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, e_g7E+6
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, +?*,J=/
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? 2<fG= I8
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? %l,p />r
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function z = zernfun(n,m,r,theta,nflag) n:`> QY
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. `DC)U1
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N e}(ws~.
% and angular frequency M, evaluated at positions (R,THETA) on the TaG'?
% unit circle. N is a vector of positive integers (including 0), and 3VB{Qj
% M is a vector with the same number of elements as N. Each element )]n:y M
% k of M must be a positive integer, with possible values M(k) = -N(k) DWHl,w;[z`
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, zYYc#N/
% and THETA is a vector of angles. R and THETA must have the same ^&h|HO-5
% length. The output Z is a matrix with one column for every (N,M) |0B h
% pair, and one row for every (R,THETA) pair. wCkhE,#-_
% }7X85@jC
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike kE UfQLbn
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), p/cVQ
% with delta(m,0) the Kronecker delta, is chosen so that the integral C \H%4p1r
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, E{_p&FF
% and theta=0 to theta=2*pi) is unity. For the non-normalized 2y,NT|jp
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 7zgU>$i
% '?v.O}
% The Zernike functions are an orthogonal basis on the unit circle. hR[Qdu6r
% They are used in disciplines such as astronomy, optics, and 9-Qub+0o
% optometry to describe functions on a circular domain. W _yVVr
% + 3aAL&
% The following table lists the first 15 Zernike functions. 1
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% Sl
% n m Zernike function Normalization S3P;@Rm
% -------------------------------------------------- "So+
% 0 0 1 1 A>xFNem
% 1 1 r * cos(theta) 2 x
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2]~-
% 1 -1 r * sin(theta) 2 iU~oPp[e
% 2 -2 r^2 * cos(2*theta) sqrt(6) +smPR
% 2 0 (2*r^2 - 1) sqrt(3) g&\A1H
% 2 2 r^2 * sin(2*theta) sqrt(6) -wW%+wH
% 3 -3 r^3 * cos(3*theta) sqrt(8) n>+M4Zb
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) )<UNiC
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) hJkIFyQ{j
% 3 3 r^3 * sin(3*theta) sqrt(8) P,j)m\|
% 4 -4 r^4 * cos(4*theta) sqrt(10) /$%apci8
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <Ktx*(D
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 'eLO#1Ipf
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Z'/:
% 4 4 r^4 * sin(4*theta) sqrt(10) |*fGG?}
% -------------------------------------------------- WDP$w(M
% wZ0$ylEX
% Example 1: 54-sb~]
% y7u"a)T
% % Display the Zernike function Z(n=5,m=1) f}Mc2PQ-
% x = -1:0.01:1; (VI4kRj
% [X,Y] = meshgrid(x,x); Zyu4!
% [theta,r] = cart2pol(X,Y); 38tRb"3zP
% idx = r<=1; bsmZR(EnU
% z = nan(size(X)); G9 ;X=c
% z(idx) = zernfun(5,1,r(idx),theta(idx)); E"b+Q
% figure l7Zqk GG]
% pcolor(x,x,z), shading interp 'hf#Q9W5
% axis square, colorbar gH,^XZe
% title('Zernike function Z_5^1(r,\theta)') f2`[skNj
% ?.LS_e_0
% Example 2: JpcG5gX^B
% Ty}'A(U
% % Display the first 10 Zernike functions [GyW1-p33w
% x = -1:0.01:1; >KNiMW^V
% [X,Y] = meshgrid(x,x); /3Zo8.
% [theta,r] = cart2pol(X,Y); T[`o$j6
% idx = r<=1; QaH32(iH
% z = nan(size(X)); @dvlSqm)
% n = [0 1 1 2 2 2 3 3 3 3]; {dH87 nt
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; [1F.
% Nplot = [4 10 12 16 18 20 22 24 26 28]; %d c=QSL
% y = zernfun(n,m,r(idx),theta(idx)); etMQy6E\
% figure('Units','normalized') B36_OH
% for k = 1:10 l:-$ulAx
% z(idx) = y(:,k); Q_$aiE
% subplot(4,7,Nplot(k)) F/tGk9v
% pcolor(x,x,z), shading interp 5V':3o;D__
% set(gca,'XTick',[],'YTick',[]) C*a>B,H
% axis square tda#9i[pkH
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 9{RCh9
% end DI{VJ&n66
% $nUhM|It
% See also ZERNPOL, ZERNFUN2. p[2`H$A
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% Paul Fricker 11/13/2006 l+HmG< P
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% Check and prepare the inputs: 0/5
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% ----------------------------- 2w_[c.
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) B5{ wSr
error('zernfun:NMvectors','N and M must be vectors.') "Rr)1x7
end -N
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if length(n)~=length(m) mhVdsa
error('zernfun:NMlength','N and M must be the same length.')
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end 'i+j;.
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n = n(:); #J~
m = m(:); !k@(}CN_*
if any(mod(n-m,2)) v+Mi"ZAd
error('zernfun:NMmultiplesof2', ... _zt)c!
'All N and M must differ by multiples of 2 (including 0).') iga.B
end "'U+T:S
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if any(m>n) 3N c#6VI
error('zernfun:MlessthanN', ... Gf71udaa
'Each M must be less than or equal to its corresponding N.') ^% ZbjJ7|j
end #0$fZ
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if any( r>1 | r<0 ) XX&4OV,^%D
error('zernfun:Rlessthan1','All R must be between 0 and 1.') eFKF9m
end 8! eYax
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) MTXh-9DA
error('zernfun:RTHvector','R and THETA must be vectors.') 8k +^jj
end !aQb
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r = r(:); (;VlK#rnC
theta = theta(:); sbv2*fno5
length_r = length(r); | KtI:n4d
if length_r~=length(theta) XM1;
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error('zernfun:RTHlength', ... %9v l
'The number of R- and THETA-values must be equal.') Jlp nR#@
end E<RPMd @a
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% Check normalization: w1Txz4JqB
% -------------------- iq^F?$gFk
if nargin==5 && ischar(nflag) Ef @
isnorm = strcmpi(nflag,'norm'); QjOO^6Fh
if ~isnorm )DB\du
error('zernfun:normalization','Unrecognized normalization flag.') H^ 'As;R
end d!{]CZ"@
else )_n=it$
isnorm = false; hKnAWKb0
end Znw3P|>B
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U7#C. Z
% Compute the Zernike Polynomials f+!k:}K
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -wa"&Q
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% Determine the required powers of r: mUj_V#v
% ----------------------------------- -*A1[Z ?
m_abs = abs(m); }1
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rpowers = []; J6J">
for j = 1:length(n) ZJe^MnE (G
rpowers = [rpowers m_abs(j):2:n(j)]; A^ofs*"Y
end %rlMjF'tG
rpowers = unique(rpowers); O!!N@Q2g
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% Pre-compute the values of r raised to the required powers, j(A>M_f;
% and compile them in a matrix: ?;VsA>PV
% ----------------------------- GQ(*k)'a
if rpowers(1)==0 H +'6*akV
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Yt[LIn-v:
rpowern = cat(2,rpowern{:}); cgnMoBIc
rpowern = [ones(length_r,1) rpowern]; nW)?cQ
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else ZIN1y;dJ
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); /!?b&N/d)
rpowern = cat(2,rpowern{:}); EXMW,
end kXV;J$1
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% Compute the values of the polynomials: L{&>,ww
% -------------------------------------- Y'{}L@"t
y = zeros(length_r,length(n)); I
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for j = 1:length(n) :i4AkBNK
s = 0:(n(j)-m_abs(j))/2; fMIRr5
pows = n(j):-2:m_abs(j); D]o=I1O?
for k = length(s):-1:1 9a[1s|>w-
p = (1-2*mod(s(k),2))* ... _\=x
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prod(2:(n(j)-s(k)))/ ... r+8)<Xt+p
prod(2:s(k))/ ... -4[eZ>$A|
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 3 [j,d]\|
prod(2:((n(j)+m_abs(j))/2-s(k))); ~!S/{Un
idx = (pows(k)==rpowers); DKJ_g.]X
y(:,j) = y(:,j) + p*rpowern(:,idx); T+^Sa
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end wFF,rUV
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if isnorm JH| D
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); -oUGmV_
end ul3~!9F5F
end !E&l=*lM.
% END: Compute the Zernike Polynomials t>Ye*eR*`U
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fv7]1EO.
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% Compute the Zernike functions: F!3p )?
% ------------------------------ ~5&B#Sm[G
idx_pos = m>0; oP`:NCj\9
idx_neg = m<0; L[ZS17;*
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z = y; ;O1jf4y
if any(idx_pos) Ypl;jkHP
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 8nng^
end /lbj!\~
if any(idx_neg) e`co:HO`#
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 8o[gzW:Q)U
end V@]SKbK}wN
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% EOF zernfun KTt+}-vP^