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, HUH=Y;
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, p)`JVq,H/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? G"]'`2.m
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? :|bPr_&U$
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function z = zernfun(n,m,r,theta,nflag) xpae0vw
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. UWz<~Vy
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 09r.0Ks
% and angular frequency M, evaluated at positions (R,THETA) on the nL9m{$Zv
% unit circle. N is a vector of positive integers (including 0), and #~"jo[
% M is a vector with the same number of elements as N. Each element CAk.2C/
% k of M must be a positive integer, with possible values M(k) = -N(k) kjH0u$n
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, C.eZcNJG
% and THETA is a vector of angles. R and THETA must have the same +]G;_/[2
% length. The output Z is a matrix with one column for every (N,M) c8h
9
% pair, and one row for every (R,THETA) pair. V<b"jCXI
% 72aj4k]^
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike xGjEEBL
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), rc"yEI-``"
% with delta(m,0) the Kronecker delta, is chosen so that the integral 5bk5EE`
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, :%R3(
&
% and theta=0 to theta=2*pi) is unity. For the non-normalized D9h\=[%e
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 6H@=O1W
% 1r$q $\
% The Zernike functions are an orthogonal basis on the unit circle. J}BS/Tr}=
% They are used in disciplines such as astronomy, optics, and _|3n h;-m
% optometry to describe functions on a circular domain. o`G@Je_}x
% xRb-m$B}L
% The following table lists the first 15 Zernike functions. {C
[7V{4(%
% >#SQDVFf
% n m Zernike function Normalization HA| YLj?|g
% -------------------------------------------------- vNP,c]:%
% 0 0 1 1 )tB mSVprl
% 1 1 r * cos(theta) 2 2|+**BxHD
% 1 -1 r * sin(theta) 2 5E$)Ip
% 2 -2 r^2 * cos(2*theta) sqrt(6) Lf3:' n
% 2 0 (2*r^2 - 1) sqrt(3) Gt' %:9r
% 2 2 r^2 * sin(2*theta) sqrt(6) ip~PF5
% 3 -3 r^3 * cos(3*theta) sqrt(8) J?HYN%
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) vV8}>
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) MbYAK-l.h
% 3 3 r^3 * sin(3*theta) sqrt(8) m3 ,i{
% 4 -4 r^4 * cos(4*theta) sqrt(10) -[Q%Vv!8
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) RV-7y^[]^
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) -3A#a_fu
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) B+ +:7!
% 4 4 r^4 * sin(4*theta) sqrt(10) |:C=j/f
% -------------------------------------------------- ,u/GA<'#M
% El,p}Bi.
% Example 1: al1Uf]xh
% ]oj
2
% % Display the Zernike function Z(n=5,m=1) [/xw5rO%
% x = -1:0.01:1; r/SV.`
k
% [X,Y] = meshgrid(x,x);
d x?4)lb
% [theta,r] = cart2pol(X,Y); "YM)bc
% idx = r<=1; R["_Mff
% z = nan(size(X)); npZ=x-ce
% z(idx) = zernfun(5,1,r(idx),theta(idx)); b k 30d
% figure ULj'DzlfH
% pcolor(x,x,z), shading interp ex1b jM7
% axis square, colorbar 1 %K^(J;
% title('Zernike function Z_5^1(r,\theta)') [;%qxAB/_
% #)z_TM07P
% Example 2: lUbQ@7a<'
% H1]G<N3
% % Display the first 10 Zernike functions (=,p"3^
% x = -1:0.01:1; VU 9w2/cM
% [X,Y] = meshgrid(x,x); s%GhjWZS
% [theta,r] = cart2pol(X,Y); CNQ>J`4
% idx = r<=1; 3+rud9T
% z = nan(size(X)); 6"b =aPTi
% n = [0 1 1 2 2 2 3 3 3 3]; 0& 54xP
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 1)U%p
% Nplot = [4 10 12 16 18 20 22 24 26 28]; @|sDb?J
% y = zernfun(n,m,r(idx),theta(idx)); uDbz`VpK
% figure('Units','normalized') N;Wm{~Zhb
% for k = 1:10 /z9oPIJ=*
% z(idx) = y(:,k); _gxI=EYi
% subplot(4,7,Nplot(k)) sE{A~{a`
% pcolor(x,x,z), shading interp bd_&=VLTC
% set(gca,'XTick',[],'YTick',[]) x8+W9i0[1
% axis square V*U{q%p(
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) eTw sh]
% end kWZ?86!
% 0rP`BK|
% See also ZERNPOL, ZERNFUN2. Sxa+"0d6
E]/` JI'%
k` cz$>
% Paul Fricker 11/13/2006 nO.RB#I$F
q$7SJ.pF
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b=(?\
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% Check and prepare the inputs: ?VaWOwWI
% ----------------------------- qVjl8%)
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) {]ie|>'=C
error('zernfun:NMvectors','N and M must be vectors.') lC):$W
end MZ]#9/
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if length(n)~=length(m) 8$4@U;Vh;
error('zernfun:NMlength','N and M must be the same length.') tn>z%6;&Z
end f}qR'ognUu
{kVhht]X
9=D09@A%e
n = n(:); W(.q.Sx>
m = m(:); a$-:F$z
if any(mod(n-m,2)) KVQ|l,E,
/
error('zernfun:NMmultiplesof2', ... AM?62
'All N and M must differ by multiples of 2 (including 0).') <Wqk5mR
end RHe'L36W
cG{>[Lf
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if any(m>n) ` 9iB`<
error('zernfun:MlessthanN', ... R{N9'2l:
'Each M must be less than or equal to its corresponding N.') P4H%pm{-
end kIR?r0_<G6
BTi:Bcv k
iY_E"$}P
if any( r>1 | r<0 ) zPWJ=T@N
error('zernfun:Rlessthan1','All R must be between 0 and 1.') k?[|8H~2C
end 1j4(/A
n_ORD@$]
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ?7n(6kmj4Q
error('zernfun:RTHvector','R and THETA must be vectors.') Wg \`!T
end yhwwF
n\
x.J%
c[Q8
N i\*<:_
r = r(:); {tqLH2cO
theta = theta(:); (rDB|kc^7
length_r = length(r); 6<E4?<O%
if length_r~=length(theta) 3JnBKh\n
error('zernfun:RTHlength', ... BM6 J
'The number of R- and THETA-values must be equal.') .~>Uh3S
end
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% Check normalization: r0 mXRZC
% -------------------- #A&(b}#:o
if nargin==5 && ischar(nflag) WcE{1&PXx
isnorm = strcmpi(nflag,'norm'); ctqXzM `
if ~isnorm ~QVN^8WPg
error('zernfun:normalization','Unrecognized normalization flag.') (+_i^SqK
end Nhq&Sn2
else "'M>%m u
isnorm = false; ze5Hg'f
end Y bX3_N&
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uWo:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <[N"W82p
% Compute the Zernike Polynomials `F)Q=
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EKwA1,Xz
7x,c)QES`
wTT_jyH)
% Determine the required powers of r: s*blZdP
% ----------------------------------- +s(JutC
m_abs = abs(m); N|G=n9p
rpowers = []; 7hQf
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for j = 1:length(n) <M//zXa
rpowers = [rpowers m_abs(j):2:n(j)]; O^tH43C
end Z33&FUU
rpowers = unique(rpowers); @I`X{oAA
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o\Ocu>:
% Pre-compute the values of r raised to the required powers, lP9XqQ(
% and compile them in a matrix: A(!nT=0o
% ----------------------------- {u/G!{N$
if rpowers(1)==0 =x8F!W}Bt<
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); YJioR4+q
rpowern = cat(2,rpowern{:}); *)PCPYB^
rpowern = [ones(length_r,1) rpowern]; %j
'_I\
else co
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rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); p^=>N9
rpowern = cat(2,rpowern{:}); .iDxq8l
end %D::$,;<<
Y6_%HYI$
uU&,KEH
% Compute the values of the polynomials: ;H%&Jht
% -------------------------------------- ^>?E1J3u
y = zeros(length_r,length(n)); XET'XJWF%
for j = 1:length(n) _;8aiZt|u
s = 0:(n(j)-m_abs(j))/2; _.E y_K_1
pows = n(j):-2:m_abs(j); dr25;L? B
for k = length(s):-1:1 }7`HJ>+m)H
p = (1-2*mod(s(k),2))* ... }Pu|%\
prod(2:(n(j)-s(k)))/ ... *6Ojv-
G|5
prod(2:s(k))/ ... 81O\BO.T
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... mPl2y3m%
prod(2:((n(j)+m_abs(j))/2-s(k))); f?-=&||f78
idx = (pows(k)==rpowers); H(eGqVAq,
y(:,j) = y(:,j) + p*rpowern(:,idx); 5IK -V)
end \
2cI=Qf
Jd].e=]pN
if isnorm 3ug|H
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ^LA.Y)4C2%
end qbrf;`
end r6B\yH2
% END: Compute the Zernike Polynomials Iyo ey
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Lk%u(duU^
A5d(L4Q]a(
[1I>Bc&o*
% Compute the Zernike functions: /}_OCuJJ,
% ------------------------------ i Sm5k:7
idx_pos = m>0; ) h*)_7
idx_neg = m<0; .zm'E<
<tT*.nM\
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z = y; &VY(W{\eY
if any(idx_pos) .EOHkhn
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); =Mg/m'QI
end &4aY5y`8+f
if any(idx_neg) oD5VE
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); s_(%1/{
end /0w?"2-
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##%R|P3
% EOF zernfun <Pg]V:=g'