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, <$Xn:B<H
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, zkw0jX~
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? >0[qi1
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? GIJV;7~
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function z = zernfun(n,m,r,theta,nflag) #nt<j2}m
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. \["1N-q b
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N B]CS2LEqh
% and angular frequency M, evaluated at positions (R,THETA) on the %DHP
% unit circle. N is a vector of positive integers (including 0), and hwG||;&/H
% M is a vector with the same number of elements as N. Each element #<^/yoH7C6
% k of M must be a positive integer, with possible values M(k) = -N(k) LKoM\g(
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Xb8:*Y1'
% and THETA is a vector of angles. R and THETA must have the same C: TuC5Sr
% length. The output Z is a matrix with one column for every (N,M) ZnxOa
% pair, and one row for every (R,THETA) pair. sP=2NqU3Q
% ,(5dQ` hA0
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike D
z]}@Z*jK
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), $]`'Mi
% with delta(m,0) the Kronecker delta, is chosen so that the integral `RL(N4H
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, JRcuw'8+q
% and theta=0 to theta=2*pi) is unity. For the non-normalized %u<&^8EL+#
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. UwzE'#Q-
% 1L(Nfkh
% The Zernike functions are an orthogonal basis on the unit circle. ;FIMCJS
% They are used in disciplines such as astronomy, optics, and 1yY'hb,0
% optometry to describe functions on a circular domain. ~Y}Z4" o
%
~gcst;
% The following table lists the first 15 Zernike functions. S(YHwH":
% 2t~7eI%d
% n m Zernike function Normalization "J0Oa?
% -------------------------------------------------- C'xU=OnA8
% 0 0 1 1 cfQh
% 1 1 r * cos(theta) 2 z;Gbqr?{{
% 1 -1 r * sin(theta) 2 '+GVozc6c"
% 2 -2 r^2 * cos(2*theta) sqrt(6) N1B$ G
% 2 0 (2*r^2 - 1) sqrt(3) .LhbhUEfn
% 2 2 r^2 * sin(2*theta) sqrt(6) Dq_{O
% 3 -3 r^3 * cos(3*theta) sqrt(8) *RqO3=
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) B "s8i{Vm
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ADJ5ZD<Q
% 3 3 r^3 * sin(3*theta) sqrt(8) EZa{C}NQ$2
% 4 -4 r^4 * cos(4*theta) sqrt(10) faKrSmE!
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 2e D\_IW
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) a#~Z5>{
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :)3$&QdHT
% 4 4 r^4 * sin(4*theta) sqrt(10) [b\lcQ8O
% -------------------------------------------------- vYTPZ@RL
% .\hib.n3
% Example 1: .w*{=x0k
% ;zxlwdfcr'
% % Display the Zernike function Z(n=5,m=1) >?uH#%C5
% x = -1:0.01:1; iTtAj~dfZ
% [X,Y] = meshgrid(x,x); XiZ Zo
% [theta,r] = cart2pol(X,Y); qS[p|*BL
% idx = r<=1; cq+M
*1;
% z = nan(size(X)); th>yi)m
% z(idx) = zernfun(5,1,r(idx),theta(idx)); >t6'8g"T
% figure \Lh<E5@]
% pcolor(x,x,z), shading interp 1rzq$, O
% axis square, colorbar K]=>F
% title('Zernike function Z_5^1(r,\theta)') |jCE9Ve#
% ]mGsNQ ].H
% Example 2: =Q8^@i4[&D
% } k%\
% % Display the first 10 Zernike functions N#6A>
% x = -1:0.01:1; :J)lC =
% [X,Y] = meshgrid(x,x); yK2*~T,6@
% [theta,r] = cart2pol(X,Y); E'kQ
% idx = r<=1; 3B_} :
% z = nan(size(X)); Y.hH
fSp
% n = [0 1 1 2 2 2 3 3 3 3]; F|ML$
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 1Mhc1MU
% Nplot = [4 10 12 16 18 20 22 24 26 28]; MZ+IorZl
% y = zernfun(n,m,r(idx),theta(idx)); g)G7
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% figure('Units','normalized') NbK?Dg8WJG
% for k = 1:10 m^s2kB4A[
% z(idx) = y(:,k); V{^fH6;[
% subplot(4,7,Nplot(k)) $vicHuX!
% pcolor(x,x,z), shading interp mWFZg.#?
% set(gca,'XTick',[],'YTick',[]) i:Ct6[
% axis square ~ !+h"%'t
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])
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% end 6+d"3-R.
% igbb=@QBJ
% See also ZERNPOL, ZERNFUN2. !JQ~r@j
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% Paul Fricker 11/13/2006 YQ`#C#Wb
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% Check and prepare the inputs: <rI$"=7
% ----------------------------- ?g*T3S"
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Da[X
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error('zernfun:NMvectors','N and M must be vectors.') (uxQBy
end |JQP7z6j]
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if length(n)~=length(m) ?GfA;O
error('zernfun:NMlength','N and M must be the same length.') JfINAaboi
end $0C/S5b
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n = n(:); n&78~@H
m = m(:); _89G2)U=C
if any(mod(n-m,2)) )Is*-
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error('zernfun:NMmultiplesof2', ... Wn#JYp
'All N and M must differ by multiples of 2 (including 0).') >2{HH\
end RV*Zi\-X
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if any(m>n) `.-k%2?/
error('zernfun:MlessthanN', ... =F-^RnO%\
'Each M must be less than or equal to its corresponding N.') !Jp.3,\?~
end cMk%]qfVo8
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if any( r>1 | r<0 ) Sqt"G6<
error('zernfun:Rlessthan1','All R must be between 0 and 1.') f?^xh
end ~:b~f]lO
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) uvM88#
error('zernfun:RTHvector','R and THETA must be vectors.') rbS=Ewk
end IL"#TKKv
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r = r(:); R~(.uV`#j
theta = theta(:); k<hO9;#qpL
length_r = length(r); _[tBLGXD
if length_r~=length(theta) @Od u.F1e
error('zernfun:RTHlength', ... s'=]a-l~
'The number of R- and THETA-values must be equal.') >c>ar>4xF
end Q>*K/%KD
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% Check normalization: ebVfny$D
% -------------------- _)"
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if nargin==5 && ischar(nflag) iW$i%`>
isnorm = strcmpi(nflag,'norm'); ^Wz{su2
if ~isnorm ZSb+92g{L$
error('zernfun:normalization','Unrecognized normalization flag.') 41D[[Gh
end )U`kU`+'
else NU*6iLIq|F
isnorm = false; (_<n0
end 4rdrl
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3Ab$
% Compute the Zernike Polynomials ;<