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, [2>yYr s_=
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}S S+>`
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? bw(a6qKK
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? &]V.S7LC#
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function z = zernfun(n,m,r,theta,nflag) XND|h#i8
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. A8xvo/n$
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N B|Du@^$
% and angular frequency M, evaluated at positions (R,THETA) on the E{`kaWmC&~
% unit circle. N is a vector of positive integers (including 0), and Ki4r<>\l{H
% M is a vector with the same number of elements as N. Each element Q`~jw>x
% k of M must be a positive integer, with possible values M(k) = -N(k) Amp#GR1CA
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, A mvEf
% and THETA is a vector of angles. R and THETA must have the same u6?Q3
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% length. The output Z is a matrix with one column for every (N,M) |<HPn4
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% pair, and one row for every (R,THETA) pair. m];]7uB5=
% u&^b~#T
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike vhe>)h*B
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 2^Eg9y'
% with delta(m,0) the Kronecker delta, is chosen so that the integral \<.+rqa!
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, V n7*JS
% and theta=0 to theta=2*pi) is unity. For the non-normalized 1=r#d-\tR
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?TM,Q
% H[{F'c[e
% The Zernike functions are an orthogonal basis on the unit circle. @V(*65b2
% They are used in disciplines such as astronomy, optics, and j -0z5|*KE
% optometry to describe functions on a circular domain. A]<y:^2])C
% <W|3\p6
% The following table lists the first 15 Zernike functions. Z"Zmo>cV4
% nx@=>E+a
% n m Zernike function Normalization l2`s! ,<>O
% -------------------------------------------------- G(Lzf(
% 0 0 1 1 \O}E7-
% 1 1 r * cos(theta) 2 FI[A[*fi
% 1 -1 r * sin(theta) 2 4<9=5 q]
% 2 -2 r^2 * cos(2*theta) sqrt(6) b $'FvZbk
% 2 0 (2*r^2 - 1) sqrt(3) +GG9^:<yr
% 2 2 r^2 * sin(2*theta) sqrt(6) jDKO}
bQ
% 3 -3 r^3 * cos(3*theta) sqrt(8) yGI;ye'U
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) qJ;jfh!
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) vY4\59]P
% 3 3 r^3 * sin(3*theta) sqrt(8) 7[w,:9& }
% 4 -4 r^4 * cos(4*theta) sqrt(10) ?b*s.
^
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) B,<da1(a
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Ed"h16j?z
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Vdtry@Q
% 4 4 r^4 * sin(4*theta) sqrt(10) .GV;+8HzS
% -------------------------------------------------- j:<n+:HC
% QU4h8}$
% Example 1: 5}:-h>
% U}& 2k
% % Display the Zernike function Z(n=5,m=1) .)RzT9sg
% x = -1:0.01:1; %+f>2U4I
% [X,Y] = meshgrid(x,x); ]*vdSr-J
% [theta,r] = cart2pol(X,Y); 34z"Pm
% idx = r<=1; YHkn2]^#A
% z = nan(size(X)); $RYa6"`
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ~kQA7;`j$
% figure Vc_'hz]Z
% pcolor(x,x,z), shading interp ao%NK<Lt
% axis square, colorbar 5pj22 s
% title('Zernike function Z_5^1(r,\theta)') Hx#;Z
% 4\y/'`xm)6
% Example 2: BZ:H`M`n
% ->sm+H-*
% % Display the first 10 Zernike functions XDsx3Ws
% x = -1:0.01:1; 2#P*,
% [X,Y] = meshgrid(x,x); 5XO;N s
% [theta,r] = cart2pol(X,Y); lU@]@_<
% idx = r<=1; qo p^;~
% z = nan(size(X)); e]`[yf
% n = [0 1 1 2 2 2 3 3 3 3]; d_CKP"TA
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ?h.wK
% Nplot = [4 10 12 16 18 20 22 24 26 28]; h^?\xm|
% y = zernfun(n,m,r(idx),theta(idx)); Gnf~u[T6
% figure('Units','normalized') yGWxpzmRS
% for k = 1:10 "*m_> IU
% z(idx) = y(:,k); m4aB*6<lq
% subplot(4,7,Nplot(k)) u2[iM d
% pcolor(x,x,z), shading interp Ge2q%
% set(gca,'XTick',[],'YTick',[]) I`p+Qt
% axis square O]lSWEe
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Ai:BEPKe
% end Y'yH;Mz
% qLw{?sH}J/
% See also ZERNPOL, ZERNFUN2. L0*nm.1X
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% Paul Fricker 11/13/2006 *i]=f6G
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% Check and prepare the inputs: y<HO:kZ8`
% ----------------------------- K&nE_.kbl
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) '>&^zgr
error('zernfun:NMvectors','N and M must be vectors.') %`OJ.:k
end ZYI{i?Te#
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if length(n)~=length(m) VSrr`B
error('zernfun:NMlength','N and M must be the same length.') bvS(@
end ,a gc
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n = n(:); Y<mej][
m = m(:); =; ^%(%Y{m
if any(mod(n-m,2)) x97
j
error('zernfun:NMmultiplesof2', ... $>GgB`
'All N and M must differ by multiples of 2 (including 0).') %1H[Wh(U
end _z'u pb&
e<=cdze
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if any(m>n) ~Q?a|mV,
error('zernfun:MlessthanN', ... zhpx"{_
'Each M must be less than or equal to its corresponding N.') T^ w36}a
end S/^"@?z,vE
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if any( r>1 | r<0 ) 'XZ)!1N
error('zernfun:Rlessthan1','All R must be between 0 and 1.') MOsl_^c
end BnCbon)
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) F">Qpgt
error('zernfun:RTHvector','R and THETA must be vectors.') "ul {d(K3
end 2ggdWg7z
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r = r(:); K@#(*."
theta = theta(:); odPL{XFj
length_r = length(r); Fb^:V4<T
if length_r~=length(theta) 6xWe=QGE
error('zernfun:RTHlength', ... Fe]B&n
'The number of R- and THETA-values must be equal.') Ys@}3\Mc
end pV20oSJNt
kBYNf =
%1O;fQL
% Check normalization: ?\$#L^;b}
% -------------------- > `n,S
if nargin==5 && ischar(nflag) <(-3_s6-
isnorm = strcmpi(nflag,'norm'); jJuW-(/4[
if ~isnorm g{8,Wx,,
error('zernfun:normalization','Unrecognized normalization flag.') D&}3$ 7>
end iTag+G4*
else QS{1CC9$
isnorm = false; r9 ui|>U"
end
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?RK]FP"A
% Compute the Zernike Polynomials GFel(cx:K
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4F{)i
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% Determine the required powers of r: 9T|IvQK8
% ----------------------------------- ]@uE#a:[
m_abs = abs(m); ZCB_
rpowers = []; J.ck~;3
for j = 1:length(n) GlbySD@
rpowers = [rpowers m_abs(j):2:n(j)]; Q\cjPc0y
end JMH8MH*
rpowers = unique(rpowers); oo=Qt(#
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% Pre-compute the values of r raised to the required powers, q o\?o
% and compile them in a matrix: -C<zF`jO
% ----------------------------- QNARkYY~|
if rpowers(1)==0 mnmP<<8C,
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); N5i+3&
rpowern = cat(2,rpowern{:}); WDg+J
rpowern = [ones(length_r,1) rpowern]; M#~Cc~oT
else NGOqy+Ty{f
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 2I&o69x?
rpowern = cat(2,rpowern{:}); SQqD:{#g"
end 1RK=,Wx
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% Compute the values of the polynomials: %,Ap7X3:QT
% -------------------------------------- J2j U4mR
y = zeros(length_r,length(n)); Q5FM8Q
for j = 1:length(n) JaK}|
s = 0:(n(j)-m_abs(j))/2; m< 3Ao^I+
pows = n(j):-2:m_abs(j); "g'jPwFG
for k = length(s):-1:1 7vABq(
p = (1-2*mod(s(k),2))* ... |7X:TfJ
prod(2:(n(j)-s(k)))/ ... LE*h9((
prod(2:s(k))/ ... r=6v`)Qr
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... zxf"87se
prod(2:((n(j)+m_abs(j))/2-s(k))); ;$a@J&
idx = (pows(k)==rpowers); DqX{'jj
y(:,j) = y(:,j) + p*rpowern(:,idx); mExVYp h
end lWqrU1Sjl
I =1+h
if isnorm l'\pk<V
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 2 Sh
end BM(]QUxRd
end :%sXO
% END: Compute the Zernike Polynomials 8G oh4T H
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jLpc
Zb,
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2AtLyN'.
% Compute the Zernike functions: Oi:<~E[kz.
% ------------------------------ vq!_^F<
idx_pos = m>0; i} N8(B(
idx_neg = m<0; 1.gG^$J d
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z = y; $d,0=Ci
if any(idx_pos) $2u 'N:o
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 2^)1N>"g
end I(9R~q
if any(idx_neg) !>>f(t4
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Qu7T[<
end ($LLl;1
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#'q<v"w
% EOF zernfun XXh6^@H=