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, E4/Dr}4
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, 3,qr-g|;jM
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? 8wFJ4v3
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 2uW;
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function z = zernfun(n,m,r,theta,nflag) X5$ Iyis
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ;dgp+
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N zHRplm+i
% and angular frequency M, evaluated at positions (R,THETA) on the >}i E(
% unit circle. N is a vector of positive integers (including 0), and U!\.]jfS
% M is a vector with the same number of elements as N. Each element _)m]_eS._
% k of M must be a positive integer, with possible values M(k) = -N(k) {hrX'2:ClT
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, I1M%J@ Cz
% and THETA is a vector of angles. R and THETA must have the same BW*rIn<?G
% length. The output Z is a matrix with one column for every (N,M) ~=l;=7 T
% pair, and one row for every (R,THETA) pair. ?IT*:A]E
% yN(%-u"
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike A$0fKko
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =m#?neop
% with delta(m,0) the Kronecker delta, is chosen so that the integral y766;
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% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]Q)OL
% and theta=0 to theta=2*pi) is unity. For the non-normalized =dYqS[kJW
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. c,+:i1IAy
% JP[K;/
% The Zernike functions are an orthogonal basis on the unit circle. LFRlzz;
% They are used in disciplines such as astronomy, optics, and -gX1-,dE
% optometry to describe functions on a circular domain. <6 Uf.u`
% }00BllJ
% The following table lists the first 15 Zernike functions. Txb#C[`
% _F|Ek ;y%
% n m Zernike function Normalization wjB:5~n50k
% -------------------------------------------------- /"Uqa,{
% 0 0 1 1 [5Mr@f4I
% 1 1 r * cos(theta) 2 'e'cb>GnA
% 1 -1 r * sin(theta) 2 B*Dz{a^.:
% 2 -2 r^2 * cos(2*theta) sqrt(6) ar+9\
% 2 0 (2*r^2 - 1) sqrt(3) z5*'{t)
% 2 2 r^2 * sin(2*theta) sqrt(6) K`fuf=
% 3 -3 r^3 * cos(3*theta) sqrt(8) M&9+6e'-F
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) $}<e|3_
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) '!~)?C<
% 3 3 r^3 * sin(3*theta) sqrt(8) -k"/X8
% 4 -4 r^4 * cos(4*theta) sqrt(10) 5MJS
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% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) z[qDkL
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) oV78Hq6
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) $c(nF01
% 4 4 r^4 * sin(4*theta) sqrt(10) wgGl[_)
% -------------------------------------------------- G
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% ^RIl
% Example 1: &E5g3lf
% ,UF_`|
% % Display the Zernike function Z(n=5,m=1) .V8Lauz8
% x = -1:0.01:1; N6i Q8P-
% [X,Y] = meshgrid(x,x); b,1ePS
% [theta,r] = cart2pol(X,Y); {9.|2%a
% idx = r<=1; lA8`l>I
% z = nan(size(X)); UH"%N)[
% z(idx) = zernfun(5,1,r(idx),theta(idx)); CB}2j
% figure [FR`Z=%
% pcolor(x,x,z), shading interp `*1p0~cu
% axis square, colorbar r$s Qf&=
% title('Zernike function Z_5^1(r,\theta)') 4ID5q~
% ' %o#q6O
% Example 2: HY:7? <r
% #Ki[$bS~6
% % Display the first 10 Zernike functions L$M9w
% x = -1:0.01:1; !%%6dB@%t
% [X,Y] = meshgrid(x,x); m^;f(IK5
% [theta,r] = cart2pol(X,Y); "oO%`:pb
% idx = r<=1; 3AN/
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% z = nan(size(X)); WCixKYq
% n = [0 1 1 2 2 2 3 3 3 3]; s`~IUNJ@P
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 'E""amIJ
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ge8ZsaiU
% y = zernfun(n,m,r(idx),theta(idx)); 3L}A3de'
% figure('Units','normalized') &6nWzF
% for k = 1:10 T1=fNF
% z(idx) = y(:,k); s?L
% subplot(4,7,Nplot(k)) Z"fJ`--
% pcolor(x,x,z), shading interp VRB;$
% set(gca,'XTick',[],'YTick',[]) dDLeSz$b
% axis square WNrk}LFof
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])
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% end /f;~X"!
% ]N F[>uiW
% See also ZERNPOL, ZERNFUN2. sLxc(d'A
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##" HF
% Paul Fricker 11/13/2006 JDT`C2-Q
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% Check and prepare the inputs: iGB}Il)
% ----------------------------- $1`2kM5
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) z-)O9PV
error('zernfun:NMvectors','N and M must be vectors.') SO0PF|{\r
end g]0_5?i
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if length(n)~=length(m) xU`p|(SS-
error('zernfun:NMlength','N and M must be the same length.') #KZBsa@p
end )\$|X}uny&
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n = n(:); U # qK.
m = m(:); E~"y$Fqe
if any(mod(n-m,2)) -(H0>Ap
error('zernfun:NMmultiplesof2', ... 1iF1GkLEq
'All N and M must differ by multiples of 2 (including 0).') 6T`i/".
end c{w2Gt!
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if any(m>n) f) L
error('zernfun:MlessthanN', ... $f7l34Sf3
'Each M must be less than or equal to its corresponding N.') },-H"Qs
end
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if any( r>1 | r<0 ) e+fN6v5pU
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 7B66]3v
end K]w'&Qm8W
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QoT;WM Z
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) LZxNAua
error('zernfun:RTHvector','R and THETA must be vectors.') |P?*5xPB
end @cXMG6:{
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{ 2f-8Z&>
r = r(:); O?#7N[7
theta = theta(:); Wmv#:U
length_r = length(r); \ @2R9,9E
if length_r~=length(theta) Ab.(7GFK
error('zernfun:RTHlength', ... U| R_OLWAg
'The number of R- and THETA-values must be equal.') a0H+.W+]
end \:LW(&[!
BnF^u5kv %
/Lr.e%
% Check normalization: FGBbO\</
% -------------------- H3-hcx54T
if nargin==5 && ischar(nflag) sc#qwQ#
isnorm = strcmpi(nflag,'norm'); 5*u+q2\F
if ~isnorm \1M4Dl5!
error('zernfun:normalization','Unrecognized normalization flag.') 'PW5ux@`<
end `C'H.g\>2Q
else U-k`s[dv
isnorm = false; +X
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end &s>Jb?_5Mx
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }3WxZv]I}
% Compute the Zernike Polynomials Ar#(psU
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +G>\-tjSD
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% Determine the required powers of r: +D*Z_Yh6
% ----------------------------------- 4Ftu
m_abs = abs(m); 42ge3>
rpowers = []; .O<obq~;C
for j = 1:length(n) AbW6x
rpowers = [rpowers m_abs(j):2:n(j)]; t4-[Z$n5
end !C.4<?*|
rpowers = unique(rpowers); }"%N4(Kd
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)%fH(ns(
% Pre-compute the values of r raised to the required powers, X1_5KH
% and compile them in a matrix: :7;@ZEe
% ----------------------------- lr&a;aZp
if rpowers(1)==0 lPAQ3t!,
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); w_V P
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rpowern = cat(2,rpowern{:}); _7y[B&g[r
rpowern = [ones(length_r,1) rpowern]; %iqD5x$OA
else vW@=<aS Z
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); KwVbbC3
rpowern = cat(2,rpowern{:}); es0hm2HT3
end Ab;.5O$y
#,'kXj
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% Compute the values of the polynomials: @IZnFHN
% -------------------------------------- m.0*NW
y = zeros(length_r,length(n)); 3=V&K-
for j = 1:length(n) ql~J8G9
s = 0:(n(j)-m_abs(j))/2; +1!ia]
pows = n(j):-2:m_abs(j); o^wqFX(Y
for k = length(s):-1:1 2MK-5Kg
p = (1-2*mod(s(k),2))* ... O^rD HFj,
prod(2:(n(j)-s(k)))/ ... u)Whr@m
prod(2:s(k))/ ... WTiD[u
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... `kSZX:=};
prod(2:((n(j)+m_abs(j))/2-s(k))); 4Wp=y
idx = (pows(k)==rpowers); hgE71H\s
y(:,j) = y(:,j) + p*rpowern(:,idx); ZYNsHcTY
end oxtay7fx
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if isnorm #T"4RrR
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); tX~w{|k
end EKN~H$.
end (^>J&[=
% END: Compute the Zernike Polynomials K:WDl;8(d
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sa8Vvzvo.
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:rP=t ,
% Compute the Zernike functions: \GU<43J2uo
% ------------------------------ UC$ppTCc?
idx_pos = m>0; $<OD31T
idx_neg = m<0; o{[qZc_%
D)}v@je"yP
^=*;X;7
z = y; !p/goqT~dY
if any(idx_pos) -tU'yKhn
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); lk =<A"^S
end *yGGBqd
if any(idx_neg) lmhLM. 2
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); dgP3@`YS
end Ws12b$
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% EOF zernfun [<TrS/,)>