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, CCoT
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, ?e=3G4N
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? m\*;Fx
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? #h=pU/R
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function z = zernfun(n,m,r,theta,nflag) ohEIr2
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. /UWv}f
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% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N %u=b_4K"j
% and angular frequency M, evaluated at positions (R,THETA) on the vciO={M
% unit circle. N is a vector of positive integers (including 0), and m8}c(GwcP
% M is a vector with the same number of elements as N. Each element % 9Jx|
% k of M must be a positive integer, with possible values M(k) = -N(k) .)(5F45Wg
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, hW<TP'Zm*
% and THETA is a vector of angles. R and THETA must have the same MS5X#B
% length. The output Z is a matrix with one column for every (N,M) ?uAq goCl
% pair, and one row for every (R,THETA) pair. bi{G
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% 7a0T]
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 0*J},#ba$
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), k2-+3zx
% with delta(m,0) the Kronecker delta, is chosen so that the integral . E8Gj'yO
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, fEJF3<UF&
% and theta=0 to theta=2*pi) is unity. For the non-normalized =w !>/#U
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. eP(|]Rk
% iQd,xr
% The Zernike functions are an orthogonal basis on the unit circle. q%S^3C&
% They are used in disciplines such as astronomy, optics, and kR0/jEz
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% optometry to describe functions on a circular domain. =|+%^)E
% ,vDSY N6
% The following table lists the first 15 Zernike functions. S7B\mv
% Mq~ g+`
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% n m Zernike function Normalization O[Yc-4
% -------------------------------------------------- k,,Bf-?
% 0 0 1 1 \+Nn>wW.
% 1 1 r * cos(theta) 2 Kcu*Z
% 1 -1 r * sin(theta) 2 qV{iUtYt
% 2 -2 r^2 * cos(2*theta) sqrt(6) fphi['X
% 2 0 (2*r^2 - 1) sqrt(3) S1I# qb
% 2 2 r^2 * sin(2*theta) sqrt(6) '"m-kor
% 3 -3 r^3 * cos(3*theta) sqrt(8) 4`Ib wg6"B
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) %\f<N1~*
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ]Uj7f4)k
% 3 3 r^3 * sin(3*theta) sqrt(8) `g+Kv&546
% 4 -4 r^4 * cos(4*theta) sqrt(10) \yhj {QS.k
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /1BqC3]tL
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) \x=j
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) pqF!1
% 4 4 r^4 * sin(4*theta) sqrt(10) MA,7|s
% -------------------------------------------------- ^ *1hz<
% *<ILSZ
% Example 1: xLShMv}
% `E2RW{$A
% % Display the Zernike function Z(n=5,m=1) P>nz8NRq
% x = -1:0.01:1; DCP
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% [X,Y] = meshgrid(x,x); IY,n7x0d
% [theta,r] = cart2pol(X,Y); Oil~QAd,
% idx = r<=1; ^k2g60]
% z = nan(size(X)); jPIOBEIG
% z(idx) = zernfun(5,1,r(idx),theta(idx)); !d
Z:Ih.[{
% figure ZH)thd9^b
% pcolor(x,x,z), shading interp =#T3p9
% axis square, colorbar >&[q`i{
% title('Zernike function Z_5^1(r,\theta)') z1m-t#v:
% rM0Idc.$&&
% Example 2: /?%1;s:'
% m)ENj6A>yP
% % Display the first 10 Zernike functions 8&wN9tPYZ
% x = -1:0.01:1; \AOHZ r
% [X,Y] = meshgrid(x,x); G'T:l("l
% [theta,r] = cart2pol(X,Y); Z,I0<ecaD
% idx = r<=1; *&BS[0;
% z = nan(size(X)); DQ.; 2W
% n = [0 1 1 2 2 2 3 3 3 3]; }X9G(`N(}
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; W`F?j-4
% Nplot = [4 10 12 16 18 20 22 24 26 28]; }B`T%(11=
% y = zernfun(n,m,r(idx),theta(idx)); |>/m{L[
% figure('Units','normalized') /_mU%fl
% for k = 1:10 Utj4f-M
% z(idx) = y(:,k); 19;Pjo8
% subplot(4,7,Nplot(k)) 6KE?@3;Om
% pcolor(x,x,z), shading interp -\&b&; _
% set(gca,'XTick',[],'YTick',[]) z7!@^!r
% axis square rqTsKrLe
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 5H2Ugk3
% end 'M35L30
% J+u z{
% See also ZERNPOL, ZERNFUN2. l.uW>AoLh
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% Paul Fricker 11/13/2006 +[":W?j
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% Check and prepare the inputs: (5GjtFojY|
% ----------------------------- 3vj1FbY
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ^WUG\@B
error('zernfun:NMvectors','N and M must be vectors.') .R_-$/ZP
end Q#PkfjXS
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if length(n)~=length(m) ;<H\{w@D
error('zernfun:NMlength','N and M must be the same length.') ;4<!vVf e
end %I#[k4,N
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n = n(:); 'wjL7PI
m = m(:); fjLS_Q
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if any(mod(n-m,2)) C zxF
error('zernfun:NMmultiplesof2', ... {YIf rM
'All N and M must differ by multiples of 2 (including 0).') Lnc>O'<5P9
end 6Ao{Aej|
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if any(m>n) }{F1Cr
error('zernfun:MlessthanN', ... #;hYJ Y
'Each M must be less than or equal to its corresponding N.') h@+(VQ
end Gg3<
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if any( r>1 | r<0 ) ~IQw?a.E
error('zernfun:Rlessthan1','All R must be between 0 and 1.') lr9s`>9
end Rv|X\Wm
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) C%Fc%}[
error('zernfun:RTHvector','R and THETA must be vectors.') aH*5(E]
end aK]H(F2#
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r = r(:); I5#zo,9
theta = theta(:); :tT6V(-W
length_r = length(r); kZNVUhW6S
if length_r~=length(theta) ]::g-&%Um
error('zernfun:RTHlength', ... h`pXUnEZ
'The number of R- and THETA-values must be equal.') fvr|<3ojo
end d[y(u<Vl
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% Check normalization: p_qJI@u8
% -------------------- A;gU@8m
if nargin==5 && ischar(nflag) z<,-:=BC"
isnorm = strcmpi(nflag,'norm'); n0opb [ ?
if ~isnorm 1Ts$kdO
error('zernfun:normalization','Unrecognized normalization flag.') />dYk Iv
end "w:?WS
else C]NL9Gq`
isnorm = false; ,m1F<Pdts
end .y>G/8_i
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ':DLv{R
% Compute the Zernike Polynomials qORRpWyx&
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -HUlB|Q8r
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% Determine the required powers of r: S(Q=2Y
% ----------------------------------- #L9F\ <K
m_abs = abs(m); .{4U]a;[
rpowers = []; .a7!*I#g
for j = 1:length(n) fm$)?E_Rp
rpowers = [rpowers m_abs(j):2:n(j)]; HD153M,
end g @qrVQv
rpowers = unique(rpowers); /Bp5^(s
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% Pre-compute the values of r raised to the required powers, L<Q>:U.@\
% and compile them in a matrix: LyG&FOf?
% ----------------------------- rA[wC%%
if rpowers(1)==0 s|IC;C|
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); XY!0yAK(!
rpowern = cat(2,rpowern{:}); eWWfUNBSLX
rpowern = [ones(length_r,1) rpowern]; wOF";0EN
else )=%TIkeF
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); =>X"
rpowern = cat(2,rpowern{:}); m.w.h^f$&
end Uq^-km#a
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% Compute the values of the polynomials: u:M)JG
% -------------------------------------- /<Yz;\:Jy
y = zeros(length_r,length(n)); Zk>#T:{h
for j = 1:length(n) ~ ^*;#[<
s = 0:(n(j)-m_abs(j))/2; +{au$v}
pows = n(j):-2:m_abs(j); # b94S?dq
for k = length(s):-1:1 J4#rOS
p = (1-2*mod(s(k),2))* ... fzjAP7 y
prod(2:(n(j)-s(k)))/ ... _ls i,kg?
prod(2:s(k))/ ... !dGSZ|YZ
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... v`Yj)
prod(2:((n(j)+m_abs(j))/2-s(k))); % 9} ?*U
idx = (pows(k)==rpowers); _p;=]#+c&
y(:,j) = y(:,j) + p*rpowern(:,idx); D] +]Br8
end FgnPh%[u
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if isnorm A8T8+M:
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 4KB>O)YNg'
end +{L=cWA"
end 'J_`CS
% END: Compute the Zernike Polynomials bPVQ-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5F$~ZDu
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% Compute the Zernike functions: ;}}k*<
Z
% ------------------------------ :N64FR#
idx_pos = m>0; 8 DPn5E#M1
idx_neg = m<0; r mJ`^6V
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z = y; `Jn2(+
if any(idx_pos) Dbw{E:pq
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); mOfTq]
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end PnZY%+[I
if any(idx_neg) 7UsU03
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); %<{1N|
end 0+Z?9$a1
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% EOF zernfun {ep.So6