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, ?1LRR
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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, N>)Db
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? iG=Di)O
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ZhC,nbM
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function z = zernfun(n,m,r,theta,nflag) 5x,/p
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. gr@Ril^
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 50T^V`6
% and angular frequency M, evaluated at positions (R,THETA) on the `9T5Dem|#
% unit circle. N is a vector of positive integers (including 0), and /wP2Wnq$
% M is a vector with the same number of elements as N. Each element & Yx12B\
% k of M must be a positive integer, with possible values M(k) = -N(k) `UqX`MFz
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, [1z.JfC :S
% and THETA is a vector of angles. R and THETA must have the same wAL}c(EHO
% length. The output Z is a matrix with one column for every (N,M) L
gy^^.
% pair, and one row for every (R,THETA) pair. zXbA$c
% AYp~;@
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike P>`|.@
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), uK ,W
% with delta(m,0) the Kronecker delta, is chosen so that the integral LPca+o|f
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, mwI7[I2q
% and theta=0 to theta=2*pi) is unity. For the non-normalized Y;
to9Kv$
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ',rK\&lL6
% OF-VVIS
% The Zernike functions are an orthogonal basis on the unit circle. MhB>bnWXR
% They are used in disciplines such as astronomy, optics, and 3od16{YH
% optometry to describe functions on a circular domain. 0y+i?y
9
% 1Lp; LY"_
% The following table lists the first 15 Zernike functions. [Q/kNK
% _8\B~;0
% n m Zernike function Normalization Ji6.-[:
% -------------------------------------------------- :l?mNm5
% 0 0 1 1 hMV>5Y[s
% 1 1 r * cos(theta) 2 dT (i*E\j
% 1 -1 r * sin(theta) 2 }u{gQlV
% 2 -2 r^2 * cos(2*theta) sqrt(6) ]IzD`
% 2 0 (2*r^2 - 1) sqrt(3) 1083p9Uh
% 2 2 r^2 * sin(2*theta) sqrt(6) o)R<sT
% 3 -3 r^3 * cos(3*theta) sqrt(8) 4GXS(
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) [8 H:5Ho
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ( 5uSqw&U
% 3 3 r^3 * sin(3*theta) sqrt(8) ooC9a>X
% 4 -4 r^4 * cos(4*theta) sqrt(10) TNK1E
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) M* {5> !\
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) cL~YQJYp
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) BL"7_phM,
% 4 4 r^4 * sin(4*theta) sqrt(10) @YG-LEh
% -------------------------------------------------- J(wFJg\/
% Htln <N
% Example 1: >Q?8tGfB
% KeXt"U
% % Display the Zernike function Z(n=5,m=1) }6=)w@v
% x = -1:0.01:1; KD H<T4#x
% [X,Y] = meshgrid(x,x); kQQDaZ8
% [theta,r] = cart2pol(X,Y); 18Ju]U
% idx = r<=1; "^;h'
% z = nan(size(X)); ^Xu4N"@
% z(idx) = zernfun(5,1,r(idx),theta(idx)); LhM$!o?W
% figure ~P;A
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% pcolor(x,x,z), shading interp ;K%/sIIke
% axis square, colorbar Z&P\}mm
% title('Zernike function Z_5^1(r,\theta)') y~VI,82*
% tV>qV\>
% Example 2: KC9e{
% 9\/oL{
% % Display the first 10 Zernike functions }&==;7,O
% x = -1:0.01:1; 4- Jwy
% [X,Y] = meshgrid(x,x); *c&|2EsZ
% [theta,r] = cart2pol(X,Y); &ODo7@v`1
% idx = r<=1; 3wcFR0f
% z = nan(size(X)); ?( z"Ub]
% n = [0 1 1 2 2 2 3 3 3 3]; m]vV.pwv
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ;[(d=6{hc]
% Nplot = [4 10 12 16 18 20 22 24 26 28]; E0EK88
% y = zernfun(n,m,r(idx),theta(idx)); R^P>yk8
% figure('Units','normalized') ffBd
% for k = 1:10 n${k^e-=
% z(idx) = y(:,k); g|7o1{
% subplot(4,7,Nplot(k)) r\Kcg~D>
% pcolor(x,x,z), shading interp sowwXrECg@
% set(gca,'XTick',[],'YTick',[]) SW'eTG
% axis square cC+2%q B
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 5,g +OY=\
% end %'Q2c'r
% 7')W+`o8eL
% See also ZERNPOL, ZERNFUN2. <c:H u{D
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% Paul Fricker 11/13/2006 |D%mWQng
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% Check and prepare the inputs: \IImxkE
% ----------------------------- Y:t?W
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) y$SUYG'v
error('zernfun:NMvectors','N and M must be vectors.') 5g/,VMe
end pt,L
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x\8|A
if length(n)~=length(m) 0<NS1y
error('zernfun:NMlength','N and M must be the same length.') a.}#nSYP
end L\E>5G;
#
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n = n(:); F P|cA^$<
m = m(:); wK#*|
if any(mod(n-m,2)) V-n{=8s
error('zernfun:NMmultiplesof2', ... 3?I!
'All N and M must differ by multiples of 2 (including 0).') qqf*g=f
end ||awNSt
n$r`s`}
#?jsC)
if any(m>n) z+{qQ!
error('zernfun:MlessthanN', ... ,_Bn{T=U
'Each M must be less than or equal to its corresponding N.') L\:m)g,F.
end Ui`{U
J&,hC%]
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if any( r>1 | r<0 ) Kw"y#Ys]
error('zernfun:Rlessthan1','All R must be between 0 and 1.') X )tH23
end MK)}zjw
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) UGy3B)
error('zernfun:RTHvector','R and THETA must be vectors.') i\ X3t5
end "g&f:[a/
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r = r(:); nH6SA1$kW
theta = theta(:); `cXLa=B)9
length_r = length(r); UNa"\
if length_r~=length(theta) 6{=U=
*
error('zernfun:RTHlength', ... tJrGRlB>
'The number of R- and THETA-values must be equal.') TZt;-t`
end T:X*
;'8P/a$
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% Check normalization: PR:B6 F8
% -------------------- |<,qnf| -
if nargin==5 && ischar(nflag) vjx'yh|
isnorm = strcmpi(nflag,'norm'); $Z#~wsw
if ~isnorm M?"4{
error('zernfun:normalization','Unrecognized normalization flag.') @tm2Y%Y!
end N'WTIM3W
else 9U6$-]J
isnorm = false; S*h^7?Bu
end
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c-XO}\?
% Compute the Zernike Polynomials *pa hZiO
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Uq#2~0n>
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% Determine the required powers of r: {] ]%0!n\
% ----------------------------------- zMbFh_dcq
m_abs = abs(m); J4::.r
rpowers = []; MLHCBRi
for j = 1:length(n) +?U[362>
rpowers = [rpowers m_abs(j):2:n(j)]; %QEBY>|lI
end g]?pY
rpowers = unique(rpowers); m1;Htw
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y)
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~j/
% Pre-compute the values of r raised to the required powers, n!/0yR2S
% and compile them in a matrix: xn2 nh@;
% ----------------------------- pS+w4gW
if rpowers(1)==0 O~V^]
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); K-TsSW$}
rpowern = cat(2,rpowern{:}); !\R5/-_UU
rpowern = [ones(length_r,1) rpowern]; Dnw^H.
else ?g+3 URpK
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); zU&Iy_Ke.
rpowern = cat(2,rpowern{:}); VtLRl0/
end #ay/VlD@
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% Compute the values of the polynomials: g@>llve{
% -------------------------------------- lu"0\}7X
y = zeros(length_r,length(n)); :VlA2Ih&q
for j = 1:length(n) u>lt}0
s = 0:(n(j)-m_abs(j))/2; Eu(QeST\
pows = n(j):-2:m_abs(j); . J O3#
for k = length(s):-1:1 )mm0PJF~q
p = (1-2*mod(s(k),2))* ... Lf5zHUH
prod(2:(n(j)-s(k)))/ ... A)]&L`s
prod(2:s(k))/ ... ]Wkgpfd56
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... zBy} > Jx
prod(2:((n(j)+m_abs(j))/2-s(k))); >va_,Y}
idx = (pows(k)==rpowers); 62R";# K
y(:,j) = y(:,j) + p*rpowern(:,idx); &wK:R,~x6
end BC.3U.
~@c<5 -`{
if isnorm hE(R[hc
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); u:p OP
end <WIIurp
end &!/>B .
% END: Compute the Zernike Polynomials %oa@2qJ^
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tI{]&dev
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-z`%x@F<&L
% Compute the Zernike functions: $f3 IO#N
% ------------------------------ h<%$?h+}
idx_pos = m>0; PSq?8.
idx_neg = m<0; LhLAQ2~
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]`$yY5 &W0
z = y; s;TB(M~i[
if any(idx_pos) T)I)r239h
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); L&kCI`Tb
end >S:(BJMo
if any(idx_neg) }2;P`s
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 0R)x"4Ww
end \o[][R#D
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% EOF zernfun BZW03e8|