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, _C@<*L=Q
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, &_,.*tha
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? 5EL&?\e
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ,soXX_Y>
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function z = zernfun(n,m,r,theta,nflag) lVgin54Q
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. I36ClOG
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N :b<<
% and angular frequency M, evaluated at positions (R,THETA) on the P7*?E*
% unit circle. N is a vector of positive integers (including 0), and 8" (j_~;
% M is a vector with the same number of elements as N. Each element n\u3$nGL1`
% k of M must be a positive integer, with possible values M(k) = -N(k) B*n_
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% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, U[6
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% and THETA is a vector of angles. R and THETA must have the same fnK H<
% length. The output Z is a matrix with one column for every (N,M) j){0>O.V
% pair, and one row for every (R,THETA) pair. 9eEA80i7
% )npvy>C'(
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike | v:fP;zc
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )zu m.6pT
% with delta(m,0) the Kronecker delta, is chosen so that the integral 51`*VR]`K
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, bM"d$tl$?'
% and theta=0 to theta=2*pi) is unity. For the non-normalized U[NQ"
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. >[4CQK`U
% wPaMYxO/
% The Zernike functions are an orthogonal basis on the unit circle. V@\A<q%jTs
% They are used in disciplines such as astronomy, optics, and Pl&x6\zL
% optometry to describe functions on a circular domain. vue=K
% VF g"AJf
% The following table lists the first 15 Zernike functions. mw~$;64;a
% ?y,z
% n m Zernike function Normalization }ssL;q
% -------------------------------------------------- a 9Kws[
% 0 0 1 1 T)MZ`dM
% 1 1 r * cos(theta) 2 vGD D
% 1 -1 r * sin(theta) 2 y(Tb=:
% 2 -2 r^2 * cos(2*theta) sqrt(6) x,#?
% 2 0 (2*r^2 - 1) sqrt(3) 3($tD*!o
% 2 2 r^2 * sin(2*theta) sqrt(6) AP0z~e
% 3 -3 r^3 * cos(3*theta) sqrt(8) (4C_Ft*~j
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) HA~BXxa/
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) W.?EjEx
% 3 3 r^3 * sin(3*theta) sqrt(8) |yi#6!}^
% 4 -4 r^4 * cos(4*theta) sqrt(10) M~5Ja0N~
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) j0A9;AP;;C
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 3j/~XT
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) a4Y43 n
% 4 4 r^4 * sin(4*theta) sqrt(10) B }
% -------------------------------------------------- ~U1M-<IX
% t ]P^6jw'
% Example 1: N==Y]Z$G
% 8-FW'bA
% % Display the Zernike function Z(n=5,m=1) (gb
vInZ
% x = -1:0.01:1; .]ZMxDZ
% [X,Y] = meshgrid(x,x); +}Qq#^:_\
% [theta,r] = cart2pol(X,Y); WJii0+8e
% idx = r<=1; ]".SW5b_
% z = nan(size(X)); i=\`f& B
% z(idx) = zernfun(5,1,r(idx),theta(idx)); B=|m._OL]n
% figure 5wa!pR\c
% pcolor(x,x,z), shading interp Kk 6i
% axis square, colorbar YkI_i(
% title('Zernike function Z_5^1(r,\theta)') jGtu>|Gj
% pZ&?uo67_
% Example 2: Us4#O&
% (RI+4V1
% % Display the first 10 Zernike functions #~`d
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% x = -1:0.01:1; }PxPJ$o
% [X,Y] = meshgrid(x,x); UI74RP
% [theta,r] = cart2pol(X,Y); s@pIcNvx
% idx = r<=1; ]I(<hDuRp
% z = nan(size(X)); @hOT<
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% n = [0 1 1 2 2 2 3 3 3 3]; Q =4~uz|
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ONm-zRx|
% Nplot = [4 10 12 16 18 20 22 24 26 28]; U&u~i
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% y = zernfun(n,m,r(idx),theta(idx)); "1ov<
% figure('Units','normalized') ^d!I{ y#
% for k = 1:10 ;
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% z(idx) = y(:,k); 3(=QY)
% subplot(4,7,Nplot(k)) MbyV_A`r_
% pcolor(x,x,z), shading interp x1`zD*{
% set(gca,'XTick',[],'YTick',[]) `_ )5K u}
% axis square zQx6r
.
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #EIcP=1m4
% end zI.:1(,
% F3&:KZ!V&m
% See also ZERNPOL, ZERNFUN2. &?3P5dy_
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% Paul Fricker 11/13/2006 YAYwrKt
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% Check and prepare the inputs: M
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% ----------------------------- QA#
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Dj x[3['
error('zernfun:NMvectors','N and M must be vectors.') x)-n[Fu
end NU.YL1
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-'RD%_
if length(n)~=length(m) *2r(!fJP=^
error('zernfun:NMlength','N and M must be the same length.') # &Z1d(!
end 2 D!$x+|
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n = n(:); Tu-I".d+
m = m(:); fP;2qho
if any(mod(n-m,2)) 4\(|V
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error('zernfun:NMmultiplesof2', ... 1'SpJL1u~
'All N and M must differ by multiples of 2 (including 0).') y.?Q
end 1-?TjR
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if any(m>n) r lXMrn
error('zernfun:MlessthanN', ... 8t1,_,2'
'Each M must be less than or equal to its corresponding N.') =xRxr@
end SOQR(UT
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if any( r>1 | r<0 ) jn#Ok@tZ
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 4L)Ox;6>
end *sq+ Vc(
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) +*KDtqZjk
error('zernfun:RTHvector','R and THETA must be vectors.') Nj`Miv o
end <77v8=as5
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r = r(:); lBfG#\rdW~
theta = theta(:); b"&1l2\ A
length_r = length(r); uU#e54^
if length_r~=length(theta) ~+O ws
error('zernfun:RTHlength', ... CUa`#
'The number of R- and THETA-values must be equal.') %y R~dt'
end uqK[p^{
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% Check normalization: t$5)6zG
% -------------------- T.iVY5^<
if nargin==5 && ischar(nflag) G,A;`:/
isnorm = strcmpi(nflag,'norm'); M;1B}x@
if ~isnorm Ar1X
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error('zernfun:normalization','Unrecognized normalization flag.') ,v>|Ub,
end ~VaO,8&+L
else
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isnorm = false; {ZD'l5jU
end ,)P6fa/
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% XyytO;XM-
% Compute the Zernike Polynomials ]6TX)1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6sl2vHzA
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% Determine the required powers of r: bB"q0{9G-
% ----------------------------------- p_l.a
m_abs = abs(m); +*P;Vb6 D
rpowers = []; -
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for j = 1:length(n) lv0}d
rpowers = [rpowers m_abs(j):2:n(j)]; D-4\AzIb
end ro*$OLc/
rpowers = unique(rpowers); <%Afa#
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% Pre-compute the values of r raised to the required powers, [wR x)F"
% and compile them in a matrix: zwpgf
% ----------------------------- g;PZ$|%&s>
if rpowers(1)==0 Y"Y+U`Qt
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); T^n0 =|
rpowern = cat(2,rpowern{:}); 34Z$a{
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rpowern = [ones(length_r,1) rpowern]; QX&1BKqWn
else xlU:&=|
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 0I
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rpowern = cat(2,rpowern{:}); /J` ZO$
end k4Ub+F
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=pR'XF%
% Compute the values of the polynomials: $Hbd:1%i
{
% -------------------------------------- @8xa"Dc
y = zeros(length_r,length(n)); &Eqa y'
for j = 1:length(n) 0R[onPU_vZ
s = 0:(n(j)-m_abs(j))/2; sFWH*kdP?
pows = n(j):-2:m_abs(j);
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for k = length(s):-1:1 b\H !\A
p = (1-2*mod(s(k),2))* ... ]^
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prod(2:(n(j)-s(k)))/ ... qGPIKu
prod(2:s(k))/ ... R2!_)Rpf
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... A *_ |/o
prod(2:((n(j)+m_abs(j))/2-s(k))); j[y,Jch
idx = (pows(k)==rpowers); q%xq\L.
y(:,j) = y(:,j) + p*rpowern(:,idx); { WW!P,w
end li
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*{e?%!Q
if isnorm <>|/U `
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); yQM<(;\O
end #+]-}v3
end mbh;oX+
% END: Compute the Zernike Polynomials KOM]7%ys1H
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #X?#v7i",D
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% Compute the Zernike functions: JK@"
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% ------------------------------ tfb_K4h6,
idx_pos = m>0; o(_~
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idx_neg = m<0; 7y2-8eL
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z = y; mfpL?N
if any(idx_pos) (fJ.o-LQ
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 9?~K"+-SI
end cw)'vAE
if any(idx_neg) 4RYvI!
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ~GZpAPg*
end 'E#;`}&Ah
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% EOF zernfun Hg}@2n)/