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, o5(p&:1M
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, %E95R8SL
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? KKR@u(+"a
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? YEZd8Y
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function z = zernfun(n,m,r,theta,nflag) -uDB#?q:W
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. &j\<UPn
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N G:f\wK[
% and angular frequency M, evaluated at positions (R,THETA) on the }t tiL
% unit circle. N is a vector of positive integers (including 0), and [b:$sR;
% M is a vector with the same number of elements as N. Each element x~Egax
% k of M must be a positive integer, with possible values M(k) = -N(k) D}SYv})Ti
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, IR (6
% and THETA is a vector of angles. R and THETA must have the same +?[,y
% length. The output Z is a matrix with one column for every (N,M) PQ`p:=~>:i
% pair, and one row for every (R,THETA) pair. Ex'6 WN~kD
% r7z8ICX'q
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike [E_eaez7#
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), mC
P*v-
% with delta(m,0) the Kronecker delta, is chosen so that the integral H[ 6L!
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, g">E it*[
% and theta=0 to theta=2*pi) is unity. For the non-normalized )$#]h]ac
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Ih*}1D)7
% gU7@}P
% The Zernike functions are an orthogonal basis on the unit circle. (U"Ub;[7
% They are used in disciplines such as astronomy, optics, and -c-#1_X5
% optometry to describe functions on a circular domain. EG<YxNX,
% \atztC{-L>
% The following table lists the first 15 Zernike functions. \ltA&}!
% s)#8>s -
% n m Zernike function Normalization GY@-}p~it
% -------------------------------------------------- 4\)"Ih
% 0 0 1 1 adG=L9
"n
% 1 1 r * cos(theta) 2 _jV(Gv'
% 1 -1 r * sin(theta) 2 I#0WN
% 2 -2 r^2 * cos(2*theta) sqrt(6) hlPZTr=a
% 2 0 (2*r^2 - 1) sqrt(3) ].f28bY
% 2 2 r^2 * sin(2*theta) sqrt(6) 'JR2@W`]]
% 3 -3 r^3 * cos(3*theta) sqrt(8) @1#QbNp#
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) -LF0%G
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) c+PT"/3
% 3 3 r^3 * sin(3*theta) sqrt(8) B3V:? #
% 4 -4 r^4 * cos(4*theta) sqrt(10) l MCoc 'ae
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0MK|spc
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) R u^v!l`!7
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) [AzQP!gi
% 4 4 r^4 * sin(4*theta) sqrt(10) (fmcWHs
% -------------------------------------------------- tETT\y|'
% 14TA( v]T
% Example 1: N zY}-:{
% c}iVBN6~.<
% % Display the Zernike function Z(n=5,m=1) 2Yd0:$a
% x = -1:0.01:1; % AqUVt9}
% [X,Y] = meshgrid(x,x); D9H(kk
% [theta,r] = cart2pol(X,Y); lv_|ws
% idx = r<=1; Nz`4q%+
% z = nan(size(X)); d,}fp)
% z(idx) = zernfun(5,1,r(idx),theta(idx)); B4^+&B#
% figure 0be1aY;m&
% pcolor(x,x,z), shading interp )clSW
% axis square, colorbar l[=7<F
% title('Zernike function Z_5^1(r,\theta)') iB[>uW
% p[BF4h{E
% Example 2: %liu[6_
% xaO9?{O
% % Display the first 10 Zernike functions 1JIL6w_
% x = -1:0.01:1; %(a<(3r
% [X,Y] = meshgrid(x,x); QUL^]6$
% [theta,r] = cart2pol(X,Y); c"OBm#
% idx = r<=1; +g_+JLQ
% z = nan(size(X)); BZy&;P
% n = [0 1 1 2 2 2 3 3 3 3]; [%(}e1T(
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; p<1z!`!P
% Nplot = [4 10 12 16 18 20 22 24 26 28]; )X{ x\
/N
% y = zernfun(n,m,r(idx),theta(idx)); Qmxe*@{`
% figure('Units','normalized') SVsLu2tVY
% for k = 1:10 %,$Ms?,n`
% z(idx) = y(:,k); fj
X~"U
% subplot(4,7,Nplot(k)) O)n LV~X
% pcolor(x,x,z), shading interp !'>(r K$
% set(gca,'XTick',[],'YTick',[]) =a>a A Z
% axis square 5Hvg%g-c
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) f}q4~NPn-
% end |4uH
% ( lbF/F>v
% See also ZERNPOL, ZERNFUN2. `n%uvo}UT
u"IYAyzL
%2Q:+6)
% Paul Fricker 11/13/2006 UpL1C~&
;-p1z%
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yM$@*od
% Check and prepare the inputs: DQ7+
% ----------------------------- O]{3aMs!Y
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) [+0rlmB
error('zernfun:NMvectors','N and M must be vectors.') N9LBji;nH
end f8`K8Y]4
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if length(n)~=length(m) cIqk=_]
error('zernfun:NMlength','N and M must be the same length.') <p"[jC2zF;
end n1OxT"tD
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n = n(:); )N[9r{3
m = m(:); dQ6:c7hp>D
if any(mod(n-m,2)) uq<kT [
error('zernfun:NMmultiplesof2', ... ([~9v@+
'All N and M must differ by multiples of 2 (including 0).') Il(p!l<Xz#
end r|$@Wsb?#
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if any(m>n) }*.S=M]y$
error('zernfun:MlessthanN', ... Sa5+_TW
'Each M must be less than or equal to its corresponding N.') eELJDSd
BV
end )eFXjnHN
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if any( r>1 | r<0 ) 0zrgK;9
error('zernfun:Rlessthan1','All R must be between 0 and 1.') '6l4MR$j&m
end VC%{qal;q
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 7_Yxz$m
error('zernfun:RTHvector','R and THETA must be vectors.') t)|*-=
end E6"+\-e
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r = r(:); #CKPNk
c
theta = theta(:); U5 X\RXy~
length_r = length(r); V+#Sb
if length_r~=length(theta) r!H'8O!
error('zernfun:RTHlength', ... Dqss/vwV
'The number of R- and THETA-values must be equal.') 0vN <0
end 7!%/vO0m
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% Check normalization: H CKD0xx
% -------------------- AY]dwKw
if nargin==5 && ischar(nflag) p;;4b@
isnorm = strcmpi(nflag,'norm'); ,;3#}OGg
if ~isnorm L $R"?O7
error('zernfun:normalization','Unrecognized normalization flag.') l=EnK"aU
end aYTVYg
else 3khsGD@
isnorm = false; KGsS2
end w>-@h>Ln
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O~3<P3W
% Compute the Zernike Polynomials !O;su~7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Gn*cphb
m|K"I3W$
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% Determine the required powers of r: ,1xX`:
% ----------------------------------- .3xpDVW^e
m_abs = abs(m); x`7Ch3`4}
rpowers = []; 3y&N}'R(F
for j = 1:length(n) 6"3-8orj
rpowers = [rpowers m_abs(j):2:n(j)]; R]dN-'U
end Ck`-<)uN
rpowers = unique(rpowers); 2o8:[3C5
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% Pre-compute the values of r raised to the required powers, a1^CpeG~
% and compile them in a matrix: }~W:3A{7;
% ----------------------------- n6A N
if rpowers(1)==0 eBlWwUy*6f
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); dO?zLc0f
rpowern = cat(2,rpowern{:}); /l.:GH36f
rpowern = [ones(length_r,1) rpowern]; rV{:'"=y-
else DIsK+1
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); { XI 0KiE
rpowern = cat(2,rpowern{:}); }j+Af["W?
end `'W/uCpl
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#%Hk-a=>)#
% Compute the values of the polynomials: -|z
]Ir
% -------------------------------------- ;$a+ >
y = zeros(length_r,length(n)); KjWF;VN*[3
for j = 1:length(n) fyt ODsb>
s = 0:(n(j)-m_abs(j))/2; C8{bqmlm@
pows = n(j):-2:m_abs(j); <x!q!;
for k = length(s):-1:1 RB\
Hl
p = (1-2*mod(s(k),2))* ... V /.Na(C~
prod(2:(n(j)-s(k)))/ ... CdEQiu
prod(2:s(k))/ ... G3.*fSY$.<
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... }na0
prod(2:((n(j)+m_abs(j))/2-s(k)));
h.Y&_=Gc
idx = (pows(k)==rpowers); M&Q