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, %_aMl
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, rJ'I>Q~x6
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? ZRUhAp'<qj
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? MZSxQ8
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function z = zernfun(n,m,r,theta,nflag) ql&*6KZ"
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 0ZPV'`KGp
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N !sA_?2$
% and angular frequency M, evaluated at positions (R,THETA) on the t.hm9}UQ
% unit circle. N is a vector of positive integers (including 0), and rt +..t\
% M is a vector with the same number of elements as N. Each element ])#\_'fg
% k of M must be a positive integer, with possible values M(k) = -N(k) MuEy>dl
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, QldzQ%4c\
% and THETA is a vector of angles. R and THETA must have the same xq-$\#O
% length. The output Z is a matrix with one column for every (N,M) %YlTF\-
% pair, and one row for every (R,THETA) pair. ? {F{;r
% D0]a\,aZ
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 2vKx]w
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ]`w}+B'/
% with delta(m,0) the Kronecker delta, is chosen so that the integral <B&R6<]T
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, snp v z1iS
% and theta=0 to theta=2*pi) is unity. For the non-normalized Q\J,}1<`6
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. fd8#Ng"1
% 4R ) |->"
% The Zernike functions are an orthogonal basis on the unit circle. w3D]~&]
% They are used in disciplines such as astronomy, optics, and 3rf#Q}"
% optometry to describe functions on a circular domain. 9-bG<`v\E
% #G,XDW2"w
% The following table lists the first 15 Zernike functions. mf|pNiQ,
% g>7Y~_}
% n m Zernike function Normalization Oz:ZQ M
% -------------------------------------------------- YirC*
% 0 0 1 1 ;
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% 1 1 r * cos(theta) 2 X`\:_|
% 1 -1 r * sin(theta) 2 kJ: 2;t=
% 2 -2 r^2 * cos(2*theta) sqrt(6) K{}4zuZ
% 2 0 (2*r^2 - 1) sqrt(3) "t&{yBQ0u
% 2 2 r^2 * sin(2*theta) sqrt(6) JF qf;3R
% 3 -3 r^3 * cos(3*theta) sqrt(8) *"G 8
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) VKLU0*2R
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ]s|lxqP
% 3 3 r^3 * sin(3*theta) sqrt(8) CYB=Uq,
% 4 -4 r^4 * cos(4*theta) sqrt(10) +~|AT+|iI
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >e8JK*Blz
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) %f[Ep 3D
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?:|YGLaB
% 4 4 r^4 * sin(4*theta) sqrt(10) ,\hYEup
% -------------------------------------------------- |r~
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% 6|;0ax4:P
% Example 1: z-0:m|=yH
% j=.g:&r)
% % Display the Zernike function Z(n=5,m=1) qCJ=Z
% x = -1:0.01:1; yCM{M
% [X,Y] = meshgrid(x,x); "L~@.W!@
% [theta,r] = cart2pol(X,Y); 9Nl*4
% idx = r<=1; 8g5V,3_6
% z = nan(size(X)); ^)cM&Bxt%
% z(idx) = zernfun(5,1,r(idx),theta(idx)); U
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% figure T~Y g5J
% pcolor(x,x,z), shading interp y-`I) w%
% axis square, colorbar C"T ,MH
% title('Zernike function Z_5^1(r,\theta)') rqvU8T7A
% .g-3e"@
% Example 2: cy:;)E>/
% owMuT^x?
% % Display the first 10 Zernike functions L/k40cEI^z
% x = -1:0.01:1; C/+nSe.
% [X,Y] = meshgrid(x,x); fJ :jk6@
% [theta,r] = cart2pol(X,Y); S.fXHtSx
% idx = r<=1; c57b f
% z = nan(size(X)); A. Nz_!
% n = [0 1 1 2 2 2 3 3 3 3]; +IsWI;lp
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; [n<.fw8$b
% Nplot = [4 10 12 16 18 20 22 24 26 28]; >
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% y = zernfun(n,m,r(idx),theta(idx)); <jL#>L%%
% figure('Units','normalized') h2}am:%mC
% for k = 1:10 "X?LAo
% z(idx) = y(:,k); A1!:BC
% subplot(4,7,Nplot(k)) `Wwh`]#"~d
% pcolor(x,x,z), shading interp O&P>x#w
% set(gca,'XTick',[],'YTick',[]) >DmRP7v
% axis square )n7)}xy#z
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) cJ4S!
% end bf^ly6ml
% xXa#J)'
% See also ZERNPOL, ZERNFUN2. lWl-@*'
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% Paul Fricker 11/13/2006 l_sg)Vr/b
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% Check and prepare the inputs: #y`k$20"
% ----------------------------- o;'4c
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) K-Y*T}?
error('zernfun:NMvectors','N and M must be vectors.') j)<[j&OWw
end ]J~g'">
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if length(n)~=length(m) {G]`1Q1DR
error('zernfun:NMlength','N and M must be the same length.') 'v`~(9'Rcj
end R mgxf/
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n = n(:); d TgM"k
m = m(:); 4jD\]Q="1
if any(mod(n-m,2)) !%)L&W_
error('zernfun:NMmultiplesof2', ... 1o)=GV1
'All N and M must differ by multiples of 2 (including 0).') z+2u-jG
end oYGUjI
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if any(m>n) Fj48quW1\P
error('zernfun:MlessthanN', ... _/8y1)I
'Each M must be less than or equal to its corresponding N.') zh
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end 2tlO"c:_/
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if any( r>1 | r<0 ) gtl;P_
error('zernfun:Rlessthan1','All R must be between 0 and 1.') I[a%a!QO
end /!o1l\i=5
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) (h%|;9tF
error('zernfun:RTHvector','R and THETA must be vectors.') =`ywd]\7
end s:G[Em1
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r = r(:); nN!vgn
j
theta = theta(:); =54Vs8.
length_r = length(r); Ty(yh(oYF`
if length_r~=length(theta) {m>~`
error('zernfun:RTHlength', ... re2Fv:4{
'The number of R- and THETA-values must be equal.') @ICejB<
end KX$qM g1j
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% Check normalization: QJniM"8v
% -------------------- FDZeIj9uF
if nargin==5 && ischar(nflag) dW:w<{a!R
isnorm = strcmpi(nflag,'norm'); oT$(<$&<
if ~isnorm @DUN;L 4
error('zernfun:normalization','Unrecognized normalization flag.') @5JLjCN
end {: Am9B
else $a)JCErN
isnorm = false; {EZFx,@t
end 0:PH[\Z
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dY4 8S{
% Compute the Zernike Polynomials X=-gAutfE=
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _wIBm2UO
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0 m)-7@
% Determine the required powers of r: h0&>GY;i
% ----------------------------------- n$}R/*
m_abs = abs(m); K*J4&5?/
rpowers = []; A}
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for j = 1:length(n) ..v@Q%
rpowers = [rpowers m_abs(j):2:n(j)]; 8T!fGzHx
end d"QM;9
rpowers = unique(rpowers); FCUVP,"T
BLsdx}
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% Pre-compute the values of r raised to the required powers, ;H D 4~3
% and compile them in a matrix: 5#N"WHz!
% ----------------------------- ir( -$*J
if rpowers(1)==0 {Zd)U "
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); F}VS)
rpowern = cat(2,rpowern{:}); ^59YfC<f
rpowern = [ones(length_r,1) rpowern]; YL0WUD_>
else (25^r
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )VV4HoH]8
rpowern = cat(2,rpowern{:}); +8?R+0P
end %M4XbSN|
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% Compute the values of the polynomials: =0jmm(:Jh
% -------------------------------------- |e.3FjTH
y = zeros(length_r,length(n)); '? !7 Be
for j = 1:length(n) w[J
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s = 0:(n(j)-m_abs(j))/2; }+QhW]nO{F
pows = n(j):-2:m_abs(j); Q8M:7#ySji
for k = length(s):-1:1 Ah8^^h|TPJ
p = (1-2*mod(s(k),2))* ... r P<d[u
prod(2:(n(j)-s(k)))/ ... `CTkx?e[
prod(2:s(k))/ ... Y3sNr)qss
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 6@,'m
prod(2:((n(j)+m_abs(j))/2-s(k))); DLg `Q0`M5
idx = (pows(k)==rpowers);
zO7lsx2=
y(:,j) = y(:,j) + p*rpowern(:,idx); 2s]]!{Z#
end *h5ld P
y~#R:&d"
if isnorm H *z0xxa
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); hhh: rmEZl
end ;_Of`C+
end )0 42?emn
% END: Compute the Zernike Polynomials fjz2m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zd*W5~xKg
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% Compute the Zernike functions: OJC*|kN-#^
% ------------------------------ Jte:l:yjtA
idx_pos = m>0; [/#k$-
idx_neg = m<0; <or>bo^
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z = y; g)zn.]
if any(idx_pos) hj m.Ath
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); x:&L?eOT
end F%ylR^H>
if any(idx_neg) !_/8!95
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ck4T#g;=
end Sv^'CpQ
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% EOF zernfun =+sIX3