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, I`H&b&
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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, y<6c*e1
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? 2#sFY/@
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? $
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function z = zernfun(n,m,r,theta,nflag) HbI{Xf[6LP
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. wbrOL(q.m
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N D
5n\h5
% and angular frequency M, evaluated at positions (R,THETA) on the "nCK%w=
% unit circle. N is a vector of positive integers (including 0), and =eUKpYI
% M is a vector with the same number of elements as N. Each element ye9GBAj
/
% k of M must be a positive integer, with possible values M(k) = -N(k) j?m(l,YD|*
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, L"vrX
% and THETA is a vector of angles. R and THETA must have the same uX3yq<lK"
% length. The output Z is a matrix with one column for every (N,M) O>qlWPht
% pair, and one row for every (R,THETA) pair. ))f@9m
% I3Z?xsa@Z
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike %W\NYSm
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), :_6o|9J\t
% with delta(m,0) the Kronecker delta, is chosen so that the integral kC%H E
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, * @]wT'
% and theta=0 to theta=2*pi) is unity. For the non-normalized Lw,}wM5X
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. &uRT/+18W3
% ~6n|GxR.[
% The Zernike functions are an orthogonal basis on the unit circle. i@nRZ$ K
% They are used in disciplines such as astronomy, optics, and zPp22
% optometry to describe functions on a circular domain. #%k_V+o3
% iv_3R}IbX
% The following table lists the first 15 Zernike functions. 9!n95
% s(3u\#P
% n m Zernike function Normalization _Sfu8k>):
% -------------------------------------------------- -LRx}Mb9
% 0 0 1 1 F$tzsz,9n
% 1 1 r * cos(theta) 2 ;&=CZ6vH
% 1 -1 r * sin(theta) 2 >8I~i:hn
% 2 -2 r^2 * cos(2*theta) sqrt(6) ^6kl4:{idE
% 2 0 (2*r^2 - 1) sqrt(3) k1xx>=md|C
% 2 2 r^2 * sin(2*theta) sqrt(6) t;VMtIW+E
% 3 -3 r^3 * cos(3*theta) sqrt(8) _]o7iqtv
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) WUie`p
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) KJoa^e;~
% 3 3 r^3 * sin(3*theta) sqrt(8) 'uL$j=vB
% 4 -4 r^4 * cos(4*theta) sqrt(10) W`9{RZ'
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) g]L8Jli
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 1q!k#Cliu
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) J' P:SC1
% 4 4 r^4 * sin(4*theta) sqrt(10) eK1l~W%
% -------------------------------------------------- qzxWv5UH
% b}Gm{;s!
% Example 1: rhPv{6Z|7
% D? %*L
% % Display the Zernike function Z(n=5,m=1) >l)x~Bkf$j
% x = -1:0.01:1; 2EpQ(G
J
% [X,Y] = meshgrid(x,x); (Ud"+a
% [theta,r] = cart2pol(X,Y); ? 3fnt"
% idx = r<=1; 5,MM`:{{
% z = nan(size(X)); <Rw2F?S~)n
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Cj~e` VRhk
% figure 81|[Y'f
% pcolor(x,x,z), shading interp 3n7>qZ.d
% axis square, colorbar I_8 n>\u
% title('Zernike function Z_5^1(r,\theta)') RjUrpS[I
% ^^ix4[1$Z
% Example 2: ,|$1(z*a{c
% SbUac<
% % Display the first 10 Zernike functions Xmmj.ZUr
% x = -1:0.01:1; KS5a8'U
% [X,Y] = meshgrid(x,x); 8hww({S2
% [theta,r] = cart2pol(X,Y); mm#UaEp
% idx = r<=1; !sI^Lh,Y
% z = nan(size(X)); VSUWX1k4%
% n = [0 1 1 2 2 2 3 3 3 3]; ImB5F'HI$
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; H'$H@Kn]-
% Nplot = [4 10 12 16 18 20 22 24 26 28]; S3HyB
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% y = zernfun(n,m,r(idx),theta(idx)); e9'0CH<
% figure('Units','normalized') X*7VDt=
% for k = 1:10 3jfAv@I ~
% z(idx) = y(:,k); /:
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% subplot(4,7,Nplot(k)) t; #@t/`
% pcolor(x,x,z), shading interp be-HF;lZe'
% set(gca,'XTick',[],'YTick',[]) >f&L7@
% axis square \uo{I~Qd
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Zr6.Nw
% end PL31(!`@d
% kene'
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% See also ZERNPOL, ZERNFUN2. y~OP9Tg
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+:aNgO#e8
% Paul Fricker 11/13/2006 mcQ
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% Check and prepare the inputs: yfqe6-8U
% ----------------------------- \AI-x$5R*
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) c*<BU6y
error('zernfun:NMvectors','N and M must be vectors.') M;AvOk|&
end ?%ltoezf
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if length(n)~=length(m) "85)2*+
error('zernfun:NMlength','N and M must be the same length.') 6e.l#
c!1}
end 'O2/PU2_
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=?]S8cth
n = n(:); ZhRdml4U2
m = m(:); Hd-g|'^K
if any(mod(n-m,2)) m#_M"B.cm
error('zernfun:NMmultiplesof2', ... OM7AK
B=S
'All N and M must differ by multiples of 2 (including 0).') N?,
end lc [)Ev
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if any(m>n) C(>g4.-p8
error('zernfun:MlessthanN', ... T~XKV`LQ
'Each M must be less than or equal to its corresponding N.') `|92!Ej
end ZcHIk{|
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if any( r>1 | r<0 ) @x[A^
error('zernfun:Rlessthan1','All R must be between 0 and 1.') \j.l1O
end Y>8JHoV
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) |,H2ge
error('zernfun:RTHvector','R and THETA must be vectors.') F]<2nb7
end dxS5-aWy9w
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r = r(:); at\u7>;.^k
theta = theta(:); >-3>Rjo>
length_r = length(r); Ll#W:~
if length_r~=length(theta) 4}*.0'Hz
error('zernfun:RTHlength', ... +.rOqkxJ
'The number of R- and THETA-values must be equal.') L0{[L
end &?xtmg<d
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% Check normalization: Gv#bd05X
% -------------------- nC?Lz1re
if nargin==5 && ischar(nflag) 7GErh,
isnorm = strcmpi(nflag,'norm'); x",ktE>9
if ~isnorm +`$$^x
error('zernfun:normalization','Unrecognized normalization flag.') w7q6v>
end zyP/'X_~:
else ,S`FxJcE
isnorm = false; ~oK0k_{~
end +nB0O/m'U
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {(tR<z)
% Compute the Zernike Polynomials /+ais3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QOPh3+.5
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% Determine the required powers of r: *.D{d0A
% ----------------------------------- -Oz! GX
m_abs = abs(m); !\Cu J5U
rpowers = []; ,R7j9#D
for j = 1:length(n) ZnuRy:
rpowers = [rpowers m_abs(j):2:n(j)]; MJH>rsTQ
end @`^Z5n.4
rpowers = unique(rpowers); \F+".X#jh
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<*/Z>Z_c2
% Pre-compute the values of r raised to the required powers, 2FO<Z %Y
% and compile them in a matrix: p u?COA
% ----------------------------- XgeUS;qtta
if rpowers(1)==0 hKnV=Ha(
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 7*WO9R/
rpowern = cat(2,rpowern{:}); x'6i9]+r
rpowern = [ones(length_r,1) rpowern]; bwszfPM
else W?ghG
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); W(-son~I
rpowern = cat(2,rpowern{:}); z 9WeOs
end Y 9st3
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% Compute the values of the polynomials: 3.ShAL
% -------------------------------------- Xw|-v$'y
y = zeros(length_r,length(n)); #i.BOQxS
for j = 1:length(n) uI9+@oV
s = 0:(n(j)-m_abs(j))/2; _oefp*iWS
pows = n(j):-2:m_abs(j); WZTv
for k = length(s):-1:1 ,u-9e4
p = (1-2*mod(s(k),2))* ... NH=@[t)P,
prod(2:(n(j)-s(k)))/ ... MFWkJbZV
prod(2:s(k))/ ... 2$o\`^dy
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... f3.oc9G
prod(2:((n(j)+m_abs(j))/2-s(k))); 2<'ol65/c
idx = (pows(k)==rpowers); K05T`+N,
y(:,j) = y(:,j) + p*rpowern(:,idx); Li 9$N"2
end >Af0S;S
ol
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if isnorm ?UeV5<TewS
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); qn}VW0!
end h^14/L=|
end ;.R)
uCd{=
% END: Compute the Zernike Polynomials mW,b#'hy
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +-5YmN'
+ kF%>F]
y Tk1
% Compute the Zernike functions: rx<P#y]3)
% ------------------------------ :n%&
idx_pos = m>0; Vk[M .=J
idx_neg = m<0; g$/7km{TP
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z = y; ^L.'At
if any(idx_pos) A2P.5EN
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 0]nveC$
end
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if any(idx_neg) .O&YdUo
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); |S:erYE,G
end iYlkc
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% EOF zernfun W!+=`[Ff