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, ?=h{`Ci^ $
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, X%._:st
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? "*E%?MG
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? R_2JP C
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function z = zernfun(n,m,r,theta,nflag) IL]VY1'#
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. yS[Z%]bvU
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N P]G`Y>#$r
% and angular frequency M, evaluated at positions (R,THETA) on the -a[]#v9
% unit circle. N is a vector of positive integers (including 0), and 8f<[Bu ze
% M is a vector with the same number of elements as N. Each element 2$O@T]
% k of M must be a positive integer, with possible values M(k) = -N(k) Bld $<uU
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, $3Ct@}=n
% and THETA is a vector of angles. R and THETA must have the same i>C:C>~
% length. The output Z is a matrix with one column for every (N,M) eiaLzI,O
% pair, and one row for every (R,THETA) pair. ^{T3lQvt
% LA.xLU3
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike u9*}@{,
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), -PSI^%TR#
% with delta(m,0) the Kronecker delta, is chosen so that the integral bt,^-gt@
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, j:9kJq>mv
% and theta=0 to theta=2*pi) is unity. For the non-normalized ^vjN$JB
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. )k8=< =s
% ;$y(Tvd;
% The Zernike functions are an orthogonal basis on the unit circle. w-%H\+J
% They are used in disciplines such as astronomy, optics, and q1Si*?2W
% optometry to describe functions on a circular domain. Oop;Y^gG}
% oO4
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% The following table lists the first 15 Zernike functions. ,xew3c'(W
% o[bG(qHZ
% n m Zernike function Normalization e%'$Vx0kA
% -------------------------------------------------- :A,V<Es}I"
% 0 0 1 1 iTt"Ik'
% 1 1 r * cos(theta) 2 _G!lQ)1
% 1 -1 r * sin(theta) 2 -T4{PM
% 2 -2 r^2 * cos(2*theta) sqrt(6) {P_~_5o_
% 2 0 (2*r^2 - 1) sqrt(3) AFWcTz6 #d
% 2 2 r^2 * sin(2*theta) sqrt(6) y#AwuC K
% 3 -3 r^3 * cos(3*theta) sqrt(8) NW`.RGLI<
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) a<%WFix
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) U/2g N
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% 3 3 r^3 * sin(3*theta) sqrt(8) }TZ5/zn.Dw
% 4 -4 r^4 * cos(4*theta) sqrt(10) )K8k3]y&
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4'W| '4'b
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) zv]-(<B
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) \*H/YByTb
% 4 4 r^4 * sin(4*theta) sqrt(10) %($qg-x
% -------------------------------------------------- ?gb"S,
% 2roPZj
% Example 1: nu] k<^I5|
% 3,bA&c3
% % Display the Zernike function Z(n=5,m=1) r3l}I6
% x = -1:0.01:1; Z1FO.[FV
% [X,Y] = meshgrid(x,x); "3{xa;c
% [theta,r] = cart2pol(X,Y); z[DUktZl
% idx = r<=1; PXcpROg56
% z = nan(size(X)); eB78z@
% z(idx) = zernfun(5,1,r(idx),theta(idx)); TR,,=3n
% figure C+Wb_
% pcolor(x,x,z), shading interp j=)Cyg3_%
% axis square, colorbar t@1e9uR
% title('Zernike function Z_5^1(r,\theta)') )^uLZMNaI
% ch<Fi%)
% Example 2: cve(pkl
% e0HG"z4
% % Display the first 10 Zernike functions R0;c'W)
% x = -1:0.01:1; $EbxV"b+
% [X,Y] = meshgrid(x,x); 36JVnW;
% [theta,r] = cart2pol(X,Y); =iRi9r'l
% idx = r<=1; 5nr}5bum
% z = nan(size(X)); |EaGKC(
% n = [0 1 1 2 2 2 3 3 3 3]; (|Am
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; !arcQ:T@G
% Nplot = [4 10 12 16 18 20 22 24 26 28]; -[s*R%w
% y = zernfun(n,m,r(idx),theta(idx)); j-ugsV`2=*
% figure('Units','normalized') [UquI "
% for k = 1:10 Z~8Xp
% z(idx) = y(:,k); R:B-4
% subplot(4,7,Nplot(k)) Qp<?[C}'W
% pcolor(x,x,z), shading interp M}}9
% set(gca,'XTick',[],'YTick',[]) qt}vM*0}V
% axis square epm
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% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 2GcQh]ohc
% end !h7`W*::
% E=w $r
% See also ZERNPOL, ZERNFUN2. XZuJ<]}X,
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% Paul Fricker 11/13/2006 1j6ZSE/*|
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% Check and prepare the inputs: |Wj)kr !|
% ----------------------------- #O^H?3Q3
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) x2/|i?ZO
error('zernfun:NMvectors','N and M must be vectors.') 3j0/&ON
end .txgb
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if length(n)~=length(m) @\Sa)
error('zernfun:NMlength','N and M must be the same length.') _x z_D12
end iBxCk^
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n = n(:); AZorz Q]s
m = m(:); x 3#1
if any(mod(n-m,2)) 5|xFY/%
error('zernfun:NMmultiplesof2', ... Iqe4O~)
'All N and M must differ by multiples of 2 (including 0).') /J3e[?78u
end dnNC
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if any(m>n) q+dY&4&u
error('zernfun:MlessthanN', ... 6YrkS;_HS
'Each M must be less than or equal to its corresponding N.') 6*kY7
end }0?642 =-
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if any( r>1 | r<0 ) 4eYj.=I
error('zernfun:Rlessthan1','All R must be between 0 and 1.') W$B>O
end i+Px &9o<9
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) waQNX7Xdn
error('zernfun:RTHvector','R and THETA must be vectors.') jr*A1y*
end sBu=@8R]y
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r = r(:); 0+;.T1?
theta = theta(:); '7
6}6G%
length_r = length(r); U[c,cdA
if length_r~=length(theta) 9HRYk13ae
error('zernfun:RTHlength', ... xRP#}i:m
'The number of R- and THETA-values must be equal.') -#Yg B5
end zbx,qctYo$
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% Check normalization: {_ww1'|A
% -------------------- ^g~Asz5]
if nargin==5 && ischar(nflag) p44d&9
isnorm = strcmpi(nflag,'norm'); aIRCz=N
if ~isnorm K4N~ApLB+
error('zernfun:normalization','Unrecognized normalization flag.') %$Wt"~WE"O
end :!N 5daK
else [}9R9G>"
isnorm = false; PsEm(.z
end b@Ik
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r{#od
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% Compute the Zernike Polynomials un6grvxr
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% WsL*P.J
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% Determine the required powers of r: hh{4r} |
% ----------------------------------- 2l{g$44
m_abs = abs(m); VDx=Tsu-
rpowers = []; dU3UCD+2y
for j = 1:length(n) ;f^.7|
rpowers = [rpowers m_abs(j):2:n(j)]; )j4]Y dJ
end V Z}^1e
rpowers = unique(rpowers); "7JO~T+v
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% Pre-compute the values of r raised to the required powers, ?<#6=
% and compile them in a matrix: <o3e0JCq
% ----------------------------- {Rc/Ten
if rpowers(1)==0 ,6}HAC $
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); C^aP)&
qt
rpowern = cat(2,rpowern{:}); YnNB#x8|
rpowern = [ones(length_r,1) rpowern]; Fm`hFBKW
else $iEM$
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Vu*yEF}
rpowern = cat(2,rpowern{:}); ot;j6eAH~E
end G{knO?BK
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c;&m}ImLe.
% Compute the values of the polynomials: s!9.o_k
% -------------------------------------- !Q*.Dw()[
y = zeros(length_r,length(n)); kmi[u8iXD_
for j = 1:length(n) SWz+.W{KQ"
s = 0:(n(j)-m_abs(j))/2; NC>rZS]
pows = n(j):-2:m_abs(j); {e/12q
for k = length(s):-1:1 q+19EJ(
p = (1-2*mod(s(k),2))* ... wlAlIvIT
prod(2:(n(j)-s(k)))/ ... ,LSF@1|Fx
prod(2:s(k))/ ... !qV{OXdrB
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Cj _Q9/
prod(2:((n(j)+m_abs(j))/2-s(k))); 54JZEc
idx = (pows(k)==rpowers); (Vf&,b@U_
y(:,j) = y(:,j) + p*rpowern(:,idx); - A
x$ Y
end 4+:Q"
z;zyk
if isnorm HN7(-ml=B
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); QjWv?tm
end MQ$[jOAqP
end eZ[CqUJ&