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, -|V#U`mwF
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, o(tJc}Mh+(
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? ,1I-%6L
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? wqG#jC!5
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function z = zernfun(n,m,r,theta,nflag) Z`kVyuQ
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. +(!/(2>~
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N u0W6u} 4;
% and angular frequency M, evaluated at positions (R,THETA) on the Z(q]rX5"
% unit circle. N is a vector of positive integers (including 0), and y{M7kYWtHV
% M is a vector with the same number of elements as N. Each element Kb]}p
% k of M must be a positive integer, with possible values M(k) = -N(k) ;gL{*gR]S
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ;=joQWNDm
% and THETA is a vector of angles. R and THETA must have the same u.A}&'H
% length. The output Z is a matrix with one column for every (N,M) 6"_pCkn;c<
% pair, and one row for every (R,THETA) pair. ;8<HB1 &,
% k9eyl)
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike f%PLR9Nh5@
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =R:O`qdC4e
% with delta(m,0) the Kronecker delta, is chosen so that the integral @:im/SE
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +o@:8!IM1
% and theta=0 to theta=2*pi) is unity. For the non-normalized `Ij EwKra
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. d%I7OBBx@
% |[~S&
% The Zernike functions are an orthogonal basis on the unit circle. fTpG>*{p
% They are used in disciplines such as astronomy, optics, and )&E]
% optometry to describe functions on a circular domain. F;_c x
% ^zTe9:hz/\
% The following table lists the first 15 Zernike functions. r\QV%09R
% iuj%.}
% n m Zernike function Normalization |fyzb=Lg
% -------------------------------------------------- @|cHDltH
% 0 0 1 1 2c]751
% 1 1 r * cos(theta) 2 8Dl(zY K;
% 1 -1 r * sin(theta) 2 ekY)?$v3
% 2 -2 r^2 * cos(2*theta) sqrt(6) _#H d2h
% 2 0 (2*r^2 - 1) sqrt(3) (Q*x"G#4>
% 2 2 r^2 * sin(2*theta) sqrt(6) r?u4[
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% 3 -3 r^3 * cos(3*theta) sqrt(8) +8xT}mX
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) n;Mk\*Cg
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 5=*i!c
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% 3 3 r^3 * sin(3*theta) sqrt(8) eV%{XR?y
% 4 -4 r^4 * cos(4*theta) sqrt(10) onmpMU7w
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) CF3x\6.q}
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) r<kgYU`
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) j|8!gW
% 4 4 r^4 * sin(4*theta) sqrt(10) _N:$|O#
% -------------------------------------------------- v6G1y[Wl
% sCJ|U6Q-
% Example 1: X9PbU1o;
% 1?w=v|b:P)
% % Display the Zernike function Z(n=5,m=1) #*rJI3
% x = -1:0.01:1; %7-(c
% [X,Y] = meshgrid(x,x); dLGHbeZ[(
% [theta,r] = cart2pol(X,Y); ogSDV
% idx = r<=1; .h4NG4FIF
% z = nan(size(X)); t{B@k[|
% z(idx) = zernfun(5,1,r(idx),theta(idx)); #qk=R7"Q
% figure MA_YMxP.'
% pcolor(x,x,z), shading interp ?f9M59(l
% axis square, colorbar Q_p&~ PNy5
% title('Zernike function Z_5^1(r,\theta)') v6DjNyg<x
% F3vywN1$,
% Example 2: O*/%zr
% $aEv*{$y
% % Display the first 10 Zernike functions G11KAq(
% x = -1:0.01:1; U:99w
% [X,Y] = meshgrid(x,x); x] `F#5j
% [theta,r] = cart2pol(X,Y); Ohgu*5!o
% idx = r<=1; cQxUEY('+
% z = nan(size(X)); 66-\}8f8a
% n = [0 1 1 2 2 2 3 3 3 3]; "*/IP9?]
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Wm" q8-<<
% Nplot = [4 10 12 16 18 20 22 24 26 28]; vN
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% y = zernfun(n,m,r(idx),theta(idx)); FO(QsR=\s
% figure('Units','normalized') "5dke^yk0
% for k = 1:10 4Th?q{X
% z(idx) = y(:,k); _'Jjt9@S
% subplot(4,7,Nplot(k)) MCTJ^ g"D
% pcolor(x,x,z), shading interp s>G]U)d<'
% set(gca,'XTick',[],'YTick',[]) ";`jS&"=
% axis square 1!V[fPJ
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ah<p_qe9|
% end |5`ecjb.
% r[^.\&-
% See also ZERNPOL, ZERNFUN2. \z6UWZ
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% Paul Fricker 11/13/2006 2gklGDJD
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% Check and prepare the inputs: 49>b]f,Vc
% ----------------------------- Z5oDj|&l}
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) C7R3W,
error('zernfun:NMvectors','N and M must be vectors.') {[:C_Up)f
end t90M]EAV
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if length(n)~=length(m) Nz3+yxv1
error('zernfun:NMlength','N and M must be the same length.') #>KiX84
end Fhllqh)
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n = n(:); YC St X)r
m = m(:); Kyk{:UnI
if any(mod(n-m,2)) ^/}4M'[ w
error('zernfun:NMmultiplesof2', ... Qp[
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'All N and M must differ by multiples of 2 (including 0).') [O ^/"Qk
end Q5dqn"?
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if any(m>n) NTX0vQG
error('zernfun:MlessthanN', ... %U}6(~
'Each M must be less than or equal to its corresponding N.') H;_Ce'oU(
end t\QLj&h}E
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if any( r>1 | r<0 ) \reVA$M[
error('zernfun:Rlessthan1','All R must be between 0 and 1.') zOMxg00
end _IOUhMo
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 1}c/l<d
error('zernfun:RTHvector','R and THETA must be vectors.') _2`b$/)-
end Op9 ^Eu%n
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r = r(:); k kD#Bb
theta = theta(:); hTO2+F*
length_r = length(r); ECM#J28D
if length_r~=length(theta) q$yg^:]2
error('zernfun:RTHlength', ... nG5\vj,zB
'The number of R- and THETA-values must be equal.') Y~I>mc]
end |[5;dt_U/
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% Check normalization: OW=3t#"7Kp
% -------------------- XW8@c2jN\7
if nargin==5 && ischar(nflag) ,KM%/;1Dm
isnorm = strcmpi(nflag,'norm'); b@4UR<
if ~isnorm .eVX/6,
error('zernfun:normalization','Unrecognized normalization flag.') eJ<P
end W\Sc ak>
else , vvfk=-
isnorm = false; '^WR5P<8c
end G8w @C
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZGX"Vn|YL
% Compute the Zernike Polynomials _nzq(m1@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [#\OCdb*3
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% Determine the required powers of r: dn])6Xl;i
% ----------------------------------- TBJ?8W(
m_abs = abs(m); 7=X6_AD
rpowers = []; x4g6Qze
for j = 1:length(n) OA9P"*
rpowers = [rpowers m_abs(j):2:n(j)]; YZOwr72VL
end FVP,$
rpowers = unique(rpowers); &Q"vXs6Gt
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% Pre-compute the values of r raised to the required powers, RaTNA W)v>
% and compile them in a matrix: \pK&gdw
% ----------------------------- 4%qmwt*p
if rpowers(1)==0
7|dm"%@
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4mp)v*z
rpowern = cat(2,rpowern{:}); (ESFR0
rpowern = [ones(length_r,1) rpowern]; _'V o3b
else t'W6Fmwkx
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )q4nyT>M
rpowern = cat(2,rpowern{:}); AriV4 +
end ]P7gEBi
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%j1 7QD8
% Compute the values of the polynomials: a}VR>!b
% -------------------------------------- 8,+T[S
y = zeros(length_r,length(n)); hF^JSCDz l
for j = 1:length(n) x2I|iA =
s = 0:(n(j)-m_abs(j))/2; r/ATZAgHP
pows = n(j):-2:m_abs(j); 9dszn^]T
for k = length(s):-1:1 V?^qW#AG
p = (1-2*mod(s(k),2))* ... og+Vrd
prod(2:(n(j)-s(k)))/ ... ?Y\WSI?i
prod(2:s(k))/ ... Jr2>D=
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 6z~ [Ay
prod(2:((n(j)+m_abs(j))/2-s(k))); (kK8
Ox fF
idx = (pows(k)==rpowers); ';v2ld 9
y(:,j) = y(:,j) + p*rpowern(:,idx); Mx93D
end oliVaavj
;2fzA<RkK
if isnorm ~/SLGyu
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); PeEaF@#k
end c??m9=OX1
end ;VCFDE{K=
% END: Compute the Zernike Polynomials *Y53bZ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1er?4
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at*DYZBjDB
% Compute the Zernike functions: v/]xdP^Z
% ------------------------------ n.5M6i/~a
idx_pos = m>0; Avljrds+7
idx_neg = m<0; 5f@&XwD9
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z = y; /N/jwLr
if any(idx_pos) v)K|{x
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); #z_.!E
end 7I(QTc)*
if any(idx_neg) 8h}1t4k
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); T|YMU?4
end MbTmdRf
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% EOF zernfun 1Nv qtVC