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, 2-~a
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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, !/947Rn
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? ^"1TPd|
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Wdo#?@m
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function z = zernfun(n,m,r,theta,nflag) 9SY(EL
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. i`+B4I8[
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N T j`y J!0
% and angular frequency M, evaluated at positions (R,THETA) on the gA_krK,Z
% unit circle. N is a vector of positive integers (including 0), and s|Zx(.EP
% M is a vector with the same number of elements as N. Each element U h.Sc:trA
% k of M must be a positive integer, with possible values M(k) = -N(k) uyFn}y62
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, sMH#BCC
% and THETA is a vector of angles. R and THETA must have the same @&5 A&(
% length. The output Z is a matrix with one column for every (N,M) Ivsb<qzG
% pair, and one row for every (R,THETA) pair. PRD_!VOW
% ;`kWpM;
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 2/@D7>F&g
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ;-Ss# &
% with delta(m,0) the Kronecker delta, is chosen so that the integral l)Zs-V!M^\
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, J='W+=N
% and theta=0 to theta=2*pi) is unity. For the non-normalized "x&3Z@q7
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. kg3ppt
% L~dC(J)@ZI
% The Zernike functions are an orthogonal basis on the unit circle. |z~LzSJv
% They are used in disciplines such as astronomy, optics, and kM!V.e[g
% optometry to describe functions on a circular domain. k(vPg,X>m
% |) Pi6Y
% The following table lists the first 15 Zernike functions. W/r^ugDV
% (S oo<.9~
% n m Zernike function Normalization /BpxKh2p
% -------------------------------------------------- Zn&k[?;Al
% 0 0 1 1 m"4B!S&Fc(
% 1 1 r * cos(theta) 2 }E; F)=E
% 1 -1 r * sin(theta) 2 S$eDnw~$
% 2 -2 r^2 * cos(2*theta) sqrt(6) DZe}y^F
% 2 0 (2*r^2 - 1) sqrt(3) F}U5d^!2
% 2 2 r^2 * sin(2*theta) sqrt(6) A62<]R)n
% 3 -3 r^3 * cos(3*theta) sqrt(8) ]>Si0%
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ''S&e
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 5h8o4
% 3 3 r^3 * sin(3*theta) sqrt(8) Z)&D`RCf
% 4 -4 r^4 * cos(4*theta) sqrt(10) g_w&"=.jBq
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `tE^jqrke5
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Fk1.iRVzi
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >|3a
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% 4 4 r^4 * sin(4*theta) sqrt(10) SMoz:J*Q(
% -------------------------------------------------- D|_V<'
% NP/>H9Q2%
% Example 1: %6ub3PLw8
% 7
({=*
% % Display the Zernike function Z(n=5,m=1) ++8_fgM
% x = -1:0.01:1; F98i*K`"
% [X,Y] = meshgrid(x,x); Y)XvlfJ,h?
% [theta,r] = cart2pol(X,Y); Pl+xH%U+?
% idx = r<=1; j'GtgT
% z = nan(size(X)); n.hElgkUOr
% z(idx) = zernfun(5,1,r(idx),theta(idx)); kIvvEh<L=
% figure phP>3f.T
% pcolor(x,x,z), shading interp !QEL"iJ6M'
% axis square, colorbar f:xWu-
% title('Zernike function Z_5^1(r,\theta)') #Qbl=o4
% NQ9Ojj{#
% Example 2: E'c%d[:H,
% -2i\G .,J
% % Display the first 10 Zernike functions }+RB=#~o
% x = -1:0.01:1; #
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% [X,Y] = meshgrid(x,x); b |7ja_
% [theta,r] = cart2pol(X,Y); lIf(6nm@
% idx = r<=1; ?4[H]BK
% z = nan(size(X)); 4vdNMV~
% n = [0 1 1 2 2 2 3 3 3 3]; dDtFx2(R
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; pCU*@c!
% Nplot = [4 10 12 16 18 20 22 24 26 28]; SwH2$:f
% y = zernfun(n,m,r(idx),theta(idx)); #Hu~}zy
% figure('Units','normalized') PlCc8Zy
% for k = 1:10 :reTJQwr
% z(idx) = y(:,k); vR>o}%`
% subplot(4,7,Nplot(k)) v6uxxsI>Hm
% pcolor(x,x,z), shading interp )1F<6R
% set(gca,'XTick',[],'YTick',[]) ;sPzOS9
% axis square *'R#4@wmP
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #cKqnk
% end [!"XcFY:a
% J]pa4C`
% See also ZERNPOL, ZERNFUN2. }
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% Paul Fricker 11/13/2006 oRq!=eUu_
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% Check and prepare the inputs: F#V q#|_)>
% ----------------------------- Cg!^S(U4
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Bw<rp-
error('zernfun:NMvectors','N and M must be vectors.') Qv#]81i(1
end >SCGK_Cr2
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if length(n)~=length(m) }zMf7<C
error('zernfun:NMlength','N and M must be the same length.') {'bip`U.
end >HTbegi
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n = n(:); g]$
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m = m(:); |)U|:F/{@
if any(mod(n-m,2)) 6*XM7'n
error('zernfun:NMmultiplesof2', ... Q9>U1]\
'All N and M must differ by multiples of 2 (including 0).') h##WA=1QZ
end py<_HyJ
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if any(m>n) 3$p#;a:=n
error('zernfun:MlessthanN', ... (ku5WWJ
'Each M must be less than or equal to its corresponding N.') ,x_Z JL
end ;b%{ilx:
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if any( r>1 | r<0 ) D`o<,Y
error('zernfun:Rlessthan1','All R must be between 0 and 1.') rT7^-B*
end |V&G81sM
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Cmd329AH
error('zernfun:RTHvector','R and THETA must be vectors.') 46,j9x
end KL3<Iz]
r%=[},JQ
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r = r(:);
<!'M} s
theta = theta(:); mJ
length_r = length(r); <'m6^]:
if length_r~=length(theta) :nGMtF
error('zernfun:RTHlength', ... 2qj{n+
'The number of R- and THETA-values must be equal.') LtKB v4
end x8N|($1
-l*g~7|j
TT}]wZ
% Check normalization: \M+L3*W
% -------------------- y{=NP
if nargin==5 && ischar(nflag) /oP^'""@je
isnorm = strcmpi(nflag,'norm'); |:q/Dt@
if ~isnorm !,&yyx.
error('zernfun:normalization','Unrecognized normalization flag.') y!Cc?$]_Y
end ~!:0iFE&H
else `rFAZcEj%
isnorm = false; hKe30#:v
end j
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k`YYZt]@
% Compute the Zernike Polynomials W)=%mdxW0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tJZc/]%`H
Dz&+PES_k
}KB[B
% Determine the required powers of r: s3 QEi^~
% ----------------------------------- Z[L5 ;
m_abs = abs(m); 2[R$RpA_
rpowers = []; :,UN8L "
for j = 1:length(n) ?9KGnOVu
rpowers = [rpowers m_abs(j):2:n(j)]; Z!{UWegun
end n^9 ?~
rpowers = unique(rpowers); *"9<TSU%m
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% Pre-compute the values of r raised to the required powers, hu@7?f_"L/
% and compile them in a matrix: M?@pN<|
% ----------------------------- ;=;JfNnbm
if rpowers(1)==0 b:MG@Hxc
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 7TWNB{
K_
rpowern = cat(2,rpowern{:}); <Oz66bTze
rpowern = [ones(length_r,1) rpowern]; 2@i;_3sv
else +x1/-J8_sg
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); =uV,bG5V1
rpowern = cat(2,rpowern{:}); i/qTFQst
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end w]!0<
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% Compute the values of the polynomials: ^R_e
% -------------------------------------- HnZPw&*
y = zeros(length_r,length(n)); );JJ2Jlkd
for j = 1:length(n) ")`S0n5e
s = 0:(n(j)-m_abs(j))/2; m_lrPY-
pows = n(j):-2:m_abs(j); +Ui_ O
for k = length(s):-1:1 Es8#]'Rk
p = (1-2*mod(s(k),2))* ... T9jw X:n
prod(2:(n(j)-s(k)))/ ... '044Vm;/
prod(2:s(k))/ ... #Z9L_gDp
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 2$0)?ZC?=
prod(2:((n(j)+m_abs(j))/2-s(k))); Zf:]Gq1
idx = (pows(k)==rpowers); A,XfD} +:Z
y(:,j) = y(:,j) + p*rpowern(:,idx); 7
.+al)hl
end xFb3O|TC
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if isnorm &Ocu#Cb
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); >)c9|e=8
end bkv/I{C>?
end u{C)qb5Pu
% END: Compute the Zernike Polynomials ~@9zil41
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ->oz#
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% Compute the Zernike functions: -!dQ)UEP
% ------------------------------ ,"G\f1
idx_pos = m>0; uMiyq<
idx_neg = m<0; BKb<2
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z = y; {;n0/
if any(idx_pos) >t#\&|9I
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); " $)yB
end ?qT(3C9p
if any(idx_neg) -c={+z "
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); A*0*sZ0
end W"qL-KW
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% EOF zernfun VS`Z_Xn