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, RQaB_bg7
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, Vjv~RNGF
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? 4r'QP .h
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? f9+J}
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function z = zernfun(n,m,r,theta,nflag) Z*AT &7
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. +[LG>
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N &E{CQ#k
% and angular frequency M, evaluated at positions (R,THETA) on the uL\b*rI
% unit circle. N is a vector of positive integers (including 0), and Xv1SRP#
% M is a vector with the same number of elements as N. Each element [r[IWy(}
% k of M must be a positive integer, with possible values M(k) = -N(k) & XS2q0-x
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, }rWEa^
% and THETA is a vector of angles. R and THETA must have the same <)hA?3J
% length. The output Z is a matrix with one column for every (N,M) 3K8#,TK3
% pair, and one row for every (R,THETA) pair. +"sjkdum1
% 4trP*u,4
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike HDmjt+3&n
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 3YKJN4
% with delta(m,0) the Kronecker delta, is chosen so that the integral pUGFQ."\
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, \&{a/e2:S
% and theta=0 to theta=2*pi) is unity. For the non-normalized RA%=_wPD
+
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (-<s[VnXP
% [`F}<L."
% The Zernike functions are an orthogonal basis on the unit circle. ?L%BD7
% They are used in disciplines such as astronomy, optics, and \wJ2>Q
% optometry to describe functions on a circular domain. 9.:]eL
% Yk;-]qi7
% The following table lists the first 15 Zernike functions. =:w]EpH"
% R6(sWN-
% n m Zernike function Normalization 1*x;jO>Hk
% -------------------------------------------------- tzTnFV
% 0 0 1 1 @r.w+E=
% 1 1 r * cos(theta) 2 R m&^[mv
% 1 -1 r * sin(theta) 2 uwL^Tq}Yh
% 2 -2 r^2 * cos(2*theta) sqrt(6) Q)/V>QW
% 2 0 (2*r^2 - 1) sqrt(3) m1 tYDZ"i
% 2 2 r^2 * sin(2*theta) sqrt(6) {^5LolCCH
% 3 -3 r^3 * cos(3*theta) sqrt(8) Io(*_3V)B
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 6UAn#d9
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) gwA+%]
% 3 3 r^3 * sin(3*theta) sqrt(8) EZ"n3#/
% 4 -4 r^4 * cos(4*theta) sqrt(10) +jEtu[ ;
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "jUM}@q5
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) {Vw\#/,
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) b p[wr
% 4 4 r^4 * sin(4*theta) sqrt(10) 1*aO2dOq
% -------------------------------------------------- a-cLy*W,~
% Daw;6f:
% Example 1: r_x|2 AoO
% s|`Z V^R
% % Display the Zernike function Z(n=5,m=1) $_ BoG
% x = -1:0.01:1; xg;o<y KF
% [X,Y] = meshgrid(x,x); PM?F;mj
% [theta,r] = cart2pol(X,Y); <Jf[N=
% idx = r<=1; QX`T-)T e
% z = nan(size(X)); %W(/W9B$/F
% z(idx) = zernfun(5,1,r(idx),theta(idx)); X([8TR
% figure ^t$xR_
% pcolor(x,x,z), shading interp j; MQ_?"iN
% axis square, colorbar ~pC\"LU`
% title('Zernike function Z_5^1(r,\theta)') sTSNu+
% 1_jd1UT
% Example 2: vG{lxPIj
% x 8/I"!gI
% % Display the first 10 Zernike functions XkEJ_;:
% x = -1:0.01:1; $(r/N"6)O2
% [X,Y] = meshgrid(x,x); ^.pd'
% [theta,r] = cart2pol(X,Y); ^[6S]Ft(
% idx = r<=1; S;8gX1Uf
% z = nan(size(X)); O
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% n = [0 1 1 2 2 2 3 3 3 3]; jU9\BYUg
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 4Zn" K}q
% Nplot = [4 10 12 16 18 20 22 24 26 28]; mm:g9j
% y = zernfun(n,m,r(idx),theta(idx)); E*Z # fa
% figure('Units','normalized') _C%:AFPP>
% for k = 1:10 3F gTM(
% z(idx) = y(:,k); `3e>JIl"0
% subplot(4,7,Nplot(k)) PB(q9gf"1}
% pcolor(x,x,z), shading interp %B~@wcI)W
% set(gca,'XTick',[],'YTick',[]) Bnfp_SM
% axis square RYyM;<9F
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) a/{M2
% end >]}c,4D(
% ^2a 63_
% See also ZERNPOL, ZERNFUN2.
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% Paul Fricker 11/13/2006 {g]Mx|5Q
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% Check and prepare the inputs: V^%P}RFMc
% ----------------------------- od-yVE&
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) g2%fla7r
error('zernfun:NMvectors','N and M must be vectors.') V%Ww;Ca]I
end "j/jhe6
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if length(n)~=length(m) HZ[.,DuW
error('zernfun:NMlength','N and M must be the same length.') gZ>)
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end xl ]1TB@
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n = n(:); e|d~&Bk0
m = m(:); phi9/tO\u
if any(mod(n-m,2)) a797'{j#PI
error('zernfun:NMmultiplesof2', ... Ih<.2
'All N and M must differ by multiples of 2 (including 0).') 6hiWgbE
end *6aIDFNl
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if any(m>n) 'Bwv-J
error('zernfun:MlessthanN', ... e0ULr!p
'Each M must be less than or equal to its corresponding N.') ~0Z.,p_
end ugzrG0=lx
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if any( r>1 | r<0 ) l.xKv$uOGR
error('zernfun:Rlessthan1','All R must be between 0 and 1.') O?t49=uB}
end +-:o+S`q~
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Kz"&:&R"
error('zernfun:RTHvector','R and THETA must be vectors.') 8l*h\p:Q
end X?.tj
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r = r(:); Ip|=NQL>
theta = theta(:); abw5Gz@Ag
length_r = length(r); vP%}XEF
if length_r~=length(theta) j@R"AP}
error('zernfun:RTHlength', ... DN;|?oNZ
'The number of R- and THETA-values must be equal.') :3[;9xCHj
end 5KTPlqm0qF
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% Check normalization: g?N^9B,$2
% -------------------- p"0Dl9
if nargin==5 && ischar(nflag) P~;1adi3
isnorm = strcmpi(nflag,'norm'); E:y^= Y
if ~isnorm
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error('zernfun:normalization','Unrecognized normalization flag.') I2WWhsNC
end q[(1zG%NbA
else <k 'zz:[c!
isnorm = false; z@?WhD
end j&[u$P*K
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #nq_R
% Compute the Zernike Polynomials ZgfhNI\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YjiMUi\V
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