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, Ks(+['*S
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, G^ZL,{
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? /QZnN?k
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? T2P0(rEz
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function z = zernfun(n,m,r,theta,nflag) >k,bHGj?
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. nU- .a5
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Py^F},?J
% and angular frequency M, evaluated at positions (R,THETA) on the /V+N
% unit circle. N is a vector of positive integers (including 0), and
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% M is a vector with the same number of elements as N. Each element M`*B/Fh2
% k of M must be a positive integer, with possible values M(k) = -N(k) <N}UwB&
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, AU)"L_
i}
% and THETA is a vector of angles. R and THETA must have the same ID
&Iz
% length. The output Z is a matrix with one column for every (N,M) 2`Ub;Nn29
% pair, and one row for every (R,THETA) pair.
oJ ~ZzW
% E{[c8l2B
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike s^TF+d?B
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), T;XEU%:LK
% with delta(m,0) the Kronecker delta, is chosen so that the integral bHH{bv~Z
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, CkE@Ll3Z
% and theta=0 to theta=2*pi) is unity. For the non-normalized ,%w_E[2
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 1&\_|2
% }QU9+<Z[r
% The Zernike functions are an orthogonal basis on the unit circle. nyWA(%N1
% They are used in disciplines such as astronomy, optics, and %6j|/|#]
% optometry to describe functions on a circular domain. +Pd&YfU9
% ?7 e|gpQ|
% The following table lists the first 15 Zernike functions. B q+RFo
% i[`nu#n/
% n m Zernike function Normalization Q.7Rv
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% -------------------------------------------------- [yM{A<\L
% 0 0 1 1 $v#Q'?jE
% 1 1 r * cos(theta) 2 1_%jDMYH
% 1 -1 r * sin(theta) 2 [X ]\^
% 2 -2 r^2 * cos(2*theta) sqrt(6) "#z4
% 2 0 (2*r^2 - 1) sqrt(3) )tl=tH/$
% 2 2 r^2 * sin(2*theta) sqrt(6) r483"k(7
% 3 -3 r^3 * cos(3*theta) sqrt(8) y:WRpCZoa
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 6^F"np{w
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) JP)/
O!
% 3 3 r^3 * sin(3*theta) sqrt(8) #Z;ziM:
% 4 -4 r^4 * cos(4*theta) sqrt(10) \j !JRD+j
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) s\_-` [B0
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) $,otW2:)
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) gRIRc4p
% 4 4 r^4 * sin(4*theta) sqrt(10) IzF7W?k
% -------------------------------------------------- ;X<#y2`
% Ck8`$x&t
% Example 1: h@=H7oV7k
% zDeh#
% % Display the Zernike function Z(n=5,m=1) eUPG){"
% x = -1:0.01:1; 'uBXSP#
% [X,Y] = meshgrid(x,x); I gcVl/d
% [theta,r] = cart2pol(X,Y); yx"xbCc#
% idx = r<=1; ks<gSCB
% z = nan(size(X)); yS p]+
% z(idx) = zernfun(5,1,r(idx),theta(idx)); {\[u2{
% figure wvvMesX<L
% pcolor(x,x,z), shading interp m:5 *:Ii.
% axis square, colorbar FKY|xG9
% title('Zernike function Z_5^1(r,\theta)') 3GUO
% ftq&<8
% Example 2: 85Zy0l
% `An|a~G1
% % Display the first 10 Zernike functions !31v@v:)
% x = -1:0.01:1; ZfM(%rx
% [X,Y] = meshgrid(x,x); Q<B=m6~
% [theta,r] = cart2pol(X,Y); fT [JU1
% idx = r<=1; _;3xG0+
% z = nan(size(X));
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% n = [0 1 1 2 2 2 3 3 3 3]; >i7zV`eK
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; U4qp?g+:
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 3ddH@Y|
% y = zernfun(n,m,r(idx),theta(idx)); He}qgE>Us
% figure('Units','normalized') Rd|};-
% for k = 1:10 ~F~g$E2 }
% z(idx) = y(:,k); sCU<1=
% subplot(4,7,Nplot(k)) /*!K4)$-*2
% pcolor(x,x,z), shading interp pE@Q
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% set(gca,'XTick',[],'YTick',[]) %$|=_K)Ks
% axis square A+w51Q
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Q!(16
% end )pLde_ k
% Ql&5fyW
% See also ZERNPOL, ZERNFUN2. GqBZWmAB
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% Paul Fricker 11/13/2006 .vYU4g]
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% Check and prepare the inputs: zC#[
% ----------------------------- E7@0,9AU
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) / =&HunaxI
error('zernfun:NMvectors','N and M must be vectors.') W- 5Z"m1I
end +LeZjA[
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if length(n)~=length(m) 'bVDm m).
error('zernfun:NMlength','N and M must be the same length.') "_t2R &A
end u^T)4~(
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n = n(:); &Iv3_T<AF
m = m(:); tQE=c7/M
if any(mod(n-m,2)) ua[ d
error('zernfun:NMmultiplesof2', ... p'z
fo!
'All N and M must differ by multiples of 2 (including 0).') Lpd q^X
end ee}&~%
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if any(m>n) },G6IuH%
error('zernfun:MlessthanN', ... Bc3(xI'>J
'Each M must be less than or equal to its corresponding N.') sT:$:=
end ``KimeA~
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if any( r>1 | r<0 ) $?RxmWsP
error('zernfun:Rlessthan1','All R must be between 0 and 1.') v&6I\1
end 60p*$Vqy
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) t@(S=i7}-
error('zernfun:RTHvector','R and THETA must be vectors.') |35"V3bs
end KY 085Fvs
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r = r(:); /NRdBN
theta = theta(:); zzOc
# /
length_r = length(r); 8U}BSM_<2
if length_r~=length(theta) 1KwUp0%&
error('zernfun:RTHlength', ... A'Q=DoE
'The number of R- and THETA-values must be equal.') pJ)PVo\cV
end '4 T}$a"i
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2_i/ F)W
% Check normalization: g=W1y
% -------------------- vzDoF0Ts*p
if nargin==5 && ischar(nflag) aVTTpMY
isnorm = strcmpi(nflag,'norm'); oAaUXkQE
if ~isnorm x?T.ItW:K
error('zernfun:normalization','Unrecognized normalization flag.') vX|i5P0)8
end K??(>0Qr}r
else w}2 ;f=
isnorm = false; fsd,q?{a:
end 'Pk14`/
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9?M>Y?4
% Compute the Zernike Polynomials P]V/<8o.53
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d@-s_gw
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% Determine the required powers of r: /;P* ?
% ----------------------------------- EPO*{bN7O
m_abs = abs(m); }t.J;(ff:
rpowers = []; PeCU V6
for j = 1:length(n) bWp40&vx
rpowers = [rpowers m_abs(j):2:n(j)]; 4-ijuqjN
end k)l*L1Y4:
rpowers = unique(rpowers); C|"BMam
uh,~CvXU]
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% Pre-compute the values of r raised to the required powers, o0SQJ1.a$
% and compile them in a matrix: St9+/Md=jQ
% ----------------------------- 9hoTxWpmy
if rpowers(1)==0 *hugQh]a
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ~r(/)w\
rpowern = cat(2,rpowern{:}); r7dvj#^
rpowern = [ones(length_r,1) rpowern]; y9<]F6TT
else ';T=kS<^_
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); vpTYfE
rpowern = cat(2,rpowern{:}); .Y@)3
end `8 Q3=^)3
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% Compute the values of the polynomials: #f*,mY|>
% -------------------------------------- \TchRSe
y = zeros(length_r,length(n)); F|Y}X|x8Q
for j = 1:length(n) u+
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s = 0:(n(j)-m_abs(j))/2; D)0pm?*5A
pows = n(j):-2:m_abs(j); funHznRR
for k = length(s):-1:1 mn5mdrv3WZ
p = (1-2*mod(s(k),2))* ... &RSUB;ymL
prod(2:(n(j)-s(k)))/ ... q ERdQ~M,
prod(2:s(k))/ ... >J!J:
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... +Ndo$|XCy]
prod(2:((n(j)+m_abs(j))/2-s(k))); D I`
M
idx = (pows(k)==rpowers); NhP&sQO
y(:,j) = y(:,j) + p*rpowern(:,idx); ;|nC;D]
end pUTC~|j%:
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if isnorm <4DSk9/
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); kqyVUfX$3
end l~cT]Ep
end |{)SLvlJl
% END: Compute the Zernike Polynomials &DUt`Dr w
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .JkcCEe{G
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% Compute the Zernike functions: r$=YhI/=
% ------------------------------ d-cK`pSB
idx_pos = m>0; /CXrxeo
idx_neg = m<0; fF~3"!1#\I
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z = y; $
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if any(idx_pos) 'yrU_k,h
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Dg:2*m_!j{
end ;p$KM-?2D
if any(idx_neg) !a(#G7zA
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); IV#kF}9$
end g%Yw Dr=0t
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% EOF zernfun ZO^Y9\L