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, ~@I@} n
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, <o:@dS
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? 4ax|Vb)D
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? FQeYx-7
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function z = zernfun(n,m,r,theta,nflag) j kn^Z":
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. MWWu@SY
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N >cOeiK
% and angular frequency M, evaluated at positions (R,THETA) on the }4c/YP"a'E
% unit circle. N is a vector of positive integers (including 0), and P-z`c\Rt
% M is a vector with the same number of elements as N. Each element <"&'>?8j
% k of M must be a positive integer, with possible values M(k) = -N(k) LhJ a)jFQ
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ;q#]-^
% and THETA is a vector of angles. R and THETA must have the same ;\b@)E}
% length. The output Z is a matrix with one column for every (N,M) *FgJ|y6gk
% pair, and one row for every (R,THETA) pair. 6p<`h^
% /Ic[N&
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike mv
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% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), {E,SHh
% with delta(m,0) the Kronecker delta, is chosen so that the integral BD;H
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, E)YVfM
% and theta=0 to theta=2*pi) is unity. For the non-normalized SX+RBVZU
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. #V 43=
% E'dX)J9e$/
% The Zernike functions are an orthogonal basis on the unit circle. (6xDu.u?A
% They are used in disciplines such as astronomy, optics, and :\}U9QfCw
% optometry to describe functions on a circular domain. L`K;IV%;
% Ky9W/dCR
% The following table lists the first 15 Zernike functions.
C B}BQd
% T |"`8mG
% n m Zernike function Normalization 13f<0wg
% -------------------------------------------------- 6}&^=^-
% 0 0 1 1 Z[IM<S9lz
% 1 1 r * cos(theta) 2 .|]IwyD
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% 1 -1 r * sin(theta) 2 zNtq"T [
% 2 -2 r^2 * cos(2*theta) sqrt(6) +l\<?
% 2 0 (2*r^2 - 1) sqrt(3) G%hO\EO
% 2 2 r^2 * sin(2*theta) sqrt(6) e@
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% 3 -3 r^3 * cos(3*theta) sqrt(8) !EFBI+?&
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) M9"Sgb`g
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8)
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% 3 3 r^3 * sin(3*theta) sqrt(8) v}`1)BUeF
% 4 -4 r^4 * cos(4*theta) sqrt(10) oX|?:MS:
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ePA;:8)_j
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) G=$}5; t
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) YOw?'+8
% 4 4 r^4 * sin(4*theta) sqrt(10) %x2b0L\g
% -------------------------------------------------- \|q-+4]@,
% YN#XmX%
% Example 1: xXOw:A'
% w~-X>~ }
% % Display the Zernike function Z(n=5,m=1) f-+.;`H)T
% x = -1:0.01:1; /yK"t<p
% [X,Y] = meshgrid(x,x); ~%P3Pp
% [theta,r] = cart2pol(X,Y); zD_HyGf
% idx = r<=1; iG-N
% z = nan(size(X)); SfDQ;1?
% z(idx) = zernfun(5,1,r(idx),theta(idx)); OOLe[P3J3
% figure "L_-}BK
% pcolor(x,x,z), shading interp S:Xs'0K_
% axis square, colorbar iwo$\
% title('Zernike function Z_5^1(r,\theta)') 'G
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% YN^jm
% Example 2: Wm>b3:
% ,>S+-L8
% % Display the first 10 Zernike functions .eTk=i[N-
% x = -1:0.01:1; b`]M|C [5
% [X,Y] = meshgrid(x,x); uGCtLA+sL
% [theta,r] = cart2pol(X,Y); FNJ!IkuR
% idx = r<=1; )*HjRTF6G
% z = nan(size(X)); t?.\|2
% n = [0 1 1 2 2 2 3 3 3 3]; b7v dk
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; %BICt @E
% Nplot = [4 10 12 16 18 20 22 24 26 28]; "WP% REE!
% y = zernfun(n,m,r(idx),theta(idx)); y< ud('D
% figure('Units','normalized') >)sqh ~P
% for k = 1:10 u_Zm1*'?B
% z(idx) = y(:,k); X7&U3v
% subplot(4,7,Nplot(k)) =LqL@5Xr
% pcolor(x,x,z), shading interp v>:=w|.HC
% set(gca,'XTick',[],'YTick',[]) Mk "vvk
% axis square w`-$-4i
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) }{=8&gA0
% end \CwtX(6.
% NxB+?
% See also ZERNPOL, ZERNFUN2. "uS7PplyO
5%'S
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% Paul Fricker 11/13/2006 ?&\h;11T
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% Check and prepare the inputs: Un\Ubqi0
% ----------------------------- D{W
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?"u'#f_
error('zernfun:NMvectors','N and M must be vectors.') T N Ist
end sSy$(%
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if length(n)~=length(m) F`3^wHw^
error('zernfun:NMlength','N and M must be the same length.') )1K! [W}t
end -O /T?H
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n = n(:); <H-tZDh5
m = m(:); -B,c B
if any(mod(n-m,2)) i.F8
error('zernfun:NMmultiplesof2', ... i<Q&
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'All N and M must differ by multiples of 2 (including 0).') iA&oLu[y3
end *F|i&2
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if any(m>n) >^;(c4C
error('zernfun:MlessthanN', ... (<
:mM
'Each M must be less than or equal to its corresponding N.') %B0w~[!4}
end yW{mK
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if any( r>1 | r<0 ) U-1VnX9m
error('zernfun:Rlessthan1','All R must be between 0 and 1.') a" ^#!G<+
end dA|Lufy#
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 8V,"Id][
error('zernfun:RTHvector','R and THETA must be vectors.') "2%y~jrDN
end r)c+".0d^
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r = r(:); F:M/z#:~
theta = theta(:); Z4\tY^NI
length_r = length(r); 4bPqmEE
if length_r~=length(theta) prqyoCfq
error('zernfun:RTHlength', ... 7KeXWW/ d
'The number of R- and THETA-values must be equal.') 4v0dd p
end +jv}\Jt
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% Check normalization: %Fs*#S
% -------------------- f 5mY;z"
if nargin==5 && ischar(nflag) o@Scz!"g
isnorm = strcmpi(nflag,'norm'); sN"p5p
if ~isnorm cO8`J&EK
error('zernfun:normalization','Unrecognized normalization flag.') q|R+x7x
end sWp{Y.
else 4u@yJ?U
isnorm = false; ?OdV1xB
end _'H2>V_
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LQ~LB'L
% Compute the Zernike Polynomials A1mYkG)l
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9 )ACgz&(
[t {vYo
])+Sc"g4k
% Determine the required powers of r: jQY>9+t
% ----------------------------------- "1_{c *ck
m_abs = abs(m); /;zZnF\e
rpowers = []; :yd=No@
for j = 1:length(n) Ngn\nkf
rpowers = [rpowers m_abs(j):2:n(j)]; C<zx'lw!
end 9"m,p
rpowers = unique(rpowers); >&*6Fqd
nrxjN(9V%+
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% Pre-compute the values of r raised to the required powers, '~ jy
% and compile them in a matrix: ]R97n|s_
% ----------------------------- pI'8>_o
if rpowers(1)==0 #k"1wSx16
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); _Jf J%YXy
rpowern = cat(2,rpowern{:}); 71K\.[ =-
rpowern = [ones(length_r,1) rpowern]; jXc5fXO
N
else _Cu[s?,kS
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); }T?i%l
rpowern = cat(2,rpowern{:}); XMjI}SPG
end (2\l i{$e
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% Compute the values of the polynomials: Y9z:xE
% -------------------------------------- ^G]KE8
y = zeros(length_r,length(n)); qkIA,Kgy
for j = 1:length(n) sV9{4T~#|
s = 0:(n(j)-m_abs(j))/2; ^4n2
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pows = n(j):-2:m_abs(j); $#6Fnhh}
for k = length(s):-1:1 e_fg s>o`(
p = (1-2*mod(s(k),2))* ... 'DaNR`9
prod(2:(n(j)-s(k)))/ ... ?7rmwy\
prod(2:s(k))/ ... &6|6J1c8
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... T{5M1r
prod(2:((n(j)+m_abs(j))/2-s(k))); |[lxV&SD.
idx = (pows(k)==rpowers); yb@X*PW/z
y(:,j) = y(:,j) + p*rpowern(:,idx); mafAC73
end BDv|~NHs
b AA'=z<
if isnorm n?TO!5RZK
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); }w|=c>'_}
end `R4W4h'I
end xEd#~`Jmr
% END: Compute the Zernike Polynomials v.~Nv@+kR
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *@b~f&Lx6
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% Compute the Zernike functions: ex;Yn{4
% ------------------------------ Mt7X<?GZm
idx_pos = m>0; FvtM~[Q
idx_neg = m<0; f_z2#,g
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z = y; BMtYM{S6
if any(idx_pos) ThT.iD[
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Q@3ld6y
end ;I0yQlx|U
if any(idx_neg) 3!ajvSOI9j
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Px^<2Q%Fs
end o$qFa9|Ec?
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% EOF zernfun x r+E