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, =.y~f A!
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, }MRd@ 0-?!
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? 0QPH}Vi5}
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? j2Tr$gx<
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function z = zernfun(n,m,r,theta,nflag) $"0M U
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. $tz;<M7B
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N WtViW=j'
% and angular frequency M, evaluated at positions (R,THETA) on the j*F`"df
% unit circle. N is a vector of positive integers (including 0), and XD |E=s
% M is a vector with the same number of elements as N. Each element XS`M-{f`
% k of M must be a positive integer, with possible values M(k) = -N(k) #Xhdn\7
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, v[#9+6P=
% and THETA is a vector of angles. R and THETA must have the same ,
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% length. The output Z is a matrix with one column for every (N,M) >2~+.WePu
% pair, and one row for every (R,THETA) pair. "
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% hJwC~HG5
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike %FXfqF9
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), NLS%S q
% with delta(m,0) the Kronecker delta, is chosen so that the integral .jS~By|r
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 8#(Q_
% and theta=0 to theta=2*pi) is unity. For the non-normalized mocI&=EF2X
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. =0^Ruh
% Q>/C*@
% The Zernike functions are an orthogonal basis on the unit circle. P8^hBv*
% They are used in disciplines such as astronomy, optics, and zXv3:uRp.
% optometry to describe functions on a circular domain. :>D[n1v
% ZZcEt
% The following table lists the first 15 Zernike functions. '3TW [!m
% Swp;HW7x
% n m Zernike function Normalization uwa~-xX6
% -------------------------------------------------- jov:]Bic
% 0 0 1 1 e?_@aa9~@{
% 1 1 r * cos(theta) 2 T^T[$26
% 1 -1 r * sin(theta) 2 "`M?R;DH
% 2 -2 r^2 * cos(2*theta) sqrt(6) :!5IW?2
% 2 0 (2*r^2 - 1) sqrt(3) M&NB/
% 2 2 r^2 * sin(2*theta) sqrt(6) PH?#)lD
% 3 -3 r^3 * cos(3*theta) sqrt(8) ?shIj;c[
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) w=j
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) I4i2+
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% 3 3 r^3 * sin(3*theta) sqrt(8) _@
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% 4 -4 r^4 * cos(4*theta) sqrt(10) ;3\3q1oX
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u}!@ ,/)
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) si&S%4(
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ##@$|6
% 4 4 r^4 * sin(4*theta) sqrt(10) COTp
% -------------------------------------------------- 356>QW'm
% {]E+~%Va
% Example 1: FDVcow*] n
% Jrg2/ee,*
% % Display the Zernike function Z(n=5,m=1) L:_bg8eD#
% x = -1:0.01:1; Bn61AFy`
% [X,Y] = meshgrid(x,x); 9uRFnzJVx
% [theta,r] = cart2pol(X,Y); PQK(0iCo4
% idx = r<=1; ]4R[<<hd
% z = nan(size(X)); \[gReaI
% z(idx) = zernfun(5,1,r(idx),theta(idx)); QmLF[\Oo_
% figure F1jglH/MF)
% pcolor(x,x,z), shading interp GP&vLt51
% axis square, colorbar r *$Ner
% title('Zernike function Z_5^1(r,\theta)') Z^]|o<.<I
% $aN-Y?U%
% Example 2: *uo'VJI7_,
% = M]iIWQ@`
% % Display the first 10 Zernike functions g.'yZvaP
% x = -1:0.01:1; ]8icBneA~'
% [X,Y] = meshgrid(x,x); P(XaTU&-
% [theta,r] = cart2pol(X,Y); 5B&;uY
% idx = r<=1; F)+{AQL
% z = nan(size(X)); %F:)5gT?
% n = [0 1 1 2 2 2 3 3 3 3]; oP!;\a( SL
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; |1ST=O7.LH
% Nplot = [4 10 12 16 18 20 22 24 26 28]; AC;V
m: @{
% y = zernfun(n,m,r(idx),theta(idx)); hQ(qbt{e
% figure('Units','normalized') SB5&A_tr
% for k = 1:10 hSFn8mpXT
% z(idx) = y(:,k); NzU,va N
% subplot(4,7,Nplot(k)) Qb)C[5a}
% pcolor(x,x,z), shading interp ]J:1P`k.
% set(gca,'XTick',[],'YTick',[]) Ma8_:7`>O
% axis square lu#LCG-.
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) )(tM/r4`c&
% end QHWBAGA
% X=Ys<TM,
% See also ZERNPOL, ZERNFUN2. {_Lgtu
#% of;mJv
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% Paul Fricker 11/13/2006 8}fu,$$5
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% Check and prepare the inputs: -s|}Rh?Y
% ----------------------------- *;m5'}jsy
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ^S)cjH`P
error('zernfun:NMvectors','N and M must be vectors.') : C b&v07
end %e`$p=m
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if length(n)~=length(m) rHf&:~
error('zernfun:NMlength','N and M must be the same length.') CBDG./
end Rb%%?*|
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n = n(:); _>=L>*
m = m(:); ?UK|>9y}Z
if any(mod(n-m,2)) 7lS#f1E
error('zernfun:NMmultiplesof2', ... ovwQ2TuK
'All N and M must differ by multiples of 2 (including 0).') f)g7
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end Fe.t/amS/
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if any(m>n) N? 5x9duK
error('zernfun:MlessthanN', ... f+|$&p%
'Each M must be less than or equal to its corresponding N.') M@3"<[g
end N<QjdD&
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if any( r>1 | r<0 ) P\]B<
error('zernfun:Rlessthan1','All R must be between 0 and 1.') @xeAc0.^
end Y!WG)u5
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ?-tVSRKQ
error('zernfun:RTHvector','R and THETA must be vectors.') MwfOy@|N
end kPQtQh]y%
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r = r(:); }hYZ"
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theta = theta(:); <BO)E(
length_r = length(r); /'Pd`Nxl.
if length_r~=length(theta) >(y<0
error('zernfun:RTHlength', ... _;4 [Q1
'The number of R- and THETA-values must be equal.') 557(EM
end %lX%8Z$v
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% Check normalization: sHc Td>xS
% -------------------- (;%|-{7e-
if nargin==5 && ischar(nflag) :K
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isnorm = strcmpi(nflag,'norm'); PlYm&
if ~isnorm -!0_:m3
error('zernfun:normalization','Unrecognized normalization flag.') 0<PR+Iv*i
end jqH3J2L
else i/b'4o=8
isnorm = false; S!PzLTc
end AW#<i_Ybf
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7 Sa1;%R
% Compute the Zernike Polynomials BS&;n
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Dfd-^N!
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% Determine the required powers of r: m~lpyAw
% ----------------------------------- p#SY /KIw
m_abs = abs(m); D0mI09=GtQ
rpowers = []; ,Rx{yf]k
for j = 1:length(n) Bm\qxQ
rpowers = [rpowers m_abs(j):2:n(j)]; UZEI:k,dv
end =&!HwOnp
rpowers = unique(rpowers); 5'w^@Rs5
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% Pre-compute the values of r raised to the required powers, BZb]SoAL
% and compile them in a matrix: 83cW=?UgA
% ----------------------------- "xAWG$b
if rpowers(1)==0 CSV;+,Vv
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 577:u<Yt
rpowern = cat(2,rpowern{:}); X%bFN
rpowern = [ones(length_r,1) rpowern]; hI pKJ&hm
else NNG}M(/V
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); okq[ o90
rpowern = cat(2,rpowern{:}); 51#"3S
end M=xQ=j?
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% Compute the values of the polynomials: q{die[J
% -------------------------------------- IMnP[WA!
y = zeros(length_r,length(n)); /D_+{dtE
for j = 1:length(n) 1! p/6
s = 0:(n(j)-m_abs(j))/2; Wk^RA_
pows = n(j):-2:m_abs(j); ^MD;"A<
for k = length(s):-1:1 Aa?I8sbc
p = (1-2*mod(s(k),2))* ... FFEfp.T1M
prod(2:(n(j)-s(k)))/ ... gPzL*6OSA
prod(2:s(k))/ ... )4xu^=N&as
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ~#}Dx
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prod(2:((n(j)+m_abs(j))/2-s(k))); `>D9P_Y"jI
idx = (pows(k)==rpowers); &V7>1kD3
y(:,j) = y(:,j) + p*rpowern(:,idx); G6K
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end #JA}3]
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if isnorm NHw x:-RH
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Pw@olG'Ah
end iA!7E;o
end t ]c{c#N/
% END: Compute the Zernike Polynomials 'mdM q=VI
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P&*sB%B
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% Compute the Zernike functions: +TeFt5[)h
% ------------------------------ gLL-VvJ[
idx_pos = m>0; iy\KzoB
idx_neg = m<0; kE;O7sN
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z = y; e=/&(Y
if any(idx_pos) 1xnLB>jP#
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); v|
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end SC#sax4N!=
if any(idx_neg) (}!C4S3#
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); +rNkN:/L
end OySy6IN]q
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% EOF zernfun Mo|wME#M