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, m5/d=k0l
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.T,fjMQ
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? R&`; C<6}D
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Xi~%,~
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function z = zernfun(n,m,r,theta,nflag) }KZt7)
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ,4&?`Q
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ][IEzeI_LN
% and angular frequency M, evaluated at positions (R,THETA) on the d@ ?++z
% unit circle. N is a vector of positive integers (including 0), and [_pw|BGp
% M is a vector with the same number of elements as N. Each element Jiv%Opo/|
% k of M must be a positive integer, with possible values M(k) = -N(k) [m9Iz!E
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, qQ%RnD9
% and THETA is a vector of angles. R and THETA must have the same |\W9$V
% length. The output Z is a matrix with one column for every (N,M) x]=s/+Y
% pair, and one row for every (R,THETA) pair. Pzl2X@{ %
% qlJzXq{|`
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 7|/Ct;oO:
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), #S*`7MvM
% with delta(m,0) the Kronecker delta, is chosen so that the integral ..5rW0lr
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, &Is}<Ew
% and theta=0 to theta=2*pi) is unity. For the non-normalized >&z=ktB
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. _3'FX#xc
% kU
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% The Zernike functions are an orthogonal basis on the unit circle. {Hu@|Q\~&
% They are used in disciplines such as astronomy, optics, and `pfZJ+
% optometry to describe functions on a circular domain. 'fGB#uBt
% "nzQ$E>?$
% The following table lists the first 15 Zernike functions. oN\IQ7oI
% qZS]eQW.
% n m Zernike function Normalization KDX1_r=Y
% -------------------------------------------------- qz@k-Jqq
d
% 0 0 1 1 ?E*;fDEC
% 1 1 r * cos(theta) 2 0S%xm'|N
% 1 -1 r * sin(theta) 2 Ddr.kXIpo
% 2 -2 r^2 * cos(2*theta) sqrt(6) Us.")GiHE
% 2 0 (2*r^2 - 1) sqrt(3) w I7iE4\vz
% 2 2 r^2 * sin(2*theta) sqrt(6) QQPT=_P]
% 3 -3 r^3 * cos(3*theta) sqrt(8) !pqfx93R*
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) D\ ;(BB
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) iaAj|:
% 3 3 r^3 * sin(3*theta) sqrt(8) L^{1dVGWNa
% 4 -4 r^4 * cos(4*theta) sqrt(10) xRI7_8Jpyn
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;Eer
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Jx jP'8
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =sYUzYm
% 4 4 r^4 * sin(4*theta) sqrt(10) ?DwI>< W
% -------------------------------------------------- g"dq;H
% XB.xIApmy
% Example 1: Hrk]6*
% zarxv|
}$
% % Display the Zernike function Z(n=5,m=1) ~v$1@DQ}
% x = -1:0.01:1; 0{q>'dv
% [X,Y] = meshgrid(x,x); )9]DJ!]&Q"
% [theta,r] = cart2pol(X,Y); WCdl 25L#
% idx = r<=1; VbG#)>"F
% z = nan(size(X)); d5z=fH9
% z(idx) = zernfun(5,1,r(idx),theta(idx)); C+m%_6<
% figure 5Qh$>R4!"
% pcolor(x,x,z), shading interp 9*&c2jh
% axis square, colorbar +I$,Y~&`>
% title('Zernike function Z_5^1(r,\theta)') vh/&KTe?:
% e2><Y<
% Example 2: 4m:D8&D_M
% ms]r1x"
% % Display the first 10 Zernike functions b4R;#rm
% x = -1:0.01:1; Mjon++>Z
% [X,Y] = meshgrid(x,x); lA/.4"nN
% [theta,r] = cart2pol(X,Y); JH|]B|3
% idx = r<=1; %A$5mi^
% z = nan(size(X)); @fc-[pv
% n = [0 1 1 2 2 2 3 3 3 3]; E-`3}"{
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; V'q?+p]
a
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 28!
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% y = zernfun(n,m,r(idx),theta(idx)); s?5vJ:M
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% figure('Units','normalized') 1
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% for k = 1:10 E^`-:L(_
% z(idx) = y(:,k); 4F`&W*x
% subplot(4,7,Nplot(k)) $A;%p6PO)
% pcolor(x,x,z), shading interp */6lyODf
% set(gca,'XTick',[],'YTick',[]) \GD\N=?~
% axis square #
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% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) $VxKv7:
% end [2P6XoI#
% Mp7X+o/
% See also ZERNPOL, ZERNFUN2. r6QshCA"
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% Paul Fricker 11/13/2006 c<a)Yqf"]
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% Check and prepare the inputs: S0'
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% ----------------------------- rQD^O4j R
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) PWBcK_4i%
error('zernfun:NMvectors','N and M must be vectors.') S?[@/35)
end <5 }
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%#<MCiaK
if length(n)~=length(m) $3=S\jyfK
error('zernfun:NMlength','N and M must be the same length.') &"?S0S>r!
end )kT.3
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n = n(:); !+u
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m = m(:); 6]sP"
if any(mod(n-m,2)) .|e8v _2J
error('zernfun:NMmultiplesof2', ... =z!^OT6eb
'All N and M must differ by multiples of 2 (including 0).') !$hi:3{U,
end 1{AK=H')
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if any(m>n) 2e9.U/9
error('zernfun:MlessthanN', ... +# 3e<+!F
'Each M must be less than or equal to its corresponding N.') PbnAY{J
end 7Fx0#cS"\
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if any( r>1 | r<0 ) 4a=QTq0p
error('zernfun:Rlessthan1','All R must be between 0 and 1.') E)`:sSd9
end 5P{[8PZxbV
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) FeZ*c~q
error('zernfun:RTHvector','R and THETA must be vectors.') p ,.6sk
end );zLgNx,
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r = r(:); oOHY+'V
theta = theta(:); M-Ek(K3SRf
length_r = length(r); ?t5<S]'r$
if length_r~=length(theta) KM+[1Ze$
error('zernfun:RTHlength', ... fx-8mf3
'The number of R- and THETA-values must be equal.') S Rk%BJ? ~
end ? G`6}NP
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% Check normalization: 2EOx],(|
% -------------------- @,j,GE%
if nargin==5 && ischar(nflag) osl\j]U8
isnorm = strcmpi(nflag,'norm'); .1}1e;f-
if ~isnorm 3RanAT.nu:
error('zernfun:normalization','Unrecognized normalization flag.') wX5q=I
end Z5p
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else T 5F)
isnorm = false; \,fa"^8
end 7=D,D+f
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L6ap|u
% Compute the Zernike Polynomials ah%Ws#&
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8{i
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% Determine the required powers of r: 1_b*j-j
% ----------------------------------- Mg2 e0}{
m_abs = abs(m); rvlvk"
rpowers = []; 1Au+X3
for j = 1:length(n) R+U$;r8l
rpowers = [rpowers m_abs(j):2:n(j)]; g60k R7;\
end v$D U
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rpowers = unique(rpowers); ''(rC38
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}h=PW'M{
% Pre-compute the values of r raised to the required powers, T-#4hY`
% and compile them in a matrix: JEk'2Htx
% ----------------------------- -r_,#LR!l
if rpowers(1)==0 ^vPsp?
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);
3%bhW9H%
rpowern = cat(2,rpowern{:}); ? 3OfiGX?
rpowern = [ones(length_r,1) rpowern]; zPxR=0|
else \+#EO%sN1%
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); b"Q8[k |d
rpowern = cat(2,rpowern{:}); tRpY+s~Fq
end n0!2-Q5U)h
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% Compute the values of the polynomials: aG83@ABx
% -------------------------------------- K2yu}F ^}
y = zeros(length_r,length(n)); pY@QR?F\
for j = 1:length(n) k#zDY*kj
s = 0:(n(j)-m_abs(j))/2; p0WUF\"
pows = n(j):-2:m_abs(j); &92/qRh7
for k = length(s):-1:1 N[e,%heR
p = (1-2*mod(s(k),2))* ... D;NL*4zt
prod(2:(n(j)-s(k)))/ ... eb}P/
prod(2:s(k))/ ... Y X^c}t}U
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... PMpq>$6b7
prod(2:((n(j)+m_abs(j))/2-s(k))); $L 8>Ha}
idx = (pows(k)==rpowers); [#C6K '
y(:,j) = y(:,j) + p*rpowern(:,idx); tc0;Ake-&
end mf3,V|>[\
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if isnorm ^8;MY5Wbs
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 5h&sdzfG
end H7GI`3o
end
^S3G%{"
% END: Compute the Zernike Polynomials Gk{ 'U
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fj;ZGbg-O
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% Compute the Zernike functions: NX,m6u
% ------------------------------ Q{|%kU"
idx_pos = m>0; Yu\$Y0 {]
idx_neg = m<0; )gEE7Ex?
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z = y; ce9P-}d
if any(idx_pos) 1oej<67PdJ
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); U?sHh2*
end a8JAJkFB
if any(idx_neg) 8Y.qP"s
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Ik$$Tn&;
end eO <N/?t
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% EOF zernfun ZK8)FmT_<O