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, O>SuZ>g+7
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, 1#>&p%P!
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? "GwWu-GS
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ;# R3k
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function z = zernfun(n,m,r,theta,nflag) k1f3?l
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. &\"Y/b]
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N [}A_uOGEP
% and angular frequency M, evaluated at positions (R,THETA) on the ){O1&|z-
% unit circle. N is a vector of positive integers (including 0), and i!SW?\
% M is a vector with the same number of elements as N. Each element ;OQ'B=uK
% k of M must be a positive integer, with possible values M(k) = -N(k) Jw:Fj{D
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, pAJ=f}",]E
% and THETA is a vector of angles. R and THETA must have the same M>?aa6@0
% length. The output Z is a matrix with one column for every (N,M) k_*XJ <S!Y
% pair, and one row for every (R,THETA) pair. ~:/%/-^
% ilDJwZg#
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /,1SE(
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), }.fL$,7a
% with delta(m,0) the Kronecker delta, is chosen so that the integral Yl)eh(\&J
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, TnN^2:cU
% and theta=0 to theta=2*pi) is unity. For the non-normalized $kxu;I
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. )3]83:lD2
% lSn5=^]q
% The Zernike functions are an orthogonal basis on the unit circle. kF(Ce{;z
% They are used in disciplines such as astronomy, optics, and
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% optometry to describe functions on a circular domain. O%YjWb
% QO5OnYh
% The following table lists the first 15 Zernike functions. I;Al?&uw
% #joF{M{
% n m Zernike function Normalization }': EJ~H
% -------------------------------------------------- n\Z^K
% 0 0 1 1 n!UMU ^
% 1 1 r * cos(theta) 2 =gW"#ZjL){
% 1 -1 r * sin(theta) 2 gf:vb*#Wa
% 2 -2 r^2 * cos(2*theta) sqrt(6) s~'9Hv9
% 2 0 (2*r^2 - 1) sqrt(3) ?*CRa$_I|
% 2 2 r^2 * sin(2*theta) sqrt(6) (y=dR1p
% 3 -3 r^3 * cos(3*theta) sqrt(8) _wm~}_Q
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) CCuxC9i7
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) !(W[!%
% 3 3 r^3 * sin(3*theta) sqrt(8) zo_k\K`{@
% 4 -4 r^4 * cos(4*theta) sqrt(10) &e%{k@
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) b%3Q$wIJ6
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ISpeV
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) qAUaF;{
% 4 4 r^4 * sin(4*theta) sqrt(10) - waX#UT=
% -------------------------------------------------- .>k=A|3G
% $|Q".dD
% Example 1: F`fGz)Mk
% 2#'rk'X,K
% % Display the Zernike function Z(n=5,m=1) a;56k
% x = -1:0.01:1; MPjr_yc]
% [X,Y] = meshgrid(x,x); &\&'L|0F
% [theta,r] = cart2pol(X,Y); &@=u+)^-{
% idx = r<=1; PASuf.U$"
% z = nan(size(X)); K{|w 43>D
% z(idx) = zernfun(5,1,r(idx),theta(idx)); (d54C(")
% figure L5R `w&Up
% pcolor(x,x,z), shading interp K1;zMh
% axis square, colorbar La\Q'0
% title('Zernike function Z_5^1(r,\theta)') &K06}[J
% vkd *ER^
% Example 2: Er`TryN|}
% W7%p^;ZQ$
% % Display the first 10 Zernike functions :[L{KFQU
% x = -1:0.01:1; fG<Dh z@
% [X,Y] = meshgrid(x,x); +<gg
% [theta,r] = cart2pol(X,Y); ~GSpl24W<
% idx = r<=1; w-J"zC
% z = nan(size(X)); a4%`"
% n = [0 1 1 2 2 2 3 3 3 3]; ,r@xPZPz:e
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ex.+'m<g
% Nplot = [4 10 12 16 18 20 22 24 26 28]; dI!8S
% y = zernfun(n,m,r(idx),theta(idx)); |drf"lX<{
% figure('Units','normalized') }|AX_=a
% for k = 1:10 6e*%\2UA
% z(idx) = y(:,k); %=y;L:S\p
% subplot(4,7,Nplot(k)) (viWY
% pcolor(x,x,z), shading interp eUYZxe :6
% set(gca,'XTick',[],'YTick',[]) dFzYOG1
% axis square !zU/Hq{wcK
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) O97VdNT8
% end Dq|GQdZ>o
% yGRR8F5>(
% See also ZERNPOL, ZERNFUN2. "";=DH
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% Paul Fricker 11/13/2006 -jFP7tEv
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% Check and prepare the inputs: FzVZs#O
% ----------------------------- z23#G>I&
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) \Ps5H5Qk;
error('zernfun:NMvectors','N and M must be vectors.') tbg*_ZQO u
end ^s=*J=k
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if length(n)~=length(m) ka/nQ~_#<
error('zernfun:NMlength','N and M must be the same length.') ?5`{7daot
end Zgy7!AF!
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n = n(:); hv3;irK]&
m = m(:); grc:Y
if any(mod(n-m,2)) a%v>eXc
error('zernfun:NMmultiplesof2', ... v_.HGGS
'All N and M must differ by multiples of 2 (including 0).')
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end Zd$JW=KR]l
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if any(m>n) 9976H\{
error('zernfun:MlessthanN', ... o OQ'*7_
'Each M must be less than or equal to its corresponding N.') d @m\f
end BGN9,ii
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if any( r>1 | r<0 ) P8<hvMF
error('zernfun:Rlessthan1','All R must be between 0 and 1.') %Uf'+!4l`
end i *'Z3Z)
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) VtWT{y5Ec
error('zernfun:RTHvector','R and THETA must be vectors.') IytDvz*|
end [3kl^TE
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r = r(:); 6Kp}_^|z
theta = theta(:); [ZD[a6(94
length_r = length(r); F{\=PCZ>7
if length_r~=length(theta) IkQe~;Y
error('zernfun:RTHlength', ... }3J=DCtS
'The number of R- and THETA-values must be equal.') x}|+sS,g
end >L=;"+B0U&
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% Check normalization: >Mw'eQ0(y
% -------------------- z0
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if nargin==5 && ischar(nflag) I|T7+{5z
isnorm = strcmpi(nflag,'norm'); <aXoB*Y
if ~isnorm nE$
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error('zernfun:normalization','Unrecognized normalization flag.') zqf[Z3
end !b63ik15O~
else |mOMRP#'
isnorm = false; 8SZK:VE@
end A?r^V2+j
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% qe%V#c
% Compute the Zernike Polynomials uXpv*i{R
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,56;4)cv
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% Determine the required powers of r: Zi*%*nX
% ----------------------------------- B`1kG Ex .
m_abs = abs(m); {OP~8e"
rpowers = []; QD4:W"i
for j = 1:length(n) jkt6/H
rpowers = [rpowers m_abs(j):2:n(j)]; S/7l/DFb
end +GeWg`
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rpowers = unique(rpowers); h/?6=D{
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% Pre-compute the values of r raised to the required powers, oj@g2H5P
% and compile them in a matrix: yb?|Eww_o
% ----------------------------- PIxjM>
if rpowers(1)==0 `HyF_m>\
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ,v7Q *3
rpowern = cat(2,rpowern{:}); bLlH//ZRH
rpowern = [ones(length_r,1) rpowern]; :,~K]G
else f3#X0.':
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); v2>Z^
rpowern = cat(2,rpowern{:}); M*`hDdS
end 8UM0vNk
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% Compute the values of the polynomials: ]0-<>
% -------------------------------------- YPEnNt+
y = zeros(length_r,length(n)); D/:3RZF
for j = 1:length(n) x<F$aXOS
s = 0:(n(j)-m_abs(j))/2; H1&RI4XC
pows = n(j):-2:m_abs(j); +zp0" ,2B
for k = length(s):-1:1 +|&0fGv;d9
p = (1-2*mod(s(k),2))* ... GTAf
prod(2:(n(j)-s(k)))/ ... g~)3WfC$[
prod(2:s(k))/ ... ArXl=s';s4
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... -Qb0:]sV#
prod(2:((n(j)+m_abs(j))/2-s(k))); ^P$7A]!
idx = (pows(k)==rpowers); zPE$
y(:,j) = y(:,j) + p*rpowern(:,idx); }-nU3{1
end $5A^'q
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if isnorm g>n0z5&TNF
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); [h-norB((
end D#0O[F@l##
end i/$SN-5}1
% END: Compute the Zernike Polynomials #>[wD#XJV
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G~!C=l
l$M +.GB<
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% Compute the Zernike functions: mi^hvks<
% ------------------------------ ]sL45k2W
idx_pos = m>0; uJ8{HB
idx_neg = m<0; h(N=V|0
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z = y; XU9'Rfp
if any(idx_pos) %VJW@S>j/
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); QO,+ps<
end %\I.DEYH
if any(idx_neg) +)gB9DoK
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); jBRPR
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end ],&\%jd<
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% EOF zernfun KyLp?!|>