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, [|oG}'Xz
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, JAd .\2%Y
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? QUn!&55
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? LYECX
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function z = zernfun(n,m,r,theta,nflag) 2PyuM=(Wt
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. +bLP+]7oZ
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N H`)eT6:|/
% and angular frequency M, evaluated at positions (R,THETA) on the Rf8Obk<
% unit circle. N is a vector of positive integers (including 0), and En9J7es_
% M is a vector with the same number of elements as N. Each element f}(4v1T
% k of M must be a positive integer, with possible values M(k) = -N(k) NMK$$0U
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, LF!KP
% and THETA is a vector of angles. R and THETA must have the same S/) ),~`4
% length. The output Z is a matrix with one column for every (N,M) e8("G[P>
% pair, and one row for every (R,THETA) pair. PL&>pM
% \Hrcf +`
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 8(Te^] v#
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 8|)!E`TKSV
% with delta(m,0) the Kronecker delta, is chosen so that the integral OU7OX]h
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, aC2Vz9e
% and theta=0 to theta=2*pi) is unity. For the non-normalized ]vz6DJs
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. JseKqJ?g
% x}K|\KXy
% The Zernike functions are an orthogonal basis on the unit circle. 7V::P_aUY
% They are used in disciplines such as astronomy, optics, and }Y.YJXum
% optometry to describe functions on a circular domain. w/o^OjwQ
% xbhHP2F|
% The following table lists the first 15 Zernike functions. sx=1pnP9`
% `oikSx$vB.
% n m Zernike function Normalization -@>]iBl
% -------------------------------------------------- vw!7f|Pg ~
% 0 0 1 1 }C_g;7*
% 1 1 r * cos(theta) 2 1gK^x^l*f
% 1 -1 r * sin(theta) 2 5*Zz_ .
% 2 -2 r^2 * cos(2*theta) sqrt(6) eK1l~W%
% 2 0 (2*r^2 - 1) sqrt(3) A+M4=
% 2 2 r^2 * sin(2*theta) sqrt(6) A4@z+ebb l
% 3 -3 r^3 * cos(3*theta) sqrt(8) {z_cczJ-
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) L]z8'n,
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) dNf9,P_}
% 3 3 r^3 * sin(3*theta) sqrt(8) !`=iKe&%E
% 4 -4 r^4 * cos(4*theta) sqrt(10) N\ Mdia
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :j3'+%'2
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) u-iQ
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) @4*eH\3
% 4 4 r^4 * sin(4*theta) sqrt(10) Hif|z[0$
% -------------------------------------------------- *(yw6(9%
% [DjlkA/Zg
% Example 1: H4
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% 2*w`l|Sx
% % Display the Zernike function Z(n=5,m=1) }GURq#
% x = -1:0.01:1; nw/g[/<;
% [X,Y] = meshgrid(x,x); hk5!$#^
% [theta,r] = cart2pol(X,Y); jG`PyIgw
% idx = r<=1; .jP|b~
% z = nan(size(X)); 1VFCK&
% z(idx) = zernfun(5,1,r(idx),theta(idx)); +sn0bi/rG
% figure ""
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% pcolor(x,x,z), shading interp 1N $OXLu
% axis square, colorbar W#g!Usf:/
% title('Zernike function Z_5^1(r,\theta)') ',[AKXJ
% 5Xxdm-0
% Example 2: ?E!M%c@,
% >wqWIw.w>
% % Display the first 10 Zernike functions uaP5(hUI
% x = -1:0.01:1; .R`_"7
% [X,Y] = meshgrid(x,x); ck
`td%
% [theta,r] = cart2pol(X,Y); [^a7l$fmi
% idx = r<=1; }KUK|p5
% z = nan(size(X)); j-J/yhWO&
% n = [0 1 1 2 2 2 3 3 3 3]; )UU`uzU;u
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; \bF<f02P
% Nplot = [4 10 12 16 18 20 22 24 26 28]; <e
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% y = zernfun(n,m,r(idx),theta(idx)); u8?$W%eW
% figure('Units','normalized') m=h/A xW
% for k = 1:10 =jm\8sl~~
% z(idx) = y(:,k); m\&99-j:@b
% subplot(4,7,Nplot(k)) ?Mo)&,__
% pcolor(x,x,z), shading interp w$&;s<0
% set(gca,'XTick',[],'YTick',[]) e`LvHU_0
% axis square #o~C0`8!B=
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) S3HyB
b
% end e@O]c"
% eW<NDI&b
% See also ZERNPOL, ZERNFUN2. NoF|j57?u'
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% Paul Fricker 11/13/2006 l2M(
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*7*cWO=
% Check and prepare the inputs: X<Xiva85
% ----------------------------- 2H8\P+
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) TT;ls<(Lg
error('zernfun:NMvectors','N and M must be vectors.') dhP")@3K;p
end g*_n|7pB
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if length(n)~=length(m) ,qB@agjvo<
error('zernfun:NMlength','N and M must be the same length.') ?)<zzL",
end _'y`hKeI[
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n = n(:); Fku<|1}&y
m = m(:); 8yOhKEPX
if any(mod(n-m,2)) uTO%O}D N
error('zernfun:NMmultiplesof2', ... !%(kMN
'All N and M must differ by multiples of 2 (including 0).') XLYGhM
end /Trbr]lWy
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if any(m>n) sc-h O9~k
error('zernfun:MlessthanN', ... }=|{"C
'Each M must be less than or equal to its corresponding N.') 8ZjRMr}
end ($UUgjv F
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if any( r>1 | r<0 ) u+6L>7t88I
error('zernfun:Rlessthan1','All R must be between 0 and 1.') /Wl8Jf7'
end
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) hhU\$'0B-
error('zernfun:RTHvector','R and THETA must be vectors.') j-i>Jd7
end S5H}
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r = r(:); `|92!Ej
theta = theta(:); TZg1,Z
length_r = length(r); 5D7k[+6
if length_r~=length(theta) i&)([C0z$
error('zernfun:RTHlength', ... ZifDU@J$t
'The number of R- and THETA-values must be equal.') i3L2N~:V
end 2zv:j7
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% Check normalization: Ne3YhCC>
% -------------------- )@tHS-Jf
if nargin==5 && ischar(nflag) Ui1s]R
isnorm = strcmpi(nflag,'norm'); d|W=_7z
if ~isnorm r1=j$G
error('zernfun:normalization','Unrecognized normalization flag.') y
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end m'
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else `q | )_
isnorm = false; fceO|mSz_
end MlS5/9m@^
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4Xwb`?}-
% Compute the Zernike Polynomials /Q89 y[
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7dE.\#6r
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% Determine the required powers of r: ]vV)$xMX
% ----------------------------------- x",ktE>9
m_abs = abs(m); +`$$^x
rpowers = []; BT$Oh4y4
for j = 1:length(n) 68<W6z
rpowers = [rpowers m_abs(j):2:n(j)]; 1IT(5Mleb
end '|Lv-7
rpowers = unique(rpowers); U1rr=h
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% Pre-compute the values of r raised to the required powers, @W==)S%O
% and compile them in a matrix: QOPh3+.5
% ----------------------------- \;Q!}_ K
if rpowers(1)==0 5'`DrTOA
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *.D{d0A
rpowern = cat(2,rpowern{:}); -Oz! GX
rpowern = [ones(length_r,1) rpowern]; !\Cu J5U
else utn,`v
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); $S2
/*
rpowern = cat(2,rpowern{:}); A9J{>f
end 0G Q8}r
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% Compute the values of the polynomials: -r~9'aEs
% -------------------------------------- <F-IF7>a
y = zeros(length_r,length(n)); B|M@o^Tf
for j = 1:length(n) Dk2Zl
s = 0:(n(j)-m_abs(j))/2; jJ'NYG
pows = n(j):-2:m_abs(j); m*i,|{UZ
for k = length(s):-1:1 E7w^A
p = (1-2*mod(s(k),2))* ... *1:kIi7_
prod(2:(n(j)-s(k)))/ ... #e@[{s7
prod(2:s(k))/ ... g
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prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... >f1fvv6
prod(2:((n(j)+m_abs(j))/2-s(k))); vD/l`Ib:
idx = (pows(k)==rpowers); C58B(Ndo
y(:,j) = y(:,j) + p*rpowern(:,idx); \TDn q!)?
end Ri::Ek3qu
nT}i&t!q8@
if isnorm p=i6~
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); =`CK`x
end TXs&*\
end o,0
Z^"|
% END: Compute the Zernike Polynomials LFYSur8
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9d=\BBNZ
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% Compute the Zernike functions: f%[xl6VE;
% ------------------------------ *7L1SjZw
idx_pos = m>0; x>A[~s"|N
idx_neg = m<0; YOvhMi
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]zIIi%
z = y; bh sCeH
if any(idx_pos) 0Xn,q]@Z
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Z\n^m^Z
=
end l!\~T"-7;:
if any(idx_neg) dAOJ:
@y
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 8R(l~
end @ @(O##(7
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% EOF zernfun ,WW=,P