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, j WerX -$
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, Y%YPR=j~ &
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\>=}7
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? x#TWZ;
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function z = zernfun(n,m,r,theta,nflag) Tp
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ,?>:Cdz4
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N m/>z}d05h
% and angular frequency M, evaluated at positions (R,THETA) on the * 57y.](w
% unit circle. N is a vector of positive integers (including 0), and *XSHzoT*
% M is a vector with the same number of elements as N. Each element 9lCZi?
% k of M must be a positive integer, with possible values M(k) = -N(k) 4XsKOv
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, h?2 :'Vu]
% and THETA is a vector of angles. R and THETA must have the same ]WP[hF
% length. The output Z is a matrix with one column for every (N,M) S!wY6z
% pair, and one row for every (R,THETA) pair. 4.0JgX
% c!}f\ ]D
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike (vqI@fB';u
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), f3Cjj]RFv
% with delta(m,0) the Kronecker delta, is chosen so that the integral %b(non*
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, @Zd/>'
% and theta=0 to theta=2*pi) is unity. For the non-normalized nolLeRE1
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. qOd*9AS'|M
% 3~Vo]wv
% The Zernike functions are an orthogonal basis on the unit circle. 2t7Hu)V
% They are used in disciplines such as astronomy, optics, and |UZhMF4/-L
% optometry to describe functions on a circular domain. nkvkHh
% p 6FPdt)
% The following table lists the first 15 Zernike functions. "vnWq=E2
% }n91aE3v
% n m Zernike function Normalization @(_M\>!%M
% --------------------------------------------------
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% 0 0 1 1 235wl
% 1 1 r * cos(theta) 2 9e:}qO5)
% 1 -1 r * sin(theta) 2 L_WVTz?`
% 2 -2 r^2 * cos(2*theta) sqrt(6) .^J7^Ky,
% 2 0 (2*r^2 - 1) sqrt(3) ]C
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% 2 2 r^2 * sin(2*theta) sqrt(6) $5;RQNhXh
% 3 -3 r^3 * cos(3*theta) sqrt(8) |p1pa4%}
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) \x_fP;ma=_
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) D3+UV+&R/
% 3 3 r^3 * sin(3*theta) sqrt(8) &J~%Nt
% 4 -4 r^4 * cos(4*theta) sqrt(10) M;i4ss,}!
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0G.y_<=
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) P_f>a?OL:
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) mVBF2F<4
% 4 4 r^4 * sin(4*theta) sqrt(10) _+\hDV>v
% -------------------------------------------------- =5-|H;da
% '"=Mw;p
% Example 1: 75pz' Cb
% ,^_aqH
% % Display the Zernike function Z(n=5,m=1) +I+7@Xi Z
% x = -1:0.01:1; {,|J?>{
% [X,Y] = meshgrid(x,x); 3 #zwY
% [theta,r] = cart2pol(X,Y); {|jG_
% idx = r<=1; u$Za hN!
% z = nan(size(X)); <A,G:&d~
% z(idx) = zernfun(5,1,r(idx),theta(idx)); #z\{BtK
% figure r"MKkSEM
% pcolor(x,x,z), shading interp VvO/
% axis square, colorbar T F !Lp:
% title('Zernike function Z_5^1(r,\theta)') `2Buf8|a,
% []{g9CO
% Example 2: &&w7-
% Xj\SJ*
% % Display the first 10 Zernike functions S:UtmS+K
% x = -1:0.01:1; ~?pF'3q
% [X,Y] = meshgrid(x,x); &S.zc@rN
% [theta,r] = cart2pol(X,Y); hwmpiyu
% idx = r<=1; od- 0wJN-m
% z = nan(size(X)); Ah2%LXdHA
% n = [0 1 1 2 2 2 3 3 3 3]; R,hX *yVq
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; NC;4
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 9oteQN{9
% y = zernfun(n,m,r(idx),theta(idx)); RN?z)9!
% figure('Units','normalized') a; Ihv#q
% for k = 1:10 &/7AW(?
% z(idx) = y(:,k); urHQb5|T}
% subplot(4,7,Nplot(k)) m6bI<C3^5
% pcolor(x,x,z), shading interp W39R)sra
% set(gca,'XTick',[],'YTick',[]) #]ii/Et#x
% axis square I3xx}^V
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) K5Fzmo a
% end &Jj^)GBU
% * xs8/?
% See also ZERNPOL, ZERNFUN2. {Ex0mw)T
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% Paul Fricker 11/13/2006 0\AYUa?RM
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% Check and prepare the inputs: 5|:=#Ql*
% ----------------------------- ru)%0Cyx
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) -t
%.I=|
error('zernfun:NMvectors','N and M must be vectors.') W K#lE&V3
end +MOUO$;fGt
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if length(n)~=length(m) L[+65ce%*
error('zernfun:NMlength','N and M must be the same length.') yPm)r2Ck
end mxjY-Kq
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n = n(:); W.o
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m = m(:); [HIg\N$I8C
if any(mod(n-m,2)) #(CI/7
-
error('zernfun:NMmultiplesof2', ... /NLpk7r[\q
'All N and M must differ by multiples of 2 (including 0).') yq[C?N &N
end U,Z.MPQ
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if any(m>n) P 2;j>=W
error('zernfun:MlessthanN', ... `
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'Each M must be less than or equal to its corresponding N.') .he%a3e
end Yk<?HNf
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if any( r>1 | r<0 ) :!3CoC.X|c
error('zernfun:Rlessthan1','All R must be between 0 and 1.') nb'],({:9
end ]=q?=%H
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ~b8U#'KD
error('zernfun:RTHvector','R and THETA must be vectors.') <JYV
G9s}
end lGjmw"/C
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r = r(:); v8
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theta = theta(:); "z< =S
length_r = length(r); Lc+wS@
if length_r~=length(theta) A(Ugam~}
error('zernfun:RTHlength', ... F7#
'The number of R- and THETA-values must be equal.') ~2V|]Y;s
end lXW.G
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f]%$HfF@
% Check normalization: M*8Ef^-U`t
% -------------------- <d$|~qS_
if nargin==5 && ischar(nflag) Po(9BRd7
isnorm = strcmpi(nflag,'norm'); noOG$P#
if ~isnorm E7oL{gU
error('zernfun:normalization','Unrecognized normalization flag.') >=6tfLQ
end "ln(EvW
else & 2>W=h
isnorm = false; z[}[:H8
end aJOhji<b#L
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 536H*HdN
% Compute the Zernike Polynomials M7fw/i
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M{3He)&
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% Determine the required powers of r: 2u *o/L+
% ----------------------------------- /F4rbL^:
m_abs = abs(m); =UM30
P/
rpowers = []; 0}PW<lU-
for j = 1:length(n) >ys>Q)
rpowers = [rpowers m_abs(j):2:n(j)]; Ym8G=KA
end bezT\F/\
rpowers = unique(rpowers); gieTkZ
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w\:-lX w
% Pre-compute the values of r raised to the required powers, UMma|9l(i
% and compile them in a matrix: 0;#%KC,
% ----------------------------- *[wy-
fu
if rpowers(1)==0 %r=uS.+hrF
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); .a8N 5{`
rpowern = cat(2,rpowern{:}); <_dyUiT$J
rpowern = [ones(length_r,1) rpowern]; X2}\i5{
else N&]v\MjI62
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); lQ<2Vw#Yl
rpowern = cat(2,rpowern{:}); cuO(*%Is1
end \3-XXq
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% Compute the values of the polynomials: &na#ES$X,
% -------------------------------------- %g5TU 6WP
y = zeros(length_r,length(n)); j&6,%s-M`a
for j = 1:length(n) .{1G"(z
s = 0:(n(j)-m_abs(j))/2; 9%S{fd\#
pows = n(j):-2:m_abs(j); WS/^WxRY
for k = length(s):-1:1 5x(`z
p = (1-2*mod(s(k),2))* ... 9c1g,:8\
prod(2:(n(j)-s(k)))/ ... OL4I}^*,
prod(2:s(k))/ ... r Lg(J|^
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... K_{f6c<
prod(2:((n(j)+m_abs(j))/2-s(k))); w17\ \[
idx = (pows(k)==rpowers); / *RDy!m
y(:,j) = y(:,j) + p*rpowern(:,idx); >6*"g{/
end MqGF~h|+
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if isnorm }__g\?Yf
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); g]EDL<b
end guz{DBlK
end u/Fa+S
% END: Compute the Zernike Polynomials ~=h]r/b< U
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%_0,z`f
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% Compute the Zernike functions: #H{<nVvg^
% ------------------------------ ] e!CH
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idx_pos = m>0; .=~beTS'Vo
idx_neg = m<0; r=Z#"68$
%Fig`qX
* t6XU
z = y; |7,|-s[R^
if any(idx_pos) VgtWT`F.I
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); cTu7U=%
end #P.jlpZk
if any(idx_neg) -CfGWO#Gbx
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); F@Y)yi?z
end -fw0bL%0
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% EOF zernfun nv1'iSEeOl