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, HJt
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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, 5-UrHbpCZ#
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? (W?t'J^#
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? l YpoS
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function z = zernfun(n,m,r,theta,nflag) ,JfP$HJ
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Q+s2S>U{v
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N +3Z+#nGtk
% and angular frequency M, evaluated at positions (R,THETA) on the nK#%Od{GF
% unit circle. N is a vector of positive integers (including 0), and <MoyL1=
% M is a vector with the same number of elements as N. Each element mSGpxZ,IE
% k of M must be a positive integer, with possible values M(k) = -N(k) 8Z3:jSgk
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, M"6J"s
% and THETA is a vector of angles. R and THETA must have the same g!^mewtd
% length. The output Z is a matrix with one column for every (N,M) p5l|qs
% pair, and one row for every (R,THETA) pair. *'@sm*
% $@84nR{>
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 4K*st8+bl-
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Nw1Bn~yx<R
% with delta(m,0) the Kronecker delta, is chosen so that the integral `>
+:38
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ,\N4tG1\
% and theta=0 to theta=2*pi) is unity. For the non-normalized \{v-Xe&d^
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. =]1cVnPI
% 6W:FT Pt44
% The Zernike functions are an orthogonal basis on the unit circle. i`=%X{9
% They are used in disciplines such as astronomy, optics, and -Ua&/Yd/}
% optometry to describe functions on a circular domain. =MwR)CI#
% W
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% The following table lists the first 15 Zernike functions. $i1:--~2\
% stiYC#b I:
% n m Zernike function Normalization zL9:e7o
% -------------------------------------------------- M>xT\
% 0 0 1 1 ^tIYr<I
% 1 1 r * cos(theta) 2 Dw$RHogb~y
% 1 -1 r * sin(theta) 2 NMUF)ksjN
% 2 -2 r^2 * cos(2*theta) sqrt(6) Q{CRy-ha
% 2 0 (2*r^2 - 1) sqrt(3) 15OzO.Ud
% 2 2 r^2 * sin(2*theta) sqrt(6) _7~q|
% 3 -3 r^3 * cos(3*theta) sqrt(8) _-2ntO<E
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 7spZe"
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) @!^Y_q
% 3 3 r^3 * sin(3*theta) sqrt(8) + WT?p]
% 4 -4 r^4 * cos(4*theta) sqrt(10) u=Xpu,q
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ZrB(!L~7
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) wN^^_
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) I'[;E.KU
% 4 4 r^4 * sin(4*theta) sqrt(10) i)$ySlEh
% -------------------------------------------------- HE>V\+
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% (G(M"S SC
% Example 1: ^m
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% HMDuP2Y
% % Display the Zernike function Z(n=5,m=1) | GN/{KH]
% x = -1:0.01:1; h6n!"z8H
% [X,Y] = meshgrid(x,x); zGy+jeH:.
% [theta,r] = cart2pol(X,Y); .`(YCn?\
% idx = r<=1; q_9 8=fyE6
% z = nan(size(X)); Q<KF<K'0hg
% z(idx) = zernfun(5,1,r(idx),theta(idx)); f4&;l|R0a
% figure ?FwHqyFVlQ
% pcolor(x,x,z), shading interp GVfRy@7n
% axis square, colorbar <$d2m6 J
% title('Zernike function Z_5^1(r,\theta)') 7|jy:F,w%
% e)m6xiZ
% Example 2: reM~q-M~o@
% c
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% % Display the first 10 Zernike functions /a)^)
% x = -1:0.01:1; kDxI7$]E
% [X,Y] = meshgrid(x,x); %oquHkX%OJ
% [theta,r] = cart2pol(X,Y); e/#6qCE
% idx = r<=1; RCoDdtMo
% z = nan(size(X)); g^7zDU&'
% n = [0 1 1 2 2 2 3 3 3 3]; *ae)<l3v
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; u J]uz%
% Nplot = [4 10 12 16 18 20 22 24 26 28]; [:qJ1^U U
% y = zernfun(n,m,r(idx),theta(idx)); UZmo?&y
% figure('Units','normalized') #p]V?
% for k = 1:10 `Q#)N0
% z(idx) = y(:,k); J<4_<.o(a
% subplot(4,7,Nplot(k)) L3'isaz&^
% pcolor(x,x,z), shading interp hwQ|'^(@O
% set(gca,'XTick',[],'YTick',[]) d$xvM
% axis square 3<N2ehi?
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) F! Cn'*
% end uI&0/
% ,-8Xb+!8I
% See also ZERNPOL, ZERNFUN2. GN=8;Kq%
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% Paul Fricker 11/13/2006 ld):Am}/o
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% Check and prepare the inputs: ZX.,<vumSy
% ----------------------------- }NRt:JC
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 49O_A[(d
error('zernfun:NMvectors','N and M must be vectors.') \2#K {
end kmo#jITa`
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\
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if length(n)~=length(m) g\?07@Zd|
error('zernfun:NMlength','N and M must be the same length.') i_+e&Bjd4j
end 5dG+>7Iy}
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n = n(:); Dq9f Fe
m = m(:); r2PN[cLu|
if any(mod(n-m,2)) H4{7,n
error('zernfun:NMmultiplesof2', ... ~k?t
'All N and M must differ by multiples of 2 (including 0).') dS \n2Qb
end \IzZJGi
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if any(m>n) fIatp
error('zernfun:MlessthanN', ... &hi][Pt
'Each M must be less than or equal to its corresponding N.') 6z/&j} (
end 3/&
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if any( r>1 | r<0 ) Z3ODZfu>
error('zernfun:Rlessthan1','All R must be between 0 and 1.') QV*la= j/
end '=Jz}F <
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 7ruWmy;j
error('zernfun:RTHvector','R and THETA must be vectors.') c\tw#;\9
end /8hjs{(;
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r = r(:); @e7_&EGR?
theta = theta(:); ZC^?ng
length_r = length(r); v{\~>1J{
if length_r~=length(theta) g K dNgU
error('zernfun:RTHlength', ... : B1
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'The number of R- and THETA-values must be equal.') [!ZYtp?Hf
end +yHzp
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% Check normalization: % %2~%FVb
% -------------------- ~$Z_#,|i?
if nargin==5 && ischar(nflag) _tO2PIL@Z
isnorm = strcmpi(nflag,'norm'); ^4saB+qm
if ~isnorm 91#n Aj%
error('zernfun:normalization','Unrecognized normalization flag.') dsbz\w3:
end Mq6_Q07
else 8mX:*$qm:
isnorm = false; ^$lsmF]^
end #P1;*m
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [@s5v
% Compute the Zernike Polynomials vF@.BM>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =E8Kacu%
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% Determine the required powers of r: Q $>SYvW
% ----------------------------------- ?Ye%k
m_abs = abs(m); /bqJ6$
rpowers = []; ]g9n#$|.
for j = 1:length(n) v8A{q
rpowers = [rpowers m_abs(j):2:n(j)]; 0 f"M-x
end \G1(r=fU
rpowers = unique(rpowers); DRi/<
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% Pre-compute the values of r raised to the required powers, #.\X%!
% and compile them in a matrix: ;4]l P
% ----------------------------- gJBk&SDgtP
if rpowers(1)==0 Bk~M ^AK@~
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); {eo?vA8SE
rpowern = cat(2,rpowern{:}); Ad`jV_z
rpowern = [ones(length_r,1) rpowern];
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else YyR~pT#ffT
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); OAz-w
rpowern = cat(2,rpowern{:}); )F35WP~
end KHXnB
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% Compute the values of the polynomials: "A9 c]
% -------------------------------------- gs77")K&
y = zeros(length_r,length(n)); x;*KRO
for j = 1:length(n) mCx6$jz
s = 0:(n(j)-m_abs(j))/2; PK*
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pows = n(j):-2:m_abs(j); D<cHa |
for k = length(s):-1:1 I^6zUVH
p = (1-2*mod(s(k),2))* ... (wIpq<%
prod(2:(n(j)-s(k)))/ ... th*E"@
prod(2:s(k))/ ... CR$5'#11)
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ?5 d3k%
prod(2:((n(j)+m_abs(j))/2-s(k))); /fc@=CO
idx = (pows(k)==rpowers); +P <Lo I
y(:,j) = y(:,j) + p*rpowern(:,idx); D*j\gI
end re/l5v,|3
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if isnorm F
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y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); %/&?t`%H
end #`4ma:Pj
end <[7.+{qfW
% END: Compute the Zernike Polynomials H;$O CDRC
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6]^}GyM!
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% Compute the Zernike functions: y8un&LP
% ------------------------------ ^1S(6'a#
idx_pos = m>0; JQ8wL _C>
idx_neg = m<0; v7/qJ9l
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z = y; D[)g-_3f6<
if any(idx_pos) i9oi}$;J
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); x,z +l-y
end |,Y(YSg.
if any(idx_neg) >T4.mB7+>
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); u%S&EuX
end Q': }'CI
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% EOF zernfun y!Q&;xO+!