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, *b/`Ya4
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, 2Yn <2U/^R
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? pDIVZC
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? SB|Qa}62
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function z = zernfun(n,m,r,theta,nflag)
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. >eB\(EP
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N S.m{eur!,E
% and angular frequency M, evaluated at positions (R,THETA) on the ^,8)iV0j_
% unit circle. N is a vector of positive integers (including 0), and *q".-u!D[
% M is a vector with the same number of elements as N. Each element |>htvDL
% k of M must be a positive integer, with possible values M(k) = -N(k) TDNQu_E
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, e<h~o!za
% and THETA is a vector of angles. R and THETA must have the same J/GSceHF
% length. The output Z is a matrix with one column for every (N,M) WP+oFkw>
% pair, and one row for every (R,THETA) pair. yXF?H"h(
% I@%t.%O Jp
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike L>%o[tS
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^1aAjYFn
% with delta(m,0) the Kronecker delta, is chosen so that the integral 2hkRd>)&5
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, A1#%`^W9
% and theta=0 to theta=2*pi) is unity. For the non-normalized $!(pF
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. J}+6UlD
% tj4VWJK
% The Zernike functions are an orthogonal basis on the unit circle. !Kj,9NX{U
% They are used in disciplines such as astronomy, optics, and j eX^}]x|%
% optometry to describe functions on a circular domain. pxf$1
% V<@ o<R
% The following table lists the first 15 Zernike functions. 7C ,UDp|
% \\7ZWp\fN
% n m Zernike function Normalization /fT+^&
% -------------------------------------------------- :1^R9yWA4
% 0 0 1 1 OJzs Q
% 1 1 r * cos(theta) 2 9;Ox;;w
% 1 -1 r * sin(theta) 2 [4C:r!
% 2 -2 r^2 * cos(2*theta) sqrt(6) (%xwl
% 2 0 (2*r^2 - 1) sqrt(3) Mt5PaTjj
% 2 2 r^2 * sin(2*theta) sqrt(6) MP 2~;T}~
% 3 -3 r^3 * cos(3*theta) sqrt(8) /)(#{i*
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Jesjtcy<*
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) rT5Ycm@
% 3 3 r^3 * sin(3*theta) sqrt(8) %V{7DA&C
% 4 -4 r^4 * cos(4*theta) sqrt(10) Qj6/[mUr~
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) $8[r9L!
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) e9[|!/./5
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )>-ibf`#?
% 4 4 r^4 * sin(4*theta) sqrt(10) <l9-;2L4
% -------------------------------------------------- ;Uu(zhbj
% Yvjc1
% Example 1: 5<j%EQN|D
% 7{qy7,Gp
% % Display the Zernike function Z(n=5,m=1) .j>hI="b
% x = -1:0.01:1; a5!Fv54
% [X,Y] = meshgrid(x,x); x,S
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% [theta,r] = cart2pol(X,Y); )
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% idx = r<=1; hQH nwr
% z = nan(size(X)); _b.qkTWUB
% z(idx) = zernfun(5,1,r(idx),theta(idx)); <_Q:'cx'
% figure A\#P*+k 0
% pcolor(x,x,z), shading interp ]U7KLUY>:
% axis square, colorbar /3:q#2'v
% title('Zernike function Z_5^1(r,\theta)') P*Tx14xe4
% 'hv k
% Example 2: )}'U`'q
% pd8Nke
% % Display the first 10 Zernike functions
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% x = -1:0.01:1; -s$F&\5by
% [X,Y] = meshgrid(x,x); /<8N\_wh
% [theta,r] = cart2pol(X,Y); QZhjb
% idx = r<=1; jDN ]3Y`
% z = nan(size(X)); k{$ ao
% n = [0 1 1 2 2 2 3 3 3 3]; aKJQm'9Ks
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 1`9xIm*9w
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ]mXLg:3B
% y = zernfun(n,m,r(idx),theta(idx)); 9Q-*@6G
% figure('Units','normalized') M7+h(\H]2
% for k = 1:10 <rL/B
k
% z(idx) = y(:,k); j"@93D~
% subplot(4,7,Nplot(k)) b-*3 2Y%
% pcolor(x,x,z), shading interp dwv 6;x
% set(gca,'XTick',[],'YTick',[]) ;6{@^
% axis square u=/CRjot
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _fP&&}
% end ]a3iEA2 (
% mA@Me7m}
% See also ZERNPOL, ZERNFUN2. (q7
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% Paul Fricker 11/13/2006 D.6,VY H
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% Check and prepare the inputs: [e4]"v`N
% ----------------------------- 3#45m+D
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) zb
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error('zernfun:NMvectors','N and M must be vectors.') *d',Vuv&[
end cl*PFQp9j
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if length(n)~=length(m) FG!X"<he
error('zernfun:NMlength','N and M must be the same length.') K[7EOXLy
end ^p/Ob'!
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n = n(:); *79m^
m = m(:); S$^RbI
if any(mod(n-m,2)) KB!|B.ChN(
error('zernfun:NMmultiplesof2', ... ]}6w#)]"
'All N and M must differ by multiples of 2 (including 0).') vHE^"l5 v
end OLj\-w^
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if any(m>n) 7RvUH-S[
error('zernfun:MlessthanN', ... P0-Fc@&Y
'Each M must be less than or equal to its corresponding N.') U70]!EaT
end T4;T6 9j;,
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if any( r>1 | r<0 ) sfVf@0g
error('zernfun:Rlessthan1','All R must be between 0 and 1.') pBC<u
end z>[tF5
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) U`x bPQ
error('zernfun:RTHvector','R and THETA must be vectors.') {3Vk p5%l
end **[Z^$)u(
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r = r(:); ^-M^gYBR
theta = theta(:); p=QYc)3F
length_r = length(r); Ih[+K#t+E
if length_r~=length(theta) }p9F#gr
error('zernfun:RTHlength', ... OlQ,Ce
'The number of R- and THETA-values must be equal.') #DkD!dW(l
end ^SfS~GQ
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% Check normalization: @oG)LT
% -------------------- 9%iFV
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if nargin==5 && ischar(nflag) cxYfZ4++m
isnorm = strcmpi(nflag,'norm'); !z
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if ~isnorm s/1 #DM"
error('zernfun:normalization','Unrecognized normalization flag.') =qvZpB7ZZ
end bO/*2oau
else WnAd5#G
isnorm = false; - n6jG}01b
end 0D(cXzQP
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b]]N{: I
% Compute the Zernike Polynomials C6&( c
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7XyOB+aQO
cUDg M
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% Determine the required powers of r: ,c)g,J9
% ----------------------------------- u>Ki$xP1
m_abs = abs(m); _hCJ|Rrln
rpowers = []; Ca $c;
for j = 1:length(n) :a<hQ|p
rpowers = [rpowers m_abs(j):2:n(j)]; 1;W=!Fx
end e"+dTq8W
rpowers = unique(rpowers); [D'Gr*5~{
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% Pre-compute the values of r raised to the required powers, ?Cci:Lin
% and compile them in a matrix: c/u_KJFF-n
% ----------------------------- i.rU&yT%
if rpowers(1)==0 /b.oEGqZX
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); PtKTm\,JL0
rpowern = cat(2,rpowern{:}); O=jN&<rb
rpowern = [ones(length_r,1) rpowern]; ur2!#bU9
else '0+$ m=
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); vg8O]
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rpowern = cat(2,rpowern{:}); LBX%H GH
end KC&`x|
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% Compute the values of the polynomials: BGjb`U#%3
% -------------------------------------- FUaNiAr[
y = zeros(length_r,length(n)); z*.v_Mx
for j = 1:length(n) a%~yol0wO7
s = 0:(n(j)-m_abs(j))/2; Z%v6xP.
pows = n(j):-2:m_abs(j); Gidkt;lj
for k = length(s):-1:1 nN ~GP"}
p = (1-2*mod(s(k),2))* ... U7%28#@
prod(2:(n(j)-s(k)))/ ... d]M[C[TOX
prod(2:s(k))/ ... FWTx&Ip
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... If}lJ6jZ
prod(2:((n(j)+m_abs(j))/2-s(k))); KP~-$NR
idx = (pows(k)==rpowers); xtJAMo>g
y(:,j) = y(:,j) + p*rpowern(:,idx); }~*rx7p
end w6EI{
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if isnorm hmGdjw t$
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); v'nHFC+p
end Uh+jt,RB`
end org*z!;.
% END: Compute the Zernike Polynomials PqhlXqX9
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% aii'}c
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% Compute the Zernike functions: YkbO&~.
% ------------------------------ &N{zkMf
idx_pos = m>0; D_aR\
idx_neg = m<0; # ,P(isEZ"
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]
z = y; MYTS3(
if any(idx_pos) U,3d) ]Zy&
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); sfC@*Y2XT
end d[U1.SNL
if any(idx_neg) 1b`G2?%
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); v>^jy8$
end )[DpK=[N^p
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\DyKtrnm%
% EOF zernfun 6 ">oo-