA must have the same Sdu@!<?B
% length. The output Z is a matrix with one column for every (N,M) W2.1xNWO
% pair, and one row for every (R,THETA) pair. )
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% AFhG{G'W
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike <n~g+ps
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 2'^OtM,
% with delta(m,0) the Kronecker delta, is chosen so that the integral BRok 89
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, (lck6v?h
% and theta=0 to theta=2*pi) is unity. For the non-normalized #&8pp8wd,}
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ]A<u eM
% _.8]7f`*Gc
% The Zernike functions are an orthogonal basis on the unit circle. PH4bM
% They are used in disciplines such as astronomy, optics, and ]3#
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% optometry to describe functions on a circular domain. Hr,gV2n
% 4y}a,
% The following table lists the first 15 Zernike functions. \,#4+&4b
% nhxd
% n m Zernike function Normalization o?hw2-mH
% -------------------------------------------------- |/Q. "d
% 0 0 1 1 ~4X!8b_
% 1 1 r * cos(theta) 2 J2k'Ke97o
% 1 -1 r * sin(theta) 2 NeZYchR
% 2 -2 r^2 * cos(2*theta) sqrt(6) }~,cCtg:o
% 2 0 (2*r^2 - 1) sqrt(3) $mg h.3z0
% 2 2 r^2 * sin(2*theta) sqrt(6) z)y(31K<1
% 3 -3 r^3 * cos(3*theta) sqrt(8) \hD
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% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) p~;z"Z
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) pC.P
% 3 3 r^3 * sin(3*theta) sqrt(8) 2<. /HH*f
% 4 -4 r^4 * cos(4*theta) sqrt(10) /@}# KP=
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Us~wv"L=UX
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) zyn =Xv@p
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6]A\8Ty
% 4 4 r^4 * sin(4*theta) sqrt(10) | BWK"G
% -------------------------------------------------- ' g!_Flk
% LO*a>9LI
% Example 1: T
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% =` >Nfa+,
% % Display the Zernike function Z(n=5,m=1) bD[W~ku
% x = -1:0.01:1; (=B7_jrl
% [X,Y] = meshgrid(x,x); .{ L m
% [theta,r] = cart2pol(X,Y); c@{^3V##T
% idx = r<=1; KFG^vmrn
% z = nan(size(X)); 3>3ZfFC
% z(idx) = zernfun(5,1,r(idx),theta(idx)); TK?N^ly
% figure `X03Q[:q"[
% pcolor(x,x,z), shading interp *jSc&{s~
% axis square, colorbar S5vMP
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% title('Zernike function Z_5^1(r,\theta)') 2>$L>2$
% (:k`wh&
% Example 2: QN5N hs
% RwHXn]1
% % Display the first 10 Zernike functions BrmFwXLP"
% x = -1:0.01:1; ?^GsR[-x
% [X,Y] = meshgrid(x,x); XE%6c3s
% [theta,r] = cart2pol(X,Y); Z+Zh;Ms
% idx = r<=1; rxA)&