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<8q 92F
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, @,GjeF]!
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? !&\meS{
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? :5G$d%O=2
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function z = zernfun(n,m,r,theta,nflag) -#daBx
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. t+jIHo
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N u9 %;{:]h
% and angular frequency M, evaluated at positions (R,THETA) on the #Af)n(
% unit circle. N is a vector of positive integers (including 0), and T 4vogoy
% M is a vector with the same number of elements as N. Each element >
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% k of M must be a positive integer, with possible values M(k) = -N(k) ` v>/
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, .$UTH@;7
% and THETA is a vector of angles. R and THETA must have the same C1n??Y[
% length. The output Z is a matrix with one column for every (N,M) e{:86C!d)
% pair, and one row for every (R,THETA) pair. S'|lU@PCl
% B U'Ki \
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike q$3HvZP
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), sN;(/O
% with delta(m,0) the Kronecker delta, is chosen so that the integral ;r%<2(
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, x
Ridc^
% and theta=0 to theta=2*pi) is unity. For the non-normalized }Z^FEd"y
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. l'W3=,G[?
% @h!U
% The Zernike functions are an orthogonal basis on the unit circle. |e~u!V\m
% They are used in disciplines such as astronomy, optics, and 2V
4`s'
% optometry to describe functions on a circular domain. 33O)k*g
% MPqY?KF
% The following table lists the first 15 Zernike functions. JN-D/s
% ;g&7*1E
% n m Zernike function Normalization y Y'gx|\
% -------------------------------------------------- $#F;xys
% 0 0 1 1 N'I?fWN!;R
% 1 1 r * cos(theta) 2 7 FEzak'
% 1 -1 r * sin(theta) 2 U`:l AG
% 2 -2 r^2 * cos(2*theta) sqrt(6) m2jwqx{G
% 2 0 (2*r^2 - 1) sqrt(3) 3D{82*&
% 2 2 r^2 * sin(2*theta) sqrt(6) /DK*yS
% 3 -3 r^3 * cos(3*theta) sqrt(8) \a\^(`3a[
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Hf;RIl2F
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) " vv$%^
% 3 3 r^3 * sin(3*theta) sqrt(8) e6Wl7&@6
% 4 -4 r^4 * cos(4*theta) sqrt(10) D7%^Ly
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) e|S+G6 :O2
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ": mCZUt
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5hlJbWJa
% 4 4 r^4 * sin(4*theta) sqrt(10) H't `Q&]a
% -------------------------------------------------- Bk\ *0B
% Sr4dY`V*:z
% Example 1: rOs)B 21/
% ?IL!
X-xx
% % Display the Zernike function Z(n=5,m=1) y.L|rRe@P
% x = -1:0.01:1; cpP.7ZR
% [X,Y] = meshgrid(x,x); a.5zdoH_
% [theta,r] = cart2pol(X,Y); Uh<H*o6e 9
% idx = r<=1; &f
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% z = nan(size(X)); N )b|
% z(idx) = zernfun(5,1,r(idx),theta(idx)); FcuEeca
% figure ,e}mR>i=e
% pcolor(x,x,z), shading interp J R8 Z6
% axis square, colorbar " 8~f
% title('Zernike function Z_5^1(r,\theta)') 8 /:X&
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% 3Yn:fsy
% Example 2: }dV9%0s!
% AJJ%gxqGq
% % Display the first 10 Zernike functions 'XC&BWJ
% x = -1:0.01:1; p{\qSPK
% [X,Y] = meshgrid(x,x); sDz)_;;%
% [theta,r] = cart2pol(X,Y); >[A65q'
% idx = r<=1; U'f$YVc
% z = nan(size(X)); d;@E~~o?B]
% n = [0 1 1 2 2 2 3 3 3 3]; e<ism?WG
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; eLe,=
% Nplot = [4 10 12 16 18 20 22 24 26 28]; \i&vOH'
% y = zernfun(n,m,r(idx),theta(idx)); 4]|9!=\
% figure('Units','normalized') t-?KKU8
% for k = 1:10 9-X{x95]
% z(idx) = y(:,k); M ,.0[+
% subplot(4,7,Nplot(k)) N,'[:{GOY
% pcolor(x,x,z), shading interp 0jip::x
% set(gca,'XTick',[],'YTick',[]) vTe$77n
% axis square Mp DdJ,
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])
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% end YUP%K!k
% {="Su{i}}
% See also ZERNPOL, ZERNFUN2. *Bb|N--jI
Y ;~~?[6
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% Paul Fricker 11/13/2006 [qjAq@@N#q
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% Check and prepare the inputs: qCxD{-9x{
% ----------------------------- =2vMw]
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 3<~2"@J
error('zernfun:NMvectors','N and M must be vectors.') 5;sQ@
end Cnc\sMDJ\B
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if length(n)~=length(m) Tvksf!ba
error('zernfun:NMlength','N and M must be the same length.') 1b
%T_a
end |R
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%FR^[H]
n = n(:); #sm_.?P
m = m(:); I!soV0VU]
if any(mod(n-m,2)) 3_jCsX
error('zernfun:NMmultiplesof2', ... ,:dEEL+>c
'All N and M must differ by multiples of 2 (including 0).') cA (e"N
end [Q.4]K2
#?b^B~ #
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if any(m>n) W2W2WyPk
error('zernfun:MlessthanN', ... =|WV^0=S'%
'Each M must be less than or equal to its corresponding N.') )68fm\t(
end #ejw@bd
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if any( r>1 | r<0 ) hB GGs
error('zernfun:Rlessthan1','All R must be between 0 and 1.') !>Qc2&ZV
end 5qtmb4R~
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ah!O&ECh
error('zernfun:RTHvector','R and THETA must be vectors.') 5[j!\d}U
end 0Z);.l^
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r = r(:); 8HRPJSO~g
theta = theta(:); e
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length_r = length(r); $ DZQdhv
if length_r~=length(theta) 1J{z}yPHc
error('zernfun:RTHlength', ... F#}1{$)%
/
'The number of R- and THETA-values must be equal.') t+4Y3*WeGF
end eDM0417O(
*_).UAP.
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% Check normalization: h@m n
GE
% -------------------- PVkN3J
if nargin==5 && ischar(nflag) -C'X4C+
isnorm = strcmpi(nflag,'norm'); 64\5v?C
if ~isnorm #G ,
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error('zernfun:normalization','Unrecognized normalization flag.') Vg,>7?]6h
end )D@n?qbG
else 6 XOu~+7
isnorm = false; %d[xr h
end zyp"*0zUr
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =u3@ Dhw
% Compute the Zernike Polynomials L5 k>;|SA
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "k1Tsd-
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% Determine the required powers of r: 4@mXtA
% ----------------------------------- $@qs(Xwr
m_abs = abs(m); k-ex<el)#
rpowers = []; On.x~t
for j = 1:length(n) =Wy`X0h
rpowers = [rpowers m_abs(j):2:n(j)]; o(>-:l i0
end jme5'FR
rpowers = unique(rpowers); PD
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b;{"lJ:+Z
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% Pre-compute the values of r raised to the required powers, Gxa.<E^k
% and compile them in a matrix: C.B}Py+
% ----------------------------- BSu)O~s
if rpowers(1)==0 6u, 0y$3
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);
pOI`,i}.
rpowern = cat(2,rpowern{:}); M7<#=pX&
rpowern = [ones(length_r,1) rpowern]; ?!
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else ;1g-z]
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 0G\myv
rpowern = cat(2,rpowern{:}); w$;*~Qc
end aLk2#1$g
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% Compute the values of the polynomials: ^\kHEM|5v
% -------------------------------------- -%V-'X5
y = zeros(length_r,length(n)); [O+^eE6h
for j = 1:length(n) |Sv #f2`
s = 0:(n(j)-m_abs(j))/2; {ZM2WFpE
pows = n(j):-2:m_abs(j); No&[ \;
for k = length(s):-1:1 iN4'jD^oP
p = (1-2*mod(s(k),2))* ... p>tdJjnt
prod(2:(n(j)-s(k)))/ ... Ww
tQ>'R"
prod(2:s(k))/ ... hG;=ci3EE
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... s1\BjSzk
prod(2:((n(j)+m_abs(j))/2-s(k))); ZUJOBjb`
K
idx = (pows(k)==rpowers); UG'U
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y(:,j) = y(:,j) + p*rpowern(:,idx); H'\ EA(v+
end LP-Q'vb<=
<.(/#=2
if isnorm J$/BH\
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); JIKxY$GS
end Bt7v[Ot
end 5"~^;O
% END: Compute the Zernike Polynomials )$4DH:WN
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (4f9wrK
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% Compute the Zernike functions: pvU oed\
% ------------------------------ NP'DuzC
idx_pos = m>0; `h3}"js
idx_neg = m<0; Jo$Dxa
z
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z = y; +mOtYfW
if any(idx_pos) <slq1
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); JsEEAM:w
end \\Tp40m+
if any(idx_neg) 6jo&i
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 6MNA.{Jdd
end *9(1:N;#
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% EOF zernfun @5G7bY7Nz