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, _Sj}~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, B0mLI%B
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? (fk5'
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? >w'$1tc?+F
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function z = zernfun(n,m,r,theta,nflag) zQuM !.
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. !G=>ve
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N #n})X,ip2
% and angular frequency M, evaluated at positions (R,THETA) on the \SYvD y]
% unit circle. N is a vector of positive integers (including 0), and -wr(vE,
% M is a vector with the same number of elements as N. Each element Px4/O~bLk
% k of M must be a positive integer, with possible values M(k) = -N(k) -@bp4Z=
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, RtF8A5ys
% and THETA is a vector of angles. R and THETA must have the same
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% length. The output Z is a matrix with one column for every (N,M) T |"`8mG
% pair, and one row for every (R,THETA) pair. 13f<0wg
% x*8O*!ZZ
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Cv TwBJy1
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), .|]IwyD
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% with delta(m,0) the Kronecker delta, is chosen so that the integral Lar r}o=
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 12gw#J/)9h
% and theta=0 to theta=2*pi) is unity. For the non-normalized $p6N|p
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5:gj&jt;)7
% P W[6/7
% The Zernike functions are an orthogonal basis on the unit circle. YF[$Q=7.
% They are used in disciplines such as astronomy, optics, and !$kR ;Q"/
% optometry to describe functions on a circular domain. .hz2&9Ow
% dtW0\^ .L
% The following table lists the first 15 Zernike functions. ToU.mM?f^
% o~~ 9!\
% n m Zernike function Normalization @ ^.*$E5
% -------------------------------------------------- sd!sus|( R
% 0 0 1 1 5+L8\V9;
% 1 1 r * cos(theta) 2 ~mA7pOHj
% 1 -1 r * sin(theta) 2 :WX0,-Gn
% 2 -2 r^2 * cos(2*theta) sqrt(6) s-6:N9-
% 2 0 (2*r^2 - 1) sqrt(3) rWr/ p^~
% 2 2 r^2 * sin(2*theta) sqrt(6) ,f<B}O
% 3 -3 r^3 * cos(3*theta) sqrt(8) !J#.!}3
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) `p|[rS>
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) #]zhZW4
% 3 3 r^3 * sin(3*theta) sqrt(8) #O|lfl>}
% 4 -4 r^4 * cos(4*theta) sqrt(10) >l2w::l%
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |cu`f{E2]
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) (Jpm
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% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~07RFR
% 4 4 r^4 * sin(4*theta) sqrt(10) 8A/>JD3^
% -------------------------------------------------- 0M\NS$u(Y
% qy9i9$8
% Example 1: -A;w$j6*
% gb_X?j%p7
% % Display the Zernike function Z(n=5,m=1) 4<Kgmy
% x = -1:0.01:1; E\!n49
% [X,Y] = meshgrid(x,x); kH2oK:lN
% [theta,r] = cart2pol(X,Y); t?.\|2
% idx = r<=1; pzSqbgfrQ
% z = nan(size(X)); nP
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% z(idx) = zernfun(5,1,r(idx),theta(idx)); 0BXs&i-TP5
% figure )k~1,
% pcolor(x,x,z), shading interp vzy!3Hiw
% axis square, colorbar Y- ~;E3(
% title('Zernike function Z_5^1(r,\theta)') uQ1jwYK`7
% ^H'kHl'F
% Example 2: "G kI5!
% xN"wF-s4?
% % Display the first 10 Zernike functions J";=d4Sd
% x = -1:0.01:1; O\JD, w
% [X,Y] = meshgrid(x,x); m+7`\|`jQ
% [theta,r] = cart2pol(X,Y); T^NJ4L4#
% idx = r<=1; :p$EiR
% z = nan(size(X)); TK %<a/
% n = [0 1 1 2 2 2 3 3 3 3]; id4]|jb
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; -fQX4'3R
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 3.~h6r5-
% y = zernfun(n,m,r(idx),theta(idx)); x
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% figure('Units','normalized') N1s.3`
% for k = 1:10 G{O\)gf
% z(idx) = y(:,k); khS/'b
% subplot(4,7,Nplot(k)) `$f2eB&
% pcolor(x,x,z), shading interp j88=f#<
% set(gca,'XTick',[],'YTick',[]) PS/W
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% axis square S|) J{~QH
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) )O -cw7 >
% end |Z!@'YB
% \N yr=<c
% See also ZERNPOL, ZERNFUN2. .xo_}Vw
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% Paul Fricker 11/13/2006 G V=OKf#
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% Check and prepare the inputs: A{M+vsL
% ----------------------------- gu!](yEgl
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) OMi02tSm
error('zernfun:NMvectors','N and M must be vectors.') qz87iJp&
end +#9xA6,AE
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if length(n)~=length(m) {9 Db9K^
error('zernfun:NMlength','N and M must be the same length.') |;~nI'0O])
end |FjBKj
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n = n(:); *^ey]),f54
m = m(:); ),MU+*`
if any(mod(n-m,2)) {dZ!I
error('zernfun:NMmultiplesof2', ... Q|Nzbmwh
'All N and M must differ by multiples of 2 (including 0).') JR!Q,7S2!N
end R/Tj^lM
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if any(m>n) uvR0TIF4
error('zernfun:MlessthanN', ... Imke/ =h
'Each M must be less than or equal to its corresponding N.') XNKtL]U}$
end *&h6*zP?
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if any( r>1 | r<0 ) ]&' jP
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ^4RO
end =obt"K%n
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) H]VsOr
error('zernfun:RTHvector','R and THETA must be vectors.') 8?A@/
end ^dheJ]n=k
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r = r(:); }!eF
theta = theta(:); o*?[_{xW
length_r = length(r); '_k >*trV
if length_r~=length(theta) QjehDwt|
error('zernfun:RTHlength', ... &qNP?>C!=
'The number of R- and THETA-values must be equal.') \)uy"+ Z`
end A>4l/
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% Check normalization: *Ro8W-+
% -------------------- Z@<q/2).|
if nargin==5 && ischar(nflag) @gs26jX~2}
isnorm = strcmpi(nflag,'norm'); N-]\oMc2
if ~isnorm O\LjtMF
error('zernfun:normalization','Unrecognized normalization flag.') JqL<$mSep
end 'Na|#tPYI
else 5JO[+>
isnorm = false; %j~9O~-
end I/u'bDq
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0Ei\VVK>
% Compute the Zernike Polynomials #&;m<%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% iSnIBs9\
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% Determine the required powers of r: hdo+Qezu:
% ----------------------------------- pA*D/P-
m_abs = abs(m); ?y+\v'3v
rpowers = []; oc!biE`u
for j = 1:length(n) {KF 7j63
rpowers = [rpowers m_abs(j):2:n(j)]; _bN))9
3
end ~5-~q0Ge
rpowers = unique(rpowers); SIKk|I)
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% Pre-compute the values of r raised to the required powers, % <8K^|w
% and compile them in a matrix: m~Lf^gbG?
% ----------------------------- {LR#(q$1
if rpowers(1)==0 c@0l-R{q
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); :6\-9m8JM
rpowern = cat(2,rpowern{:}); [V!^\g\6
rpowern = [ones(length_r,1) rpowern]; ?AqrlR]5
else f]@[4<N y
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 'DaNR`9
rpowern = cat(2,rpowern{:}); ?7rmwy\
end O[i2A(
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