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, gEr4zae
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, a
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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? 2qZa9^}
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? )p$\gwr=2
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function z = zernfun(n,m,r,theta,nflag) kAZC"qM%i
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. $uEJn&n7}
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N v!NB~"LQ
% and angular frequency M, evaluated at positions (R,THETA) on the "s F Xl
% unit circle. N is a vector of positive integers (including 0), and hq/J6 M
% M is a vector with the same number of elements as N. Each element c%|vUAq*
% k of M must be a positive integer, with possible values M(k) = -N(k) Dh2:2Rz=#7
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, gw_|C|!P
% and THETA is a vector of angles. R and THETA must have the same g3|BE2?
% length. The output Z is a matrix with one column for every (N,M) #*!+b
% pair, and one row for every (R,THETA) pair. &EAk
z
% v"z(JF
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike _9D|u<D
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), H4M{_2DO
% with delta(m,0) the Kronecker delta, is chosen so that the integral }qc#lz
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, zuUT S[
% and theta=0 to theta=2*pi) is unity. For the non-normalized a
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. X'cf&>h
% K!3{M!B
% The Zernike functions are an orthogonal basis on the unit circle. m)s
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% They are used in disciplines such as astronomy, optics, and No:^hY:F8
% optometry to describe functions on a circular domain. )-=2w-ZX
% X?tj$
% The following table lists the first 15 Zernike functions. B{s]juPG
% rmOQ{2}
% n m Zernike function Normalization H76E+AY
% -------------------------------------------------- n
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% 0 0 1 1 `vudS?
% 1 1 r * cos(theta) 2 +0VG[c\8
% 1 -1 r * sin(theta) 2 t,RyeS/
% 2 -2 r^2 * cos(2*theta) sqrt(6) Tdg6kkJ
% 2 0 (2*r^2 - 1) sqrt(3) @u,+F0Yd
% 2 2 r^2 * sin(2*theta) sqrt(6) I0!j<G
% 3 -3 r^3 * cos(3*theta) sqrt(8) M]c7D`%s
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Z.!g9fi8>
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) `)"tO&Fn
% 3 3 r^3 * sin(3*theta) sqrt(8) 5v"Y\k+1
% 4 -4 r^4 * cos(4*theta) sqrt(10) j5kA^MTG
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Vl<`|C>
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) hQXxG/yFm
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9o EpPL5
% 4 4 r^4 * sin(4*theta) sqrt(10) aC`Li^
% -------------------------------------------------- =M/qV
% gW kjUz)
% Example 1: ji}#MBac
% L#n}e7Y9
% % Display the Zernike function Z(n=5,m=1) +4Q[N;[+*
% x = -1:0.01:1; h%'
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% [X,Y] = meshgrid(x,x); /mc*Hc8R8
% [theta,r] = cart2pol(X,Y); 0A.PD rM:
% idx = r<=1; >;,gGH
% z = nan(size(X)); pDGT@qJ
% z(idx) = zernfun(5,1,r(idx),theta(idx)); =nzFd-P
% figure _74UdD{^o
% pcolor(x,x,z), shading interp R;r|cep
% axis square, colorbar KGu= ;
% title('Zernike function Z_5^1(r,\theta)') >rKhlUD
% ?9p$XG
% Example 2: Mq@}snp"S
% mmHJh\2v
% % Display the first 10 Zernike functions GA\2i0ow
% x = -1:0.01:1; D i+4Eb
% [X,Y] = meshgrid(x,x); Uj,g]e8e
% [theta,r] = cart2pol(X,Y); wazP,9W?
% idx = r<=1; F99A;M8(
% z = nan(size(X)); 8
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% n = [0 1 1 2 2 2 3 3 3 3]; 8)pB_en3sO
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; VgA48qZ
% Nplot = [4 10 12 16 18 20 22 24 26 28]; k d+l k:
% y = zernfun(n,m,r(idx),theta(idx)); >Fyu@u
% figure('Units','normalized') _%%yV
% for k = 1:10 _lP4}9p
% z(idx) = y(:,k); )A"jVQjI%w
% subplot(4,7,Nplot(k)) pw3(t
% pcolor(x,x,z), shading interp ;|!MI'Af
% set(gca,'XTick',[],'YTick',[]) AFGwT%ZD
% axis square zka?cOmYF[
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) bEd?^h
% end 8b7;\C~$p
% 8"i/wMP]
% See also ZERNPOL, ZERNFUN2. F$h'p4$T
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% Paul Fricker 11/13/2006 dQ<(lzS~
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% Check and prepare the inputs: N<liS3>
% ----------------------------- lUHtjr
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) f*<ps
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error('zernfun:NMvectors','N and M must be vectors.') B'p5M.6d#:
end 9 #Y2`pT
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if length(n)~=length(m) c|lU(Tf
error('zernfun:NMlength','N and M must be the same length.') `VZZ^K9zR
end VhvTBo<cw
>)^NJ2Fd
#h N.=~
n = n(:); (;UP%H>
m = m(:); skR,-:"8
if any(mod(n-m,2)) ]_u`EvEx6
error('zernfun:NMmultiplesof2', ... SKR;wu
'All N and M must differ by multiples of 2 (including 0).') g\&2s,
end ,dcg?48
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if any(m>n) cy3M^_5B<
error('zernfun:MlessthanN', ... 1Nj=B_T
'Each M must be less than or equal to its corresponding N.') fa{@$ppx
end [))JX"a
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if any( r>1 | r<0 ) H)5QqZ8
error('zernfun:Rlessthan1','All R must be between 0 and 1.') =/9<(Tt%m
end OQFi.8
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ffk4mhH
error('zernfun:RTHvector','R and THETA must be vectors.') a#y{pT2 b
end s}(X]Gx1
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r = r(:); {ma;G[!
theta = theta(:); t$ZkdF
length_r = length(r); J=*K"8Qr
if length_r~=length(theta) e$|VG*
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error('zernfun:RTHlength', ... ,I`_F,
'The number of R- and THETA-values must be equal.') .zSD`v@[
end |I^y0Q:K
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% Check normalization: dy:d=Z
% -------------------- Y<Q\d[3^F
if nargin==5 && ischar(nflag) Ae49n4J
isnorm = strcmpi(nflag,'norm'); {/ &B!zvl
if ~isnorm |$e:*
error('zernfun:normalization','Unrecognized normalization flag.') 0S.?E.-&0
end 4seciz0?
else GN%(9N'W
isnorm = false; >^3zU
end FH*RU1Z
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d(S}NH
% Compute the Zernike Polynomials #DUh(:E'`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V;93).-$
%
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% Determine the required powers of r: C) .2gQ
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% ----------------------------------- f1Zt?=
m_abs = abs(m); zZ,Yfd|W
rpowers = []; 7Fl-(Nv`
for j = 1:length(n) /s[DI;M$o
rpowers = [rpowers m_abs(j):2:n(j)]; -t4
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end 0x5xLg;Q
rpowers = unique(rpowers); >IY,be6>P
Y=Hz;Ni
XDYosC:
% Pre-compute the values of r raised to the required powers, >Z\BfH
% and compile them in a matrix: DB@EVH
% ----------------------------- >}SRSqJu
if rpowers(1)==0 X/+OF'po
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ;fGx;D
rpowern = cat(2,rpowern{:}); 'm O2t~n
rpowern = [ones(length_r,1) rpowern]; 8#59iQl
else R0<< f]
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); yVS\Q,:J9
rpowern = cat(2,rpowern{:}); de YyaV
end s;{K!L@
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% Compute the values of the polynomials: 9#rt:&xo0
% -------------------------------------- H?U't
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y = zeros(length_r,length(n)); m mw-a0
for j = 1:length(n) tt4+ m>/T
s = 0:(n(j)-m_abs(j))/2; 7>-yaL{
pows = n(j):-2:m_abs(j); >n!ni(
for k = length(s):-1:1 SxMj,u%X/
p = (1-2*mod(s(k),2))* ... k/lFRi-i
prod(2:(n(j)-s(k)))/ ... cwynd=^nC
prod(2:s(k))/ ... Q2\
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... nY^Nbh0
prod(2:((n(j)+m_abs(j))/2-s(k))); ZnXejpj)D
idx = (pows(k)==rpowers); @2' %o<lF
y(:,j) = y(:,j) + p*rpowern(:,idx); ^ vbWRG~
end <k]qH-v4
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if isnorm Z2p> n`D
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 3s(Ia^
end 8A{6j
end wUp)JI
% END: Compute the Zernike Polynomials _;e\:7<m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,7,;twKz
T[.[
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% Compute the Zernike functions: $']VQ4tZ
% ------------------------------ \6sQJq
idx_pos = m>0; Eark)
idx_neg = m<0; 8/Rm!.8+~
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z = y; 8tQ|-l*
if any(idx_pos) .3wY\W8Dr-
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Iql5T#K+
end 0BTLcEqgZ
if any(idx_neg) ^M
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z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 3"rkko?A
end Y}.Ystem
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% EOF zernfun $qm~c[x%