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, 1'.SHY|
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, CPJ8G}4
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? 5%H(AaG*q
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? <2b&AF{En
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function z = zernfun(n,m,r,theta,nflag) [k1N-';;;
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. #cHH<09rl
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N CC{*'p6
% and angular frequency M, evaluated at positions (R,THETA) on the kV>[$6
% unit circle. N is a vector of positive integers (including 0), and b&q!uFP
% M is a vector with the same number of elements as N. Each element m+66x {M2c
% k of M must be a positive integer, with possible values M(k) = -N(k) UZcsMMKH
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, e6?iQ0
% and THETA is a vector of angles. R and THETA must have the same ^\<nOzU?
% length. The output Z is a matrix with one column for every (N,M) :P,g,
% pair, and one row for every (R,THETA) pair. z{wW6sgPr
% Vq8 G( <77
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike }~W:3A{7;
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), :/rl \woA>
% with delta(m,0) the Kronecker delta, is chosen so that the integral zN3[W`q+m
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, eBlWwUy*6f
% and theta=0 to theta=2*pi) is unity. For the non-normalized dO?zLc0f
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. /l.:GH36f
% '3%J hG)#
% The Zernike functions are an orthogonal basis on the unit circle. ;_$Q~X
% They are used in disciplines such as astronomy, optics, and 5OHg% ^
% optometry to describe functions on a circular domain. *}F>c3x]
% @`Fv}RY{
% The following table lists the first 15 Zernike functions.
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% #%Hk-a=>)#
% n m Zernike function Normalization -|z
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% -------------------------------------------------- ;$a+ >
% 0 0 1 1 KjWF;VN*[3
% 1 1 r * cos(theta) 2 fyt ODsb>
% 1 -1 r * sin(theta) 2 C8{bqmlm@
% 2 -2 r^2 * cos(2*theta) sqrt(6) <x!q!;
% 2 0 (2*r^2 - 1) sqrt(3) %
w\
% 2 2 r^2 * sin(2*theta) sqrt(6) 8
x=J&d
% 3 -3 r^3 * cos(3*theta) sqrt(8) _sp,,gz
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) vl`Qz"Xy
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) }na0
% 3 3 r^3 * sin(3*theta) sqrt(8)
h.Y&_=Gc
% 4 -4 r^4 * cos(4*theta) sqrt(10) $ol]G`+
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~^{>!wU+
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) $&25hvK,
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) [c^!;YBp)
% 4 4 r^4 * sin(4*theta) sqrt(10) XC(:O(jdA2
% -------------------------------------------------- .2Q4EbM2
% t]3> X
% Example 1: <wH"{G3?
% hQeGr2gMq
% % Display the Zernike function Z(n=5,m=1) &nV/XLpG
% x = -1:0.01:1; 1;*4yJ2
% [X,Y] = meshgrid(x,x); &6feR#~A
% [theta,r] = cart2pol(X,Y); 3# g"Z7/
% idx = r<=1; IZ/PZ"n_(
% z = nan(size(X)); PFKl6_(
% z(idx) = zernfun(5,1,r(idx),theta(idx)); DX2_}|$!
% figure ]Cc3}+(s
% pcolor(x,x,z), shading interp m&PB5s\=
% axis square, colorbar bmOK8
% title('Zernike function Z_5^1(r,\theta)') /IxoS
% cv{icz,%w
% Example 2: bcR";cE
% t !8(I R
% % Display the first 10 Zernike functions ; Sd== *
% x = -1:0.01:1; Aaw]=8 OI
% [X,Y] = meshgrid(x,x); @3w6!Sgh
% [theta,r] = cart2pol(X,Y); N&uRL_X.
% idx = r<=1; H9\,;kM)
% z = nan(size(X)); a1>Tz
% n = [0 1 1 2 2 2 3 3 3 3]; !V'~<&
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; h]Y,gya[yk
% Nplot = [4 10 12 16 18 20 22 24 26 28]; q90
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% y = zernfun(n,m,r(idx),theta(idx)); Bq5-L}z
% figure('Units','normalized') WaPuJ5;e
% for k = 1:10 FUP0X2P
% z(idx) = y(:,k); D03QisH=
% subplot(4,7,Nplot(k)) B:>>D/O
% pcolor(x,x,z), shading interp zv-9z
% set(gca,'XTick',[],'YTick',[]) d[\$a4G+
% axis square !b"2]Qv
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) yMz dM&a!*
% end 4wkmgS
% * lJkk
% See also ZERNPOL, ZERNFUN2. /HE{8b7n3F
u}">b+{!
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% Paul Fricker 11/13/2006 ;=UkTn}N?l
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% Check and prepare the inputs: H
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% ----------------------------- Kr'Yz!
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Hmx
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error('zernfun:NMvectors','N and M must be vectors.') JLbmh1'
end NY
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if length(n)~=length(m) V}MRdt7
error('zernfun:NMlength','N and M must be the same length.') ;d.gVR_V
end I vX+yU
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n = n(:); )zydD=,bu
m = m(:); l[6lXR&|
if any(mod(n-m,2)) Sc?q}tt^C
error('zernfun:NMmultiplesof2', ... &u4;A[-R
'All N and M must differ by multiples of 2 (including 0).') >rYkVlv
end ;LC?3.
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if any(m>n) [;Jq=G8&t
error('zernfun:MlessthanN', ... _l+8[\v
'Each M must be less than or equal to its corresponding N.') 4$y P_3
end #l
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if any( r>1 | r<0 ) hs*n?vxp3
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ,FwJ0V
end L%<DLe^P`l
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) iEx4va-j
error('zernfun:RTHvector','R and THETA must be vectors.') FEi@MJJ\e
end $>zqCi2tB<
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r = r(:); )S`=y-L$
theta = theta(:); txiX1o!/L
length_r = length(r); #fDM{f0]R
if length_r~=length(theta) \cdns;
error('zernfun:RTHlength', ... RgVnx] IF
'The number of R- and THETA-values must be equal.') !tSh9L;<O
end )XDbg>
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% Check normalization: j XYr&F
% -------------------- hlfdmh?/
if nargin==5 && ischar(nflag) "H]R\xp
isnorm = strcmpi(nflag,'norm');
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if ~isnorm "TVmxE%(
error('zernfun:normalization','Unrecognized normalization flag.') 8v)iOPmDC
end :m<#\!?
else ,F n-SrB:
isnorm = false; 7M~/[f7Z{
end #itZ~tol
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C%|m[,Gx
% Compute the Zernike Polynomials m%b#B>J,n
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !gcea?I
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% Determine the required powers of r: 7gx?LI_e
% ----------------------------------- a+j"8tHu$
m_abs = abs(m); dl(!{tZ#
rpowers = []; 0]zMb^wo
for j = 1:length(n) lx7]rkWo|a
rpowers = [rpowers m_abs(j):2:n(j)]; 4HpKKhv"
end et/v/Hvw1
rpowers = unique(rpowers); yG;@S8zC
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A1*\ \[
% Pre-compute the values of r raised to the required powers, r^{Bw1+
% and compile them in a matrix: h@TP=
% ----------------------------- Yy;BJ_
if rpowers(1)==0 #|T2`uYotf
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); P26"z))~d
rpowern = cat(2,rpowern{:}); 211V'|a_>
rpowern = [ones(length_r,1) rpowern]; 5}b)W>3@`
else xz~Y
%Y|Z
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); u'^kpr`y
rpowern = cat(2,rpowern{:}); j<k-w
end DG}s`'
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% Compute the values of the polynomials: *m2?fP\
% -------------------------------------- T^A[m0mk
y = zeros(length_r,length(n)); bn7g!2
for j = 1:length(n) M@Ti$=
s = 0:(n(j)-m_abs(j))/2; Xpt9$=d
pows = n(j):-2:m_abs(j); Mcq!QaO}&
for k = length(s):-1:1 [NV/*>"j&
p = (1-2*mod(s(k),2))* ... //RD$e?h~
prod(2:(n(j)-s(k)))/ ... *U$%mZS]1
prod(2:s(k))/ ... 8c>xgFWp9
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Vt,P.CfdC
prod(2:((n(j)+m_abs(j))/2-s(k))); Xkk 8#Y":
idx = (pows(k)==rpowers); ;%k C?Vzi
y(:,j) = y(:,j) + p*rpowern(:,idx); D]5j?X'
end YI`BA`BQ8
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if isnorm 5tk7H2K^<
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); <8YvsJ
end h
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end saDu'SmYV
% END: Compute the Zernike Polynomials LIKQQ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R}IuMMx
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`Y^l.%AZZ
% Compute the Zernike functions: +(W7hK4ip
% ------------------------------ 3`)ej`
idx_pos = m>0; c`/=)IO4%
idx_neg = m<0; oS}fr?
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z = y; >UV}^OO
if any(idx_pos) ZAn9A>5_
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); `sgW0Uf
end |>1#)cONW
if any(idx_neg) ,`YIcrya:
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); @sW!g;\T
end )3<>H!yG}
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% EOF zernfun i29a1nD4Hm