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, :6LOb f\01
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, ;@K,>$ur-
这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? ><iE VrpN
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? X?$Eb
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function z = zernfun(n,m,r,theta,nflag) `xe[\Z2
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. l ,)l"6OV
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N jM
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% and angular frequency M, evaluated at positions (R,THETA) on the {npKdX
% unit circle. N is a vector of positive integers (including 0), and P,AS`=z
% M is a vector with the same number of elements as N. Each element pfg"6P
% k of M must be a positive integer, with possible values M(k) = -N(k) ,G1|]
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% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, aq"E@fb
% and THETA is a vector of angles. R and THETA must have the same :YjOv
% length. The output Z is a matrix with one column for every (N,M) 4,f[D9|:
% pair, and one row for every (R,THETA) pair. )Y~q6D K
% 7d/wT+f
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 93fKv
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9<<$uf.B
% with delta(m,0) the Kronecker delta, is chosen so that the integral Ed#%F-1sX
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, M4M
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% and theta=0 to theta=2*pi) is unity. For the non-normalized `{I,!to
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. H_;Dq*
% F']Vg31c
% The Zernike functions are an orthogonal basis on the unit circle. 8s8q`_.)(
% They are used in disciplines such as astronomy, optics, and 3f's>+,#%
% optometry to describe functions on a circular domain. 3leg,qd
% #f.@XIt'
% The following table lists the first 15 Zernike functions. 05*_h0}
% .5L/<
% n m Zernike function Normalization 9 N=KU
% -------------------------------------------------- m|~,# d@
% 0 0 1 1 R2Tvo?xI7
% 1 1 r * cos(theta) 2 O~d!*A
% 1 -1 r * sin(theta) 2 ~2@U85"o
% 2 -2 r^2 * cos(2*theta) sqrt(6) T'XAcH
% 2 0 (2*r^2 - 1) sqrt(3) $';'MoS
% 2 2 r^2 * sin(2*theta) sqrt(6) G+[>or}
% 3 -3 r^3 * cos(3*theta) sqrt(8) R ;5w*e}?5
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) \+GXUnkj
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ~\<ZWU<BE
% 3 3 r^3 * sin(3*theta) sqrt(8) #2yOqUO\
% 4 -4 r^4 * cos(4*theta) sqrt(10) B>X+eK
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) T<zonx1
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) tP!sOvQ:
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) g/z9bOgIX
% 4 4 r^4 * sin(4*theta) sqrt(10) 1:YDN.*
% -------------------------------------------------- U
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% Zpb3>0<R
% Example 1: a^[io1}-
% >;xEzc!W3*
% % Display the Zernike function Z(n=5,m=1) EUuMSDp
% x = -1:0.01:1; 6El%T]^
% [X,Y] = meshgrid(x,x); w#PaN83+
% [theta,r] = cart2pol(X,Y); od^ha
% idx = r<=1; =5Q;quKu^5
% z = nan(size(X)); Rz=]KeZu
% z(idx) = zernfun(5,1,r(idx),theta(idx)); tY# F8a&
% figure m$LZ3=v%8
% pcolor(x,x,z), shading interp D4#,9?us
% axis square, colorbar 5jNBt>.0
% title('Zernike function Z_5^1(r,\theta)') w5n>hz_5
% "6KOql3
% Example 2: ,7%(Jj$
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% ^"buF\3L
% % Display the first 10 Zernike functions HwST^\Ao
% x = -1:0.01:1; I}:>M!w
% [X,Y] = meshgrid(x,x); '3hvR4P
% [theta,r] = cart2pol(X,Y); 3'/wRK l
% idx = r<=1; mz\m^g3
% z = nan(size(X)); GUN<ZOYb=
% n = [0 1 1 2 2 2 3 3 3 3]; bjT0Fi0-
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 8#Z$}?W
% Nplot = [4 10 12 16 18 20 22 24 26 28]; +'#d*r91@
% y = zernfun(n,m,r(idx),theta(idx)); ZN4&:9M
% figure('Units','normalized') cQ+,F2
% for k = 1:10 Be]o2N;J
% z(idx) = y(:,k); W2^eE9
% subplot(4,7,Nplot(k)) .{ x5(bi0S
% pcolor(x,x,z), shading interp 7H>dv'
% set(gca,'XTick',[],'YTick',[]) pu>LC6m3a
% axis square tQl=
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) n,HWVo>([
% end T >-F~?7Sv
% MPL2#YU/a
% See also ZERNPOL, ZERNFUN2. _v$mGZpGY
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% Paul Fricker 11/13/2006 R>0ta
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M]_vb,=1
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% Check and prepare the inputs: O-'T*M>
% ----------------------------- Ahwu'mgnC
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Hd2_Cg FB
error('zernfun:NMvectors','N and M must be vectors.') XqwdJND
end r}5GJ|p0
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if length(n)~=length(m) _|Dt6
error('zernfun:NMlength','N and M must be the same length.') jyT(LDsS
end :iWV:0)P
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n = n(:); )eG&"3kFe!
m = m(:); #M>E{w9
if any(mod(n-m,2)) =VSieh
error('zernfun:NMmultiplesof2', ... eo,]b1C2n
'All N and M must differ by multiples of 2 (including 0).') 9q5[W=|
end 1%:A9%O)t
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if any(m>n) pKJ[e@E^
error('zernfun:MlessthanN', ... #,9|Hr%
'Each M must be less than or equal to its corresponding N.') s`TBz8QO$
end ujSzm=_P
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if any( r>1 | r<0 ) *K 7L5.
error('zernfun:Rlessthan1','All R must be between 0 and 1.') FG(`&S+,
end l00D|W_9
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ~iSW^mi
error('zernfun:RTHvector','R and THETA must be vectors.') Af%?WZlOq
end eyG.XAP
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r = r(:); S`^W#,rj
theta = theta(:); iUKj:q:
length_r = length(r); WT)")0)[
if length_r~=length(theta) *~"`&rM(
error('zernfun:RTHlength', ... CNz[@6-cYU
'The number of R- and THETA-values must be equal.')
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end ]aR4U`
D0P% .r"v
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% Check normalization: z~_\onC
% -------------------- b(VU{cf2d
if nargin==5 && ischar(nflag) GwycSb1
isnorm = strcmpi(nflag,'norm'); -$q/7,os
if ~isnorm uj@<_|7
error('zernfun:normalization','Unrecognized normalization flag.') {MtB!x
end aVb]H0
else E6gEP0b
isnorm = false; [ bW=>M
end KWUz]>Z
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >KPJ74R
% Compute the Zernike Polynomials
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z^l!y5s/H
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% Determine the required powers of r: jJvNN -^
% ----------------------------------- f0s
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m_abs = abs(m); X
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rpowers = []; yDC97#%3u
for j = 1:length(n) 1sjn_fPz
rpowers = [rpowers m_abs(j):2:n(j)]; #V6
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end #77UKYj2L-
rpowers = unique(rpowers); |DD?3#G01
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% Pre-compute the values of r raised to the required powers, .7FI%
% and compile them in a matrix: dWy1=UQfP
% ----------------------------- {1%ZyY
if rpowers(1)==0 uH[0kh
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 3Y-v1.^j
rpowern = cat(2,rpowern{:}); E2|iAT+=.
rpowern = [ones(length_r,1) rpowern]; 5m42Bqy"
else -#6*T,f0P(
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); l,FoK76G
rpowern = cat(2,rpowern{:}); Jf$wBPg
end DcA'{21
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% Compute the values of the polynomials: P{eRDQ=
% -------------------------------------- J"rwWIxO*
y = zeros(length_r,length(n)); #:|?t&On
for j = 1:length(n) l`&6W?C
s = 0:(n(j)-m_abs(j))/2; J36@Pf]h
pows = n(j):-2:m_abs(j); &|'6-wD.
for k = length(s):-1:1 ?8@*q6~8
p = (1-2*mod(s(k),2))* ... h\d($Ki
prod(2:(n(j)-s(k)))/ ... U_'q- *W
prod(2:s(k))/ ... =7fh1XnW
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... vs|6ww
prod(2:((n(j)+m_abs(j))/2-s(k))); g{hA,-3
idx = (pows(k)==rpowers); !^fR8Tp9
y(:,j) = y(:,j) + p*rpowern(:,idx); ; ZV^e
end HDyZzjgG
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if isnorm gc3 U/
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y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); f+Me dc~
end {K4t8T]
end 2bnIT>(
% END: Compute the Zernike Polynomials ~@ b}=+n
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YBIe'(p
NB5B$q_'#
Wmxw!
% Compute the Zernike functions: )]>
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% ------------------------------ So%1RY{)
idx_pos = m>0; h<ct W>6v
idx_neg = m<0; x[W]?`W3r~
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z = y; BON""yIC
if any(idx_pos) 3dDQz#
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); EJaaW&>[
end \w[ZY$/
if any(idx_neg) H0 n@kKr
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 8sF0]J[g{
end p]|ME
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% EOF zernfun GnrW{o