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, b#e]1Q
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, V@Ax}<$A
这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? ?l(nM+[kSL
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? w8O hJv
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function z = zernfun(n,m,r,theta,nflag) )eVzS j>MT
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. <. ezw4ju
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N makaI0M
% and angular frequency M, evaluated at positions (R,THETA) on the n<=y"*
% unit circle. N is a vector of positive integers (including 0), and r}Ltv?4
% M is a vector with the same number of elements as N. Each element =P1RdyP
% k of M must be a positive integer, with possible values M(k) = -N(k) hjw4Xzju
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, gfV]^v
% and THETA is a vector of angles. R and THETA must have the same \A` gK\/h
% length. The output Z is a matrix with one column for every (N,M) D\@e{.$MZ|
% pair, and one row for every (R,THETA) pair. w 7Cne%J8
% dvC0 <*V
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike H^ESAs6
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 7?+5%7-
% with delta(m,0) the Kronecker delta, is chosen so that the integral 5aa}FdUq
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, KsZ@kTs
% and theta=0 to theta=2*pi) is unity. For the non-normalized 7sCR!0
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 6Wf*>G*h
% :P HUsy
% The Zernike functions are an orthogonal basis on the unit circle. ys:1Z\$P
% They are used in disciplines such as astronomy, optics, and ,xm;JXJ
% optometry to describe functions on a circular domain. ]r"31.w(
% cb\jrbj6
% The following table lists the first 15 Zernike functions. 9yO{JgKA
% lO%MyP
% n m Zernike function Normalization -bv>iIC
% -------------------------------------------------- c(QG4.)m
% 0 0 1 1 &8pCHGmV)
% 1 1 r * cos(theta) 2 l~`txe
% 1 -1 r * sin(theta) 2 PWADbu{+
% 2 -2 r^2 * cos(2*theta) sqrt(6) Tnzco
% 2 0 (2*r^2 - 1) sqrt(3) =1%zI%
% 2 2 r^2 * sin(2*theta) sqrt(6) MtMvpHk
% 3 -3 r^3 * cos(3*theta) sqrt(8) Z&AHM &,yj
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 45]Ym{]
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) n$XMsl.>
% 3 3 r^3 * sin(3*theta) sqrt(8) Bl>_&A)
% 4 -4 r^4 * cos(4*theta) sqrt(10) %i;r]z-
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0sq=5 BnO
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 67Af} >Q
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) W;xW:
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% 4 4 r^4 * sin(4*theta) sqrt(10) O|A~dj`
% -------------------------------------------------- UchALR^5
% ]#vvlM>/
% Example 1: w`H.ey
% o[5=S,'
% % Display the Zernike function Z(n=5,m=1) $O;N/N:m
% x = -1:0.01:1; 0X] ekq
% [X,Y] = meshgrid(x,x); V+4k!
% [theta,r] = cart2pol(X,Y); Xq=!"E
% idx = r<=1; F{a0X0ru~
% z = nan(size(X)); jhjW*F<u
% z(idx) = zernfun(5,1,r(idx),theta(idx)); =:t@;y
% figure EM>c%BH<N
% pcolor(x,x,z), shading interp ^.pE`l%1}
% axis square, colorbar /K2.V@T
% title('Zernike function Z_5^1(r,\theta)') |TQedC
% P#vv+]/
% Example 2: @p9e:[
% Zztt)/6*
% % Display the first 10 Zernike functions ECmHy@(
% x = -1:0.01:1; a}[=_vb}K
% [X,Y] = meshgrid(x,x); /-G qG)PX
% [theta,r] = cart2pol(X,Y); DK#65H'
% idx = r<=1; ZNL;8sI?>
% z = nan(size(X)); 0-;DN:>
% n = [0 1 1 2 2 2 3 3 3 3]; O+{pF.P#V
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ]yj4~_&O
% Nplot = [4 10 12 16 18 20 22 24 26 28]; !Vp,YN+yN
% y = zernfun(n,m,r(idx),theta(idx)); Egjk^:@
% figure('Units','normalized') 7gZVg@
% for k = 1:10 _D7HQ
% z(idx) = y(:,k); teQaHe#
% subplot(4,7,Nplot(k)) T@d_t
% pcolor(x,x,z), shading interp Mc#O+'](f
% set(gca,'XTick',[],'YTick',[]) tF;& x
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% axis square @4 Os?_gJ\
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) -Y
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% end 82Z[eo
% Y*5@|Q
% See also ZERNPOL, ZERNFUN2. R%]9y]HQ
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% Paul Fricker 11/13/2006 PJSDY1T
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% Check and prepare the inputs: (to/9OrG
% ----------------------------- Z CQt1;
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 0T{c:m~QXe
error('zernfun:NMvectors','N and M must be vectors.') 98b9%Z'2f
end 5 vu_D^Q
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if length(n)~=length(m) H>X\C;X[
error('zernfun:NMlength','N and M must be the same length.') {g:/BFLr#
end 0c\|S>g[
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n = n(:); `1[Sv"
m = m(:); Hq"<vp
if any(mod(n-m,2)) E^EU+})Ujr
error('zernfun:NMmultiplesof2', ... kj<D 4)
'All N and M must differ by multiples of 2 (including 0).') @6i8RmOu}
end hI>rtaY_
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if any(m>n) z7.C\l
error('zernfun:MlessthanN', ... Q
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'Each M must be less than or equal to its corresponding N.') _'v }=:X
end Y+"hu2aPkY
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if any( r>1 | r<0 ) %9-#`
error('zernfun:Rlessthan1','All R must be between 0 and 1.') F)<G]i8n~
end hiK[!9r
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ]\9B?W(#
error('zernfun:RTHvector','R and THETA must be vectors.') 1R+ )T'in
end M;vlQ"Yl'
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r = r(:); HBga'xJ
theta = theta(:); nGJIjo_I
length_r = length(r); ]stLC; nI
if length_r~=length(theta) BqEubP(si
error('zernfun:RTHlength', ... >s 8:1l
'The number of R- and THETA-values must be equal.') )r6SGlE[Y
end xO9]yULgu
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% Check normalization: v YmtpKNj%
% -------------------- GT\s!D;<
if nargin==5 && ischar(nflag) 7^t(RNq
isnorm = strcmpi(nflag,'norm'); .jGsO0
if ~isnorm hZ\W ?r
error('zernfun:normalization','Unrecognized normalization flag.') L};;o+5uJD
end U37?P7i's
else M?4r 5R
isnorm = false; ao";5m
end ]R0A{+]n
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q"K >ML>0
% Compute the Zernike Polynomials 8$jT#\_
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uA/.4 b
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% Determine the required powers of r: :mtw}H 'F8
% ----------------------------------- % x*Ec[l
m_abs = abs(m); DEwtP
rpowers = []; F+y`4>x
for j = 1:length(n) 5@Lxbe(
q
rpowers = [rpowers m_abs(j):2:n(j)]; ESf7b `tS
end 46?F+,Rzl
rpowers = unique(rpowers); I&NpN~AU
)! *M
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% Pre-compute the values of r raised to the required powers, 16> >4U:Y
% and compile them in a matrix: ,vdP
#:
% ----------------------------- 3w:Z4]J
if rpowers(1)==0 tDLk ZCP
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @G$<6CG\
rpowern = cat(2,rpowern{:});
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rpowern = [ones(length_r,1) rpowern]; ut5!2t$c
else W*DIW;8p
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ~md|k
rpowern = cat(2,rpowern{:}); 1 l*(8!_
end tfKeo|DM"
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MJ/%$
% Compute the values of the polynomials: ]%Yis=v
% -------------------------------------- /uz5V/i0
y = zeros(length_r,length(n)); 68GGS`&
for j = 1:length(n) t-x"(
s = 0:(n(j)-m_abs(j))/2; +2fJ
pows = n(j):-2:m_abs(j); `"b7y(M
for k = length(s):-1:1 Z
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p = (1-2*mod(s(k),2))* ... ;I))gY-n
prod(2:(n(j)-s(k)))/ ... pBnf^Ew1
prod(2:s(k))/ ... >h( rd1
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ...
:E&T}RN
prod(2:((n(j)+m_abs(j))/2-s(k))); yz$1qEII`q
idx = (pows(k)==rpowers); U9[A(
y(:,j) = y(:,j) + p*rpowern(:,idx); yGG\[I;7
end _xL&sy09t
/FV6lR!0^
if isnorm vrnj}f[h
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); m'"VuH?^
end ow$l!8
end 9}0Jc(B/x
% END: Compute the Zernike Polynomials mS&\m#s<
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fX$4TPy(h
C(*@-Npf[
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% Compute the Zernike functions: o4'4H y
% ------------------------------ F20-!b
idx_pos = m>0; @=#s~ 3
idx_neg = m<0; }ZVv
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_v=WjN
z = y; 9x^
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if any(idx_pos) :O+b4R+
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); m1o65FsY08
end `/ReJj&~
if any(idx_neg)
x Bw.M{
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 2LH;d`H[0
end fvMhq:Bu
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% EOF zernfun |Pl{Oo+