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, MBg[hu%
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, ecs 0iW-,
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? ISNL='%
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? T#-;>@a}
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function z = zernfun(n,m,r,theta,nflag) /_l\7MeI
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. =J]WVA,GqA
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ]w6Q? %'9
% and angular frequency M, evaluated at positions (R,THETA) on the .c-a$39
% unit circle. N is a vector of positive integers (including 0), and U)bv,{-q
% M is a vector with the same number of elements as N. Each element wUCxa>h'
% k of M must be a positive integer, with possible values M(k) = -N(k) \PE;R.v_:
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, IANSpWea?
% and THETA is a vector of angles. R and THETA must have the same T3P9
% length. The output Z is a matrix with one column for every (N,M) viAAb
% pair, and one row for every (R,THETA) pair. >E<ib[vK[
% 'M >m$cCMZ
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike FoK2h!_
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi),
.fl r
% with delta(m,0) the Kronecker delta, is chosen so that the integral 4`#Q
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 7v%c.
% and theta=0 to theta=2*pi) is unity. For the non-normalized -n05Z@7
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5&n{QE?Um
% D8Fi{?A#FV
% The Zernike functions are an orthogonal basis on the unit circle. ^MvuFA,C
% They are used in disciplines such as astronomy, optics, and aL;!BlU8v
% optometry to describe functions on a circular domain. Z71m(//*}
% Z#d#n!Lz
% The following table lists the first 15 Zernike functions. n6%`
% <R$ 2x_
% n m Zernike function Normalization Kb?{^\FiU
% -------------------------------------------------- @[3c1B6K
% 0 0 1 1 EhHxB
fAQ
% 1 1 r * cos(theta) 2 U0_^6zd_
% 1 -1 r * sin(theta) 2 3
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% 2 -2 r^2 * cos(2*theta) sqrt(6) ]X>yZec
% 2 0 (2*r^2 - 1) sqrt(3) Eu?z!
% 2 2 r^2 * sin(2*theta) sqrt(6) 0y9 b0G
% 3 -3 r^3 * cos(3*theta) sqrt(8) [
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% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) xN-,gT'!
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 5^ Qa8yA>7
% 3 3 r^3 * sin(3*theta) sqrt(8) rz "$zc.)
% 4 -4 r^4 * cos(4*theta) sqrt(10) 4 ThFC
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :k/Xt$`
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Ki;SONSV~|
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) p)IL(_X)
% 4 4 r^4 * sin(4*theta) sqrt(10) dDPQDIx
% -------------------------------------------------- G>V6{g2Q
% {.:$F3T
% Example 1: p
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% @z@%vr=vX
% % Display the Zernike function Z(n=5,m=1) y+(\:;y$7
% x = -1:0.01:1; n[ B~C
% [X,Y] = meshgrid(x,x); sT\:**
% [theta,r] = cart2pol(X,Y); [r/zBF-.
% idx = r<=1; 5BhR4+1J
% z = nan(size(X)); NHGTV$T`1
% z(idx) = zernfun(5,1,r(idx),theta(idx)); PE%$g\#?
% figure V"4Z9Qg}
% pcolor(x,x,z), shading interp Vx_33";S\
% axis square, colorbar @[n#-!i
% title('Zernike function Z_5^1(r,\theta)') UPh#YV 0/,
% K!-OUm5A
% Example 2: <gp?}Lk
% TLdlPBnr8
% % Display the first 10 Zernike functions 3"y 6|e/5
% x = -1:0.01:1; bHwEd%f
% [X,Y] = meshgrid(x,x); i5 rkP`)j
% [theta,r] = cart2pol(X,Y); \/NF??k,jk
% idx = r<=1; T D_@0Rd
% z = nan(size(X)); Q7s@,c!m_
% n = [0 1 1 2 2 2 3 3 3 3]; 5f-b>=02
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ~ nsb
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Gnkar[oa&
% y = zernfun(n,m,r(idx),theta(idx)); Kw
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% figure('Units','normalized') 5> x_G#W
% for k = 1:10 k +-w%
% z(idx) = y(:,k); `geHSx_
% subplot(4,7,Nplot(k)) }E
'r?N
% pcolor(x,x,z), shading interp ~G!JqdKJ0
% set(gca,'XTick',[],'YTick',[]) >@YefNX6
% axis square _;1{feR_
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ,;)ZF
% end &|hK79D
% ^xZh@e5
% See also ZERNPOL, ZERNFUN2. ;5Sdx5`_
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% Paul Fricker 11/13/2006 jJ,_-ui
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% Check and prepare the inputs:
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% ----------------------------- c3vb~l)
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) % MHb
error('zernfun:NMvectors','N and M must be vectors.') -=ZL(r
1
end b9.M'P\
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if length(n)~=length(m) ^J'_CA
error('zernfun:NMlength','N and M must be the same length.') )Z}AhX
end ,lyW'<~gA
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n = n(:); 8Lo#{`
m = m(:); {0zn~+
if any(mod(n-m,2)) 4.RQ3SoDa
error('zernfun:NMmultiplesof2', ... f-b],YE
'All N and M must differ by multiples of 2 (including 0).') !gsvF\XDM
end &.?XntI9O
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if any(m>n) T{YZ`[
error('zernfun:MlessthanN', ... * QgKo$IF
'Each M must be less than or equal to its corresponding N.') Uzu6>yT
end bF'rK'',
%`Re{%1;
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if any( r>1 | r<0 ) JAPr[O&
error('zernfun:Rlessthan1','All R must be between 0 and 1.') yIMqQSt79z
end #/)t]&n
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) wQkM:=t5
error('zernfun:RTHvector','R and THETA must be vectors.') @-N` W9
end *b~6 B M$
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r = r(:); nwRltK
theta = theta(:); f:T?oR>2
length_r = length(r); sDY~jP[Oa
if length_r~=length(theta) gq?:n.;TY
error('zernfun:RTHlength', ... TkbaoD
'The number of R- and THETA-values must be equal.') PNU(;&2<
end Szu s*YL7
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5j%G7.S\
% Check normalization: ,$P,x
% -------------------- Jd2.j?P=
if nargin==5 && ischar(nflag) jG5HW*>k0
isnorm = strcmpi(nflag,'norm'); 4w4B\Na>l
if ~isnorm *7BfK(9T
error('zernfun:normalization','Unrecognized normalization flag.') [}RoZB&I
end jN=<dq
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else 2z.ot'
isnorm = false; 2Xb,
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end ]S|FK>U[
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zM0NRERi
% Compute the Zernike Polynomials }[*'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% bp1AN9~
4ls:BO;k]
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% Determine the required powers of r: i;[y!U
% ----------------------------------- p 7?
m_abs = abs(m); G)3I+uxn
rpowers = []; M[uWX=
for j = 1:length(n) EeIDlm0o
rpowers = [rpowers m_abs(j):2:n(j)]; IRdt:B|@
end ~MpikBf
rpowers = unique(rpowers); J
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% Pre-compute the values of r raised to the required powers, K%^n.
% and compile them in a matrix: (!j#u)O
% ----------------------------- xU
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if rpowers(1)==0 ngY%T5-
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); /
)0hsQs
rpowern = cat(2,rpowern{:}); k[=qx{Osx%
rpowern = [ones(length_r,1) rpowern]; 8~=*\
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else c:R?da
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); XtF
m5\U
rpowern = cat(2,rpowern{:}); lame/B&nc
end U"oNJ8&%|
|<%!9Z
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% Compute the values of the polynomials: jDzQw>TX
% -------------------------------------- voWH.[n^_
y = zeros(length_r,length(n)); "kg`TJf=
for j = 1:length(n) #-hO\
QdC
s = 0:(n(j)-m_abs(j))/2; nHK(3Z4G
pows = n(j):-2:m_abs(j); Qm%F]nyy
for k = length(s):-1:1 yNu_>!Cp5
p = (1-2*mod(s(k),2))* ... *zfgO pK
prod(2:(n(j)-s(k)))/ ... P
rt}
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prod(2:s(k))/ ... .nV2n@SR
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ZWs
prod(2:((n(j)+m_abs(j))/2-s(k))); f^c+M~\JKj
idx = (pows(k)==rpowers); )U^=`* 7
y(:,j) = y(:,j) + p*rpowern(:,idx); Et>#&Nw8
end 3? {AGJ1
-(VJ,)8t2
if isnorm .Po"qoGy
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 0^;2
end |diI(2w
end L"_XWno
% END: Compute the Zernike Polynomials =KRM`_QShg
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7 WJ\nK
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ZsGvv]P
% Compute the Zernike functions: @SQsEq+A?\
% ------------------------------ gLiJ&H
idx_pos = m>0; Dc9uq5l
idx_neg = m<0; \0$+*ejz
'H1~Zhv
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z = y; 0zt]DCdY
if any(idx_pos) ,GbmL8P7Y
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); OV>&`puL
end &(F
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if any(idx_neg) 8f@}-
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); %Ymi,o>
end OvfluFu7
>7U/TVd&
G5ATR<0m
% EOF zernfun g?j)p y