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, lSy_cItF
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, 4j(*%da
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? vcZ"4%w
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? "$3~):o
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function z = zernfun(n,m,r,theta,nflag) %r@:7/
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 4 g8t
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N +E+I.}sOB
% and angular frequency M, evaluated at positions (R,THETA) on the U^Iq]L
% unit circle. N is a vector of positive integers (including 0), and vvLzUxV
% M is a vector with the same number of elements as N. Each element [;#^h/5E
% k of M must be a positive integer, with possible values M(k) = -N(k) pS8`OBenA
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, (e32oP"
% and THETA is a vector of angles. R and THETA must have the same 'X~CrgQl
% length. The output Z is a matrix with one column for every (N,M) N_p^DP
% pair, and one row for every (R,THETA) pair. xv7nChB
% g@m__
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +D?Re%HI
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =j@8/
% with delta(m,0) the Kronecker delta, is chosen so that the integral SJlL!<i$
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, y(j vl|z[
% and theta=0 to theta=2*pi) is unity. For the non-normalized u"(2Xer
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. :eBp`dmn
% LbnF8tj}h
% The Zernike functions are an orthogonal basis on the unit circle. ~g *`E!2
% They are used in disciplines such as astronomy, optics, and 3=_to7]
% optometry to describe functions on a circular domain. .p'\@@o5
% R4XcWx*pQ
% The following table lists the first 15 Zernike functions. 7H. HiyppW
% E6xWo)`%5s
% n m Zernike function Normalization N8Un42
% -------------------------------------------------- h[]3#
% 0 0 1 1 !6_tdZ
% 1 1 r * cos(theta) 2 a61?G!]
% 1 -1 r * sin(theta) 2 OKCX>'j:S
% 2 -2 r^2 * cos(2*theta) sqrt(6) /?C6oj1
% 2 0 (2*r^2 - 1) sqrt(3) _2eL3xXha.
% 2 2 r^2 * sin(2*theta) sqrt(6) )J&!>GP
% 3 -3 r^3 * cos(3*theta) sqrt(8) _p| KaT``
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 7T?7KS
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) BgwZZ<B
% 3 3 r^3 * sin(3*theta) sqrt(8) ^Y^5 @x=
% 4 -4 r^4 * cos(4*theta) sqrt(10) #Y>d@
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) S4%MnT6Uy
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) BtP*R,>
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) tHo/Vly6Z
% 4 4 r^4 * sin(4*theta) sqrt(10) }J:WbIr0!
% -------------------------------------------------- 5O"wPsl
% `=#ry*E^:
% Example 1: jqy?Od)
% l5_%Q+E_
% % Display the Zernike function Z(n=5,m=1) LiD-su
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% x = -1:0.01:1; 7h.:XlUm|
% [X,Y] = meshgrid(x,x); yGPi9j{QXq
% [theta,r] = cart2pol(X,Y); XXZ$^W&
% idx = r<=1; +isaqfy/
% z = nan(size(X)); i{2rQy+
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 7033#@_
% figure o#F0 3
% pcolor(x,x,z), shading interp [>f4&yY
% axis square, colorbar .g6(07TyV
% title('Zernike function Z_5^1(r,\theta)') fpvzx{2
% Q"H1(kG|
% Example 2: kx3]A"]>'
% ,_yf5 a
% % Display the first 10 Zernike functions N%`Eq@5
% x = -1:0.01:1; 2BIOA#@t
% [X,Y] = meshgrid(x,x); V~qlg1h
% [theta,r] = cart2pol(X,Y); \JEI+A PY*
% idx = r<=1; pi?U|&.1z
% z = nan(size(X)); <S
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% n = [0 1 1 2 2 2 3 3 3 3]; 5>[j^g+@
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; eVy\)dCsU
% Nplot = [4 10 12 16 18 20 22 24 26 28]; W=
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% y = zernfun(n,m,r(idx),theta(idx)); !Pb39[f
% figure('Units','normalized') B\Y!5$
% for k = 1:10 }[I|oV5*+&
% z(idx) = y(:,k); ;/-#oW@gQ
% subplot(4,7,Nplot(k)) ~0@+8%^>;
% pcolor(x,x,z), shading interp w`OHNwXh#I
% set(gca,'XTick',[],'YTick',[]) Xa32p_|5~
% axis square kT6EHuB
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) k`Ifd:V.y
% end YNi3oG]h
% !U!}*clYL
% See also ZERNPOL, ZERNFUN2. c{t(),nAA
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% Paul Fricker 11/13/2006 G#A& Y$
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% Check and prepare the inputs: 5zH?1Z~*
% ----------------------------- x?|
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7|Tu@0XXA
error('zernfun:NMvectors','N and M must be vectors.') $?u ^hMU=
end W:16qbK
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if length(n)~=length(m) W=EvEx^?%
error('zernfun:NMlength','N and M must be the same length.') ul$YV9[\
end ]n:)W.|`R
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n = n(:); ~IKPi==@,
m = m(:); hOSkxdi*^
if any(mod(n-m,2)) K}U}h>N
error('zernfun:NMmultiplesof2', ... O2Mo ~}
'All N and M must differ by multiples of 2 (including 0).') N5=;
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end nEM>*;iE
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if any(m>n) AhARBgf<
error('zernfun:MlessthanN', ... 217KJ~)'
'Each M must be less than or equal to its corresponding N.') Whq@>pX8
end dviL5Eaj
/*bS~7f1
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if any( r>1 | r<0 ) Hs+VA$$*
error('zernfun:Rlessthan1','All R must be between 0 and 1.') l*]*.?m/5
end e/m,PE
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) D/Y .'P:j
error('zernfun:RTHvector','R and THETA must be vectors.') p_jDnb#
end %jY/jp=R
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r = r(:); 'Z+~G
theta = theta(:); 1TKOvy_
length_r = length(r); 4cql?W (D
if length_r~=length(theta) Q-%Q7n'c
error('zernfun:RTHlength', ... "}]1OL S V
'The number of R- and THETA-values must be equal.') lV-7bZ
end #s1O(rLRl
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% Check normalization: Dm6}$v'0
% -------------------- Cd#>,,\z
if nargin==5 && ischar(nflag) ]}cai1
isnorm = strcmpi(nflag,'norm'); OCF\*Sx
if ~isnorm )>Oip
error('zernfun:normalization','Unrecognized normalization flag.') H'$g!Pg
end vS:%(Y"!<
else 9/MUzt
isnorm = false; 7{:| )
end $L.0$-je4
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dx@#6Fhy
% Compute the Zernike Polynomials rO/mK$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <$n%h/2%
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% Determine the required powers of r: SG8H~]CO)
% ----------------------------------- 50(/LV1
m_abs = abs(m); qu8i Jq
rpowers = []; b1jh2pG(V
for j = 1:length(n) viAvD6e
rpowers = [rpowers m_abs(j):2:n(j)]; FK{YRt
end W?G4\ubM3<
rpowers = unique(rpowers); rB|D^@mG
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% Pre-compute the values of r raised to the required powers, |^Z1 D TAw
% and compile them in a matrix: %lV&QQa
% ----------------------------- _h7+.U=
if rpowers(1)==0 N<:5 r
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); {SW104nb
rpowern = cat(2,rpowern{:}); J
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rpowern = [ones(length_r,1) rpowern]; S)z
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else fSl+;|Kn
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); !'B.ad
rpowern = cat(2,rpowern{:}); :KZI+
end t/_w}
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% Compute the values of the polynomials: qI<6% ^i
% -------------------------------------- ,Z#t-?
y = zeros(length_r,length(n)); Vy{=Y(cpF2
for j = 1:length(n) LDW":k|
s = 0:(n(j)-m_abs(j))/2; X_|8CD-@6
pows = n(j):-2:m_abs(j); AShJtxxa
for k = length(s):-1:1 0[xum
p = (1-2*mod(s(k),2))* ... &7T0nB/)
prod(2:(n(j)-s(k)))/ ... PX[taDN
prod(2:s(k))/ ... {LY$
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ?
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prod(2:((n(j)+m_abs(j))/2-s(k))); N6$pOQ
idx = (pows(k)==rpowers); z}s0D]$+x
y(:,j) = y(:,j) + p*rpowern(:,idx); 8=T;R&U^M
end vAq`*]W+
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if isnorm +XJj:%yt
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Mvrc[s+o
end s9~W( Wi
end 4
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% END: Compute the Zernike Polynomials DL|,:2`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f$iv+7<B^
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% Compute the Zernike functions: q>%KIBh(
% ------------------------------ $/5Jc[Ow
idx_pos = m>0;
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idx_neg = m<0; /Bid:@R
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z = y; bec n$R
if any(idx_pos) gf2l19aP
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); B1JdkL 3h
end ,4jkTQ*@2
if any(idx_neg) O!lZ%j@%
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); `&4L'1eF{
end mgL~ $
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% EOF zernfun j =r`[Bm