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, txEN7!
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, . _+cvXy
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? lpi"@3
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Y S3~sA
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function z = zernfun(n,m,r,theta,nflag) aEf3hB* ~
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. b'wy{~l@
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 9nY`rF8@
% and angular frequency M, evaluated at positions (R,THETA) on the LhG\)>Y%
% unit circle. N is a vector of positive integers (including 0), and
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% M is a vector with the same number of elements as N. Each element F:/x7]7??Z
% k of M must be a positive integer, with possible values M(k) = -N(k) =gF035
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, |JkfAnrN$I
% and THETA is a vector of angles. R and THETA must have the same zw#n85=
% length. The output Z is a matrix with one column for every (N,M) qV=:2m10x
% pair, and one row for every (R,THETA) pair. Na@bXcz)
% ,ye}p1M
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike cb-IRGF
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), MkW=sD_
% with delta(m,0) the Kronecker delta, is chosen so that the integral tE%g)hL-
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, d==0 @`
% and theta=0 to theta=2*pi) is unity. For the non-normalized MKbcJZe
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. QC'Ru'8S
% ;R=n<=Axa
% The Zernike functions are an orthogonal basis on the unit circle. ?j&hG|W9<z
% They are used in disciplines such as astronomy, optics, and xLed];2G
% optometry to describe functions on a circular domain. S(@kdL
% |GMo"[
% The following table lists the first 15 Zernike functions. iM!Ya!
% ")KqPD6k
% n m Zernike function Normalization _DxHJl
% -------------------------------------------------- -k + jMH
% 0 0 1 1 hh4R
% 1 1 r * cos(theta) 2 ?22U0UF
% 1 -1 r * sin(theta) 2 gWgp:;Me
% 2 -2 r^2 * cos(2*theta) sqrt(6) ILr=<j
% 2 0 (2*r^2 - 1) sqrt(3) 1 b7jNkQ
% 2 2 r^2 * sin(2*theta) sqrt(6) k'r} @-X
% 3 -3 r^3 * cos(3*theta) sqrt(8) Y. J!]|
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Mbc&))A
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) a~Dk@>+P>
% 3 3 r^3 * sin(3*theta) sqrt(8) G^B>C
% 4 -4 r^4 * cos(4*theta) sqrt(10) 9(t(sP_
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |ufL s
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) <M\&zHv
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Tdh(J",d
% 4 4 r^4 * sin(4*theta) sqrt(10) RP$u/x"b
% -------------------------------------------------- yF\yxdUX#
% \me5"ZU
% Example 1: Q
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% , >aa2
% % Display the Zernike function Z(n=5,m=1) jyD~ER}J
% x = -1:0.01:1; -ED}
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% [X,Y] = meshgrid(x,x); * WV=X p
% [theta,r] = cart2pol(X,Y); J4ZHE\
% idx = r<=1; R?u(aY)P
% z = nan(size(X)); ' pgPQM<
% z(idx) = zernfun(5,1,r(idx),theta(idx)); a4UwhbH
% figure \ Bj{.jL
% pcolor(x,x,z), shading interp u<8b5An;
% axis square, colorbar dnomnY(*<
% title('Zernike function Z_5^1(r,\theta)') $y6 <2w%b
% A|LO!P,w
% Example 2: n
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% #OPEYJ;*9d
% % Display the first 10 Zernike functions d<d3j9u(#
% x = -1:0.01:1; inh:b .,B
% [X,Y] = meshgrid(x,x); 8#;=>m%
% [theta,r] = cart2pol(X,Y); zg3kU65PJE
% idx = r<=1; \dJhDR
% z = nan(size(X)); PP{9Y Vr
% n = [0 1 1 2 2 2 3 3 3 3]; 7tWC<#
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; A:Wr5`FJ
% Nplot = [4 10 12 16 18 20 22 24 26 28]; E"9(CjbQ[
% y = zernfun(n,m,r(idx),theta(idx)); <y8oYe_!
% figure('Units','normalized') ntLEk fK{
% for k = 1:10 dV[G-p
% z(idx) = y(:,k); f2[R2sto@
% subplot(4,7,Nplot(k)) ?fH1?Z\'K
% pcolor(x,x,z), shading interp hu$eO'M_
% set(gca,'XTick',[],'YTick',[]) $M)SsD~
% axis square hlL$3.]
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) =s!0EwDH3
% end ~bkO8tn
% 2b7-=/[6
% See also ZERNPOL, ZERNFUN2.
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% Paul Fricker 11/13/2006 C W#:'
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% Check and prepare the inputs: [3bPoAr\
% ----------------------------- lv=q( &
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) g;=VuQuP|
error('zernfun:NMvectors','N and M must be vectors.') ic`BDkNO
end rwJU;wy
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%JrZMs>
if length(n)~=length(m) (Ff}Y.4
error('zernfun:NMlength','N and M must be the same length.') ~2\Sn-`
end EA(4xj&:U
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n = n(:); Wz.iDRFl
m = m(:); }O7sP^
if any(mod(n-m,2)) {,JO}Dmu5
error('zernfun:NMmultiplesof2', ... QP.Lq}
'All N and M must differ by multiples of 2 (including 0).') rlR!Tc>
end (9RfsV4^
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if any(m>n) Hh Q0>
error('zernfun:MlessthanN', ... ;+XrCy!.)L
'Each M must be less than or equal to its corresponding N.') Lo'pNJH;$
end zEU[u7%
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if any( r>1 | r<0 ) +ZNOvcsV
error('zernfun:Rlessthan1','All R must be between 0 and 1.') z*h:Nt%.
end iGSJ\
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) =gI41Y]
error('zernfun:RTHvector','R and THETA must be vectors.') <~5O-.G]
end I+H~ 5zq.
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r = r(:); UgAp9$=z
theta = theta(:); iGhvQmd(/*
length_r = length(r); 6Yn>9llo}=
if length_r~=length(theta) v^ @)&,
error('zernfun:RTHlength', ... |Jn|GnM
'The number of R- and THETA-values must be equal.') g0j)k6<6(Y
end c+3`hVV
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g6%]uCFB
% Check normalization: ns>$
% -------------------- 3`yO&upk
if nargin==5 && ischar(nflag) QUW`Yc
isnorm = strcmpi(nflag,'norm'); 0 YFXF
if ~isnorm 12U]=
error('zernfun:normalization','Unrecognized normalization flag.') uQvTir*e
end ]6B9\C.2-_
else eR \duZ!`
isnorm = false; _ +DL
end ]0* aE
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F2}Fuupb.
% Compute the Zernike Polynomials ]]K?Q
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fX`u"`o5
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% Determine the required powers of r: `[5xncZ-
% ----------------------------------- &zF>5@fM
m_abs = abs(m); n7bVL#Sq[
rpowers = []; ((A@VcX
for j = 1:length(n) F%-@_IsG#
rpowers = [rpowers m_abs(j):2:n(j)]; y\^zxG*]'
end "b`#RohCi
rpowers = unique(rpowers); e2c'Wab
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p
% Pre-compute the values of r raised to the required powers, tcj"rV{G
% and compile them in a matrix: Zzjx;SF
% ----------------------------- Dst;sLr[,
if rpowers(1)==0 wA$7SWC
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Eh8GqFEM
rpowern = cat(2,rpowern{:}); Bbs1U
rpowern = [ones(length_r,1) rpowern]; OU%"dmSDk
else P?V+<c{
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); :G 5p`;hGo
rpowern = cat(2,rpowern{:}); #a=]h}&1?
end #B~;j5
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% Compute the values of the polynomials: n R, QG8
% -------------------------------------- NW6;7nWb
y = zeros(length_r,length(n)); (E0WZ$f}
for j = 1:length(n) h>!h|Ma
s = 0:(n(j)-m_abs(j))/2; :;Z/$M16B
pows = n(j):-2:m_abs(j); esTL3 l{[
for k = length(s):-1:1 Ne+Rs+~4
p = (1-2*mod(s(k),2))* ... L-E &m* %
prod(2:(n(j)-s(k)))/ ... [!%5(Ro_
prod(2:s(k))/ ... /E<Q_/'Z
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ThX3@o
prod(2:((n(j)+m_abs(j))/2-s(k))); xBxiBhqzF
idx = (pows(k)==rpowers); aU;X&g+_)
y(:,j) = y(:,j) + p*rpowern(:,idx); }}k%.Qb
end 3\Xk)a_
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if isnorm ,Dv*<La`\
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 17'd~-lE
end <!m.+
end v+x<X5u
% END: Compute the Zernike Polynomials ]Y]]X[@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9`92
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% Compute the Zernike functions: } OIe!
% ------------------------------ -sv%A7i
idx_pos = m>0; ,$t1LV;o=
idx_neg = m<0; 392(N(
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z = y; y$fMMAN7
if any(idx_pos) rOLZiE T
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); VTL_I^p
end . h)VR
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if any(idx_neg) )kjQ W&)g
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); " TCJT390
end uM'n4 oH
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% EOF zernfun <=|^\r
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